ࡱ> *,%&'()spEGXDz } J X g ` `bjbjss 7&/ $ TTTP(UV$ ^ 2g"TgTgTgxrzlz8$hQ ztxzz TgTg ===z Tg Tg=z==NDL @ (Tg^ `' T}\0Ĭ0aa`(a (=zzzߗ^zzzzzzz CtPd tP   Student Manual for Fundamental Statistics for the Behavioral Sciences (7th edition) David C. Howell The University of Vermont Contents Chapter 1  HYPERLINK "RMChapter 1.doc" Introduction Chapter 2  HYPERLINK "RMChapter 2.doc" Basic Concepts Chapter 3  HYPERLINK "RMChapter 3.doc" Displaying Data Chapter 4  HYPERLINK "RMChapter 4.doc" Measures of Central Tendency Chapter 5  HYPERLINK "RMChapter 5.doc" Measures of Variability Chapter 6  HYPERLINK "RMChapter 6.doc" The Normal Distribution Chapter 7  HYPERLINK "RMChapter 7.doc" Basic Concepts of Probability Chapter 8  HYPERLINK "RMChapter 8.doc" Sampling Distributions and Hypothesis Testing Chapter 9  HYPERLINK "RMChapter 9.doc" Correlation Chapter 10  HYPERLINK "RMChapter 10.doc" Regression Chapter 11  HYPERLINK "RMChapter 11.doc" Multiple Regression Chapter 12  HYPERLINK "RMChapter 12.doc" Hypothesis Tests Applied to Means: One Sample Chapter 13  HYPERLINK "RMChapter 13.doc" Hypothesis Tests Applied to Means: Two Related Samples Chapter 14 HYPERLINK "RMChapter 14.doc"Hypothesis Tests Applied to Means: Two Independent Samples Chapter 15  HYPERLINK "RMChapter 15.doc" Power Chapter 16  HYPERLINK "RMChapter 16.doc" One-way Analysis of Variance Chapter 17  HYPERLINK "RMChapter 17.doc" Factorial Analysis of Variance Chapter 18  HYPERLINK "RMChapter 18.doc" Repeated-Measures Analysis of Variance Chapter 19  HYPERLINK "RMChapter 19.doc" Chi-Square Chapter 20  HYPERLINK "RMChapter 20.doc" Nonparametric and Distribution-Free Statistical Tests Chapter 21  HYPERLINK "RMChapter 21.doc" Choosing the Appropriate Analysis Preface The purpose of this manual is to provide answers to students using the accompanying text, Fundamental Statistics for the Behavioral Sciences, 7th ed. I have provided complete answers to all of the odd-numbered questions. I am often asked for answers to even-numbered exercises as well. I do not provide those because many instructors want to have exercises without answers. I am attempting to balance the two competing needs. You may find on occasion that you do not have the same answer that I do. Much of this will depend on the degree to which you or I round off intermediate steps. Sometimes it will make a surprising difference. If your answer looks close to mine, and you did it the same way that I did, then dont worry about small differences. It is even possible that I made an error. I know that there will be errors in some of these answers. There always are. Even the most compulsive problem solver is bound to make errors, and it has been a long time since anyone accused me of being compulsive. I do try, honest I do, but something always slips pastsometimes they even slip past while I am correcting another error. So I maintain a page on the web listing the errors that I and other have found. If you find an error (minor and obvious typos dont count unless they involve numbers), please check there and let me know if it is a new one. Some classes even compete to see who can find the most errorsits rough when you have to compete with a whole class. The address for the main web page, is  HYPERLINK "http://www.uvm.edu/~dhowell/fundamentals/" http://www.uvm.edu/~dhowell/fundamentals/ , and the link to the Errata is there. Important note: Due to the way hypertext links are shown by Microsoft Word, the underlining often obscures a single underline character, as in More_Stuff. If you see a space in an address, it is often really a _. Chapter 1-Introduction 1.1 A good example is the development of tolerance to caffeine. People who do not normally drink caffeinated coffee are often startled by the effect of one or two cups of regular coffee, whereas those who normally drink regular coffee see no such effect. To test for a context effect of caffeine, you would first need to develop a dependent variable measuring the alerting effect of caffeine, which could be a vigilance task. You could test for a context effect by serving a group of users of decaffeinated coffee two cups of regular coffee every morning in their office for a month, but have them drink decaf the rest of the time. The vigilance test would be given shortly after the coffee, and tolerance would be seen by an increase in errors over days. At the end of the month, they would be tested after drinking caffeinated coffee in the same and in a different setting. The important points here are: Tolerance is shown by an increase in errors on the vigilance task. To see the effect of context, subjects need to be presented with caffeine in two different contexts. There needs to be a difference between the vigilance performance in the two contexts. 1.3 Contexts affects peoples response to alcohol, to off-color jokes, or to observed aggressive behavior. 1.5 The sample would be the addicts that we observe. 1.7 Not all people in the city are listed in the phone book. In particular, women and children are underrepresented. A phone book is particularly out of date as a random selection device with the increase in the use of cell phones. Many telephone surveys really miss the general population, and instead focus on a restricted population, dominated by male adults. 1.9 In the tolerance study discussed in the text, we really do not care what the mean length of paw-lick latency is. No one would be excited to know that a mouse can stand on a surface at 105 degrees for 3.2 seconds without licking its paws. But we do very much care that the population mean of paw-lick latencies for morphine-tolerant mice is longer in one context than in another. 1.11 I would expect that your mother would continue to wander around in a daze, wondering what happened. 1.13 Three examples of measurement data: performance on a vigilance task; typing speed, blood alcohol level. 1.15 Relationship: The relationship between stress and susceptibility to disease; the relationship between driving speed and accident rate. 1.17 You could have one group of mice trained and tested in the same condition, one group trained in one condition and tested in the other, and a group given a placebo in the training context but given morphine in the testing condition. 1.19 This is an Internet search exercise without a fixed answer. The Statistics Homepage is an online statistics text. Various departments offer data sets, computing advice, and clarifying examples. Chapter 2-Basic Concepts 2.1 Nominal: names of students in the class; Ordinal: the order in which students hand in their first exam; Interval: the students grade on that first exam; Ratio: the amount of time that the student spent studying for that exam. 2.3 If the rat lies down to sleep in the maze, after performing successfully for several trials, this probably says little about what the animal has learned in the task. It may say more about the animals level of motivation. In this exercise I am trying to get the students to see that there is often quite a difference between what you and I think our variable is measuring and what it actually measures. Just because we label something as a measure of learning does not make it so. Just because the numbers increase on a ratio scale (twice as much time in the maze) doesnt mean that what those numbers are actually measuring is ratio (twice as much learning). 2.5 We have to assume the following at the very least (and I am sure I left out some) Mice are adequate models for human behavior. Morphine tolerance effects in mice are like heroin tolerance effects in humans, Time on a warm surface is in some way analogous to a human response to heroin. A context shift for mice is analogous to a context shift for humans. A drug overdose is analogous to pain tolerance. 2.7 The independent variables are the sex of the subject and the sex of the other person. 2.9 The experimenter expected to find that women would eat less in the presence of a male partner than in the presence of a female partner. Men, on the other hand, were not expected to vary the amount that they ate as a function of sex of their partner. 2.11 We would treat a discrete variable as if it were continuous if it had many different levels and were at least ordinal. 2.13 When I drew 50 numbers 3 times I obtained 29, 26, and 19 even numbers, respectively. For my third drawing only 38 percent of my numbers were even, which is probably less than I might have expectedespecially if I didnt have a fair amount of experience with similar exercises. 2.15 Eyes level condition: X3 = 2.03; X5 = 1.05; X8 = 1.86 "X = 14.82  EMBED Equation.2  2.17 Eyes level condition: ("X)2 = 14.822 = 219.6324; "X2 = 1.652 + ... + 1.732 = 23.22 "X/N = 14.82/10 = 1.482 This is the mean, a type of average. The above answers are the variance and standard deviation of Y. You really aren t going to do much more calculation that this. 2.19 Putting the two sets of data together: Multiply pairwise "XY = 22.27496 "X"Y = 14.82*14.63 = 216.82 SXY `" SXSY. They do differ, as you would expect.  EMBED Equation.2  2.21 X 5 7 3 6 3 "X = 24 X + 4 9 11 7 10 7 "(X + 4) = 44 = (24 + 5*4) 2.23 In the text I spoke about room temperature as an ordinal scale of comfort (at least up to some point). Room temperature is a continuous measure, even though with respect to comfort it only measures at an ordinal level. 2.25 The Beth Perez story: The dependent variable is the weekly allowance, measured in dollars and cents, and the independent variable is the sex of the child. We are dealing with a selected samplethe children in her class. The age of the students would influence the overall mean. The fact that these children are classmates could easily lead to socially appropriate responsesor what the children deem to be socially appropriate in their setting. At least within her school, Beth could randomly sample by taking a student roster, assigning each student a number, and matching those up with numbers drawn from a random number table. Random assignment to Sex would obviously be impossible. I dont see negative aspects of the lack of random assignment here because that is the nature of the variable under consideration. It would be better if we could randomly assign a child to a sex and see the result, but we clearly cant. The outcome of the study could be influenced by the desire of some children to exaggerate their allowance, or to minimize it so as not to appear too different from their peers. I would suspect that boys would be likely to exaggerate. The descriptive features of the study are her statements that the boys in her class received $3.18 per week in allowance, on average, while the girls received an average of $2.63. The inferential aspects are the inferences to the population of all children, concluding that boys get more than girls. 2.27 I would record the sequence number of each song that is played and then plot them on a graph. I cant tell if they are truly random, but if I see a pattern to the points I can be quite sure that they are not random. I think that it is important for students to become involved with the Internet early on. There is so much material out there that will be helpful, and you have to start finding it now. I find it impossible to believe that my explanations of concepts are always the best explanations that could be given and that they serve each student equally well. If one explanation doesnt make sense, you can find others that may. Chapter 3-Displaying Data 3.1 Katz et al (1990) No Passage Group:  There is too little data to say very much about the shape of this distribution, but it certainly isnt looking normally distributed. 3.3 I would use stems of 3*, 3., 4*, 4. 5*, and 5. for this display. 3.5 Compared to those who read the passages: Almost everyone who read the passages did better than the best person who did not read them. Certainly knowing what you are talking about is a good thing (though not always practiced). 4 | 3* | b) 68966 | 3. | 44343 | 4* | 6669697 | 4. | 42102 | 5* | 57557 | 5. | 5669 | 6* | | 6. | 66 | 7* | 21232231 | 7. | 5 | HI | 91 93 Notice that I have entered the data in the order in which I encountered them, rather than in increasing order. It makes it easier. c) It is obvious that the two groups are very different in their performance. We would be worried if they werent. d) This is an Internet exercise with no fixed answer. That source is far more advanced than the students would be at this time, but I think that they should be able to read it if they just skip over what they dont understand. 3.7 The following is a plot (as a histogram) of reaction times collapsed across all variables.  SHAPE \* MERGEFORMAT  3.9 Histogram of GPA scores  3.11 (1) Mexico has very many young people and very few old people, while Spain has a more even distribution. (2) The difference between males and females is more pronounced at most ages in Spain than it is in Mexico. (3) You can see the high infant mortality rate in Mexico. 3.13 The distribution of those whose attendance is poor is far more spread out than the distribution of normal attendees. This would be expected because a few very good students can score well on tests even when they dont attend, but most of the poor attenders are generally poor students who would score badly no matter what. The difference between the average grades of these two groups is obvious. 3.15 As the degree of rotation increases, the distribution of reaction time scores appears to move from left to rightwhich is also an increase. I think it is a good idea to really think through this problem, rather than to just take the answer as given. It is important to see that looking at data can lead to conclusions to scientific questions, even without formal statistical tests. Many students have a hard time seeing the relationship between data and a question they would like to ask. (Probably many older adults do as well.) 3.17 The data points are probably not independent in that data set. As time went on, there would be changes in the subjects performance. At first he might get better with practice, but then fatigue would start to set in. Since the data are given in the order in which they were collected, at least within each condition, data nearer in time should be more similar than data farther apart in time. 3.19 The amount of shock that a subject delivers to a white participant does not vary as a function of whether or not that subject has been insulted by the experimenter. However, the black participants do suffer more shocks when the subject has been insulted. 3.21 Wikipedia gives an excellent set of data on HIV/AIDS prevalence at  HYPERLINK "http://en.wikipedia.org/wiki/List_of_countries_by_HIV/AIDS_adult_prevalence_rate" http://en.wikipedia.org/wiki/List_of_countries_by_HIV/AIDS_adult_prevalence_rate 3.23 There is a tremendous increase in Downs Syndrome in children born to older mothers. This increase doesnt really take off until mothers are in their 40s, but with parents delaying having children, this is a potential problem.  3.25 Smoking and low birthweight: The data are given as the percentage of births for each group that were less than 2500 grams.  The relationship is unlikely to be a fluke because it is so consistent year after year. You can see that within each group there is very little variability. Students often wonder why behavioral scientists care about what appears to be a public health problem. But public health problems are very often behavioral problems. Psychologists spend a great deal of time dealing with the behavioral consequences of low birthweight, and trying to find ways of lowering the rate, and with addictions such as smoking. 3.27 White females have a longer life expectancy than black females, but the difference has shrunk considerably since 1920, though recent changes have been modest. Chapter 4-Measures of Central Tendency 4.1 Mode = 72; Median = 72, Mean = 70.18 4.3 Even without reading the passage, students are still getting about twice as many items correct as they would by chance. This suggests that the test, while testing reading comprehension, is also testing something else. I am not surprised at these results because most students can guess at better than chance levels. I think it is worth pointing out that these data suggest that the test measures something other than reading comprehension. Most students just say they were able to guess intelligently, without realizing that this means that the test is somehow measuring guessing ability. This will become more obvious when we talk about correlation in Chapter 9. Any positively skewed distribution will have a mean greater than the median. 4.5 The mean falls above the median. 4.7 Rats running a straight alley maze:  EMBED Equation.2  4.9 Multiplying by a constant (5): Original data 8 3 5 5 6 2 Mean = 4.833, Mode = 5, Median = 5 Revised data 40 15 25 25 30 10 Mean = 24.17 = 54.833, Mode = 25, Median = 25 4.11 Measures of central tendency for ADDSC and GPA: ADDSC Mode = 50 Median = 50 Mean = 4629/88 = 52.6 GPA Mode = 3.00 Median = 2.635 Mean = 216.15/88 =2.46 4.13 The means are very nearly the same for the two conditions.  4.15 The only measure that is acceptable for nominal data is the mode, because the mode is the only one that does not depend on the relationships among the points on the scale. 4.17 Class attendance: Regular Attendees Mean = 276.42; Median = 276 Poor Attendees Mean = 248.33; Median = 256 The two groups were 20 points apart in terms of the medians, and about 25 points apart in terms of means. Clearly, those students who come to class do better. Because this is not a true experiment (we dont assign subjects to groups at random), we dont know exactly what it means. I would like to think that students did poorly because they didnt hear my brilliant presentations, but it could also be that poorer students in general are less likely to come to class. This is an issue of confounding, and it is a good example making the preference for random assignment apparent in a situation with which most students can identify. 4.19 This is an Internet activity in which there is no fixed answer. 4.21 a) mean = 46.57; 10% trimmed mean = 46.67. b) mean = 28.4; 10% trimmed mean = 25.0 c) Trimming was more effective in the second example because the second distribution was quite positively skewed. 4.23 The Male Optimists had a mean of 1.016, while the Male Pessimists had a mean of 0.945. This difference is very reliable. Chapter 5-Measures of Variability 5.1 Variability of NoPassage group: Range = 57 34 = 23 St. Dev. = 6.83 Variance = 46.62 5.3 The variability of the NoPassage group is much smaller than the variability of the Passage group. If this difference turns out to be reliable, it could possibly be explained by the fact that the questions for the Passage group are asking for more than guessing and test-taking skills, and there may be greater variability due to variability in knowledge. On the other hand, it is not uncommon to find one standard deviation equal to two to three times another in small samples. 5.5 Percentages within two standard deviations in Exercise 5.2 s = 10.61  EMBED Equation.2  + 2(10.61) = 70.18 + 21.22 = 48.96 91.4 16 scores (or 94%) lie within 2 standard deviations of the mean 5.7 Multiplying or dividing by a constant: Original 2 3 4 4 5 5 9  EMBED Equation.2  1= 4.57 s1 = 2.23 X * 2 4 6 8 8 10 10 18  EMBED Equation.2  2= 9.14 s2 = 4.45 X / 2 1 1.5 2 2 2.5 2.5 4.5  EMBED Equation.2  3 = 2.29 s3 = 1.11 5.9 Convert revised data to mean = 0 Since adding or subtracting a constant will not change the standard deviation, but will change the mean, I can subtract 3.27 from every score for X2 in Exercise 5.8, making the mean = 0, and keeping s2 = 1.0. the new values are X3 -0.889 0.539 -1.842 0.539 -0.413 1.016 1.016  EMBED Equation.2 1= 0 s1 = 1 5.11 Boxplot for Exercise 5.1: Median location = (N + 1)/2 = 29/2 = 14.5 Median = 46 Hinge location = (median location +1)/2 = 15/2 = 7.5 Hinge = 43 and 52 H-spread = 52 43 = 9 Inner fences = hinges + 1.5*H-spread = hinges + 1.5*9 = hinges + 13.5 = 29.5 and 65.5 Adjacent values = 34 and 57 30 35 40 45 50 55 60   5.13 Boxplot for ADDSC: Median location = (N + 1)/2 = 89/2 = 44.5 Median = 50 Hinge location = (median location +1)/2 = 45/2 = 22.5 Hinge = 44.5 and 60.5 H-spread = 60.5 44.5 = 16 Inner fences = hinges + 1.5*H-spread = hinges + 1.5*16 = hinges + 24 = 20.5 and 85.5 Adjacent values = 26 and 78 30 35 40 45 50 55 60 70 80 90 100   5.15 Variance when you add a score equal to the mean.  EMBED Equation.2  Note that the new variance is (1-1/N) times the old variance. The point that I was trying to make here is that adding scores that dont deviate from the mean actually decrease the variance because they decrease the average deviation from the mean. 5.17 Angle of rotation:  SHAPE \* MERGEFORMAT  5.19 The following is a cut-and-paste from the JMP help screen. (I dont expect students to make all of these distinctions from what they are given, because many of the lines overlap.)  5.21 Treatment of anorexia: I would hypothesize that the two treatment groups would show more of a weight gain than the control group, but I have no reason to predict which treatment group would do better. I would assume that the variability would be about the same within each group. Complete (Before and After) data for the three groupsfrom which difference scores were derived: Cognitive BehavioralFamily TherapyControlMean Median St. Dev. 3.01 1.40 7.31 7.26 9.00 7.16 -.45 -.35 7.99  EMBED StaticEnhancedMetafile   EMBED StaticEnhancedMetafile   EMBED StaticEnhancedMetafile  If we look at the weight gain or loss, it would appear that the Control group remained stable, but the two treatment groups gained weight. The gain is greater for the Family Therapy group. 5.23 The descriptive statistics from SPSS are given below. The variable labels should be clear.  Notice that the Winsorized variance is considerably greater than the trimmed variance, as it should be. However, it is lower than the variance of the original data, reflecting the fact that the extreme values have been replaced. Cognitive behavior scores were positively skewed, with several quite high values and one or two low values. Trimming and Winsorizing reduced the influence of those values. This causes the Winsorized variance to be considerably smaller than the original variance. The trimmed mean is considerably smaller than the original mean, but the Winsorized mean is only slightly smaller. Chapter 6-The Normal Distribution 6.1 Distribution of original values:  For the first distribution the abscissa would take on the values of: 1 2 3 4 5 6 7 For the second distribution the values would be: -3 -2 -1 0 1 2 3 For the third distribution the values would be: -1.90 -1.27 -0.63 0 0.63 1.27 1.90 In these calculations I used the parameters as given, rather than the statistics calculated on the sample. 6.3 Psychology 1 exam grades:  EMBED Equation.2  The percentage between 165 and 225 is the percentage between z = -1.0 and z = 1.0. This is twice the area between z = 0 and z = 1 = 20.3413 = .6826. The percentage below 195 is just the percentage below z = 0 = .500. The percentage below z = 1 is the percentage in the larger portion = .8413. 6.5 Guessing on the Psychology 1 exam: a) We know the mean and standard deviation if the students guess; they are 75 and 7.5, respectively. We also know that a z score of 1.28 cuts off the upper 10%. We simply need to convert z = 1.28 to a raw score.  EMBED Equation.2  b) For the top 25% of the students the logic is the same except that z = 0.675.  c) For the bottom 5% the cutoff will be z = -1.645.  EMBED Equation.2  d) I would conclude that students were not just guessing, and could make use of test-taking skills that they had acquired over the years. There is a difference between Exercises 6.3 and 6.4 on the one hand, and 6.5 on the other. In the first two we are talking about performance on the test if students take it normally. There the mean is 195. In Exercise 6.5 we are talking about performance if the students just guessed purely at random without seeing the questions, but only the answers. Here the mean is 75, with a standard deviation of 7.5. These parameters are given by the binomial distribution with N = 300, p = .25, and q = .75, though the students would certainly not be expected to know this. 6.7 Reading scores for fourth and ninth grade children: a)  To do better than the average 9th grade student, the 4th grader would have to have a score of 30 or higher.  EMBED Equation.2  The probability that a fourth grader would exceed a score of 30 is the probability of a z greater than 1.00 = .1587. c) The probability of a 9th grader doing worse than the average 4th grader is the probability of a 9th grader getting a score below 25, which is the probability of being more than half a standard deviation below the mean, which is .3085. 6.9 Diagnostically meaningful cutoffs:  EMBED Equation.2  A T score of 62.8 is the score that cuts off the top 10% of the distribution, and is therefore a diagnostically meaningful cutoff. 6.11 Seat belt study:  b) We need the probability of getting a 62 if the mean is 44 with a standard deviation of 7.  EMBED Equation.2  The probability of z > 2.57 = .0051. This is such a small probability that we will probably conclude that the student just made up the data, rather than collecting them honestly. 6.13 Distribution of correct responses Distribution     b) The easiest way to find the cutoff for the lowest 10% is to simply take the sample data and count them, empirically finding the point with 10% of the scores below it. 6.15 Reaction time data:  EMBED Equation.2  For a normal distribution we would expect 75% of the scores to be equal to or less than 2.06 seconds. In our data the 75th percentile is 1.88 seconds. 6.17 Identifying the highest 2% of Behavior Problem scores: The upper 2% is cut off by z = 2.05  EMBED Equation.2  The critical cutoff is a score of 70.5. 6.19 The statisticians were upset because, by defining overweight as weighing more than 95% of peers (i.e. above the 95th percentile), the article seemed to be suggesting that there were 22% of children in the top 5%. Moreover, the article says that in 1986 only 8% of children were in the top 15%. That is just sillyit is analogous to all of the children are above average. I assume that they meant to say that 22% (etc.) were above what the 95th percentile was some years ago, but that is a different thing. Even if that is the case, the results still look too extreme to be likely. 6.21 Histogram of combined data on emotional stability  Notice that we have combined two normal distributions with the same mean, but the resulting distribution is not normal, as can be seen by comparing it to the superimposed normal curve. If the means were very different the distribution would become bimodal. Chapter 7-Basic Concepts of Probability 7.1 Views of probability: AnalyticIf two tennis players are exactly equally skillful so that the outcome of their match is random, the probability is .50 that Player A will win the upcoming match. Relative FrequencyIf in past matches Player A has beaten Player B on 13 of the 17 occasions they have played, then, unless something has changed, Player A has a probability of 13/17 = .76 of winning their upcoming match. SubjectivePlayer As coach feels that she has a probability of .90 of winning her upcoming match with Player B. 7.3 More raffle tickets: The probability winning second prize given that you did not win first is 1/999 = .001. The probability that mom comes in first and you are second = 1/1000 * 1/999 = .000001. The probability of you first and mom second = 1/1000 * 1/999 = .000001 The probability that the two of you will take the top two prizes is .000001 + .000001 = .000002. 7.5 Part a) of Exercise 7.3 dealt with conditional probabilities. 7.7 What is the probability that you will feel better about your life given that you seek psychological counseling? The research hypothesis is that those who seek help when they need it feel better about life than those who refuse to seek help. 7.9 The mother and child are both sleeping for 11 hours, so the probabilities must be based on the remaining 13 hours. p(mom looking) = 2/13 = .154; p(baby looking) = 3/13 = .231; p(both looking) = .154*.231 = .036. 7.11 We would expect 3.33 percent of the fliers to end up in the trash if the message and the behavior were independent. In fact, Geller et al. found 4.5 percent of those fliers in the trash. This may look like a very small difference, but given the number of fliers that were handed out, it is a reliable one. It would appear that having a message on a flier increases its probability of being disposed of properly. 7.13 A continuous variable that is routinely treated as if it were discrete is childrens learning abilities, where placement in classes often assumes that the child falls within one category or another. 7.15 If we assume that we know nothing about the applicant, the probability of their being admitted is the probability that they fall above the 80th percentile (which equals .20) times the probability that they will be admitted if they do, which is 10/100 = .10. The probability is .20*.10 = .02. Alternatively, we know that 10 out of 500 are admitted, so we could take the probability as being 10/500 = .02, which is the same thing. 7.17 ADDSC N = 88  EMBED Equation.2  = 52.6 s = 12.42 [calculated from data set]  EMBED Equation.2  The probability associated with z = -.21 is .5832. 7.19 Dropouts with ADDSC > 60: p(dropout|ADDSC > 60) = 7/25 = .28 7.21 Conditional and unconditional probability of dropping out: p(dropout) = 10/88 = .11 p(dropout|ADDSC > 60) = .28 Students are much more likely to drop out of school if they scored at or above ADDSC = 60 in elementary school. 7.23 If there is no discrimination in housing, then a persons race and whether or not they are offered a particular unit of housing are independent events. We could calculate the probability that a particular unit (or a unit in a particular section of the city) will be offered to anyone in a specific income group. We can also calculate the probability that the customer is a member of an ethnic minority. We can then calculate the probability of that person being shown the unit assuming independence and compare that answer against the actual proportion of times a member of an ethnic minority was offered such a unit. 7.25 The data again would appear to show that the U. S. Attorneys are more likely to request the death penalty when the victim was White than when the victim was Non-white. (This finding is statistically significant, though we wont address that question until Chapter 19.) 7.27 In this situation we begin with the hypothesis that African Americans are fairly represented in the population. If so, we would expect 0.43% of the pool of 2124 people from which juries are drawn are African American. That comes out to be an expectation of 9.13 people. But the pool actually only had 4 African Americans. We would not expect exactly 9 peoplewe might have 7 or 8. But 4 sounds awfully small That is such an unlikely event if the pool is fair that we would probably conclude that the pool is not a fair representation of the population of Vermont. An important point here is that this is a conditional probability. If the pool is fair the probability of this event is only .05an unlikely result. Chapter 8-Hypothesis Testing 8.1 Last nights hockey game: Null hypothesis: The game was actually an NHL hockey game. On the basis of that null hypothesis I expected that each team would earn somewhere between 0 and 6 points. I then looked at the actual points and concluded that they were way out of line with what I would expect if this were an NHL hockey game. I therefore rejected the null hypothesis. Notice that I havent drawn a conclusion about what type of game it actually was, because that is not what I set out to test. 8.3 A Type I error would be concluding that I was shortchanged when in fact I was not. 8.5 The rejection region is the set of outcomes for which we would reject the null hypothesis. The critical value would be the minimum amount of change below which I would reject the null. It is the border of the rejection region. 8.7 For the Mode test I would draw a very large number of samples and calculate the mode, range, and their ratio (M). I would then plot the resulting values of M. 8.9 Guessing the height of the chapel. a) The null hypothesis is that the average of two guesses is as accurate as one guess. The alternative hypothesis is that the average guess is more accurate than the single guess. b) A Type I error would be to reject the null hypothesis when the two kinds of guesses are equally accurate. A Type II error would be failing to reject the null hypothesis when the average guess is better than the single guess. c) I would be tempted to use a one-tailed test simply because it is hard to image that the average guess would be less accurate, on average, than the single guess. 8.11 A sampling distribution is just a special case of a general distribution in which the thing that we are plotting is a statistic which is the result of repeated sampling. 8.13 Magen et al (2008) study a) The null hypothesis is that the phrasing of the question will not effect the outcomethe means of the two groups are equal in the population. The alternative hypothesis is that the mean will depend on which condition the person is in. b) I would compare the two group means. c) If the difference is significant I would conclude that the phrasing of the choice makes a real difference in the outcome. 8.15 Rerunning Exercise 8.14 for ( = .01: We first have to find the cutoff for ( = .01 under a normal distribution. The critical value of z = 2.33 (one-tailed), which corresponds to a raw score of 42.69 (from a population with m = 59 and s = 7). We then find where 42.69 lies relative to the distribution under H1:  EMBED Equation.2  From the appendix we find that .85.08% of the scores fall above this cutoff. Therefore ( = .851. 8.17 To determine whether there is a true relationship between grades and course evaluations I would find a statistic that reflected the degree of relationship between two variables. (The students will see such a statistic (r) in the next chapter.) I would then calculate the sampling distribution of that statistic in a situation in which there is no relationship between two variables. Finally, I would calculate the statistic for a representative set of students and classes and compare my sample value with the sampling distribution of that statistic. 8.19 Allowances for fourth-grade students: The null hypothesis in this case would be the hypothesis that boys and girls receive the same allowance on average. I would use a two-tailed test because I want to reject the null whenever there is a difference in favor of one gender over the other. I would reject the null whenever the obtained difference between the average allowances were greater than I would be lead to expect if they were paid the same in the population. I would increase the sample size and get something other than a self-report of allowances. 8.21 Hypothesis testing and the judicial system The judicial system operates in ways similar to our standard logic of hypothesis testing. However, in a court we are particularly concerned with the danger of convicting an innocent person. In a trial the null hypothesis is equivalent to the assumption that the accused person is innocent. We set a very small probability of a Type I error, which is far smaller than we normally do in an experiment. Presumably the jury tries to set that probability as close to 0 as they reasonably can. By setting the probability of a Type I error so low, they knowingly allow the probability of a Type II error (releasing a guilty person) to rise, because that is thought to be the lesser evil. Chapter 9-Correlation 9.1 Low birthweight statistics:  The two outliers would appear to have a distorting effect on the correlation coefficient. However, if you replot the data without those points the relationship is still apparent and the correlation only drops to -.54. 9.3 With 24 degrees of freedom, and two-tailed test at a = .05 would require r > .388. 9.5 We can conclude that infant mortality is closely tied to both income and the availability of contraception. Infants born to people living in poverty are much more likely to die before their first birthday, and the availability of contraception significantly reduces the number of infants put at risk in the first place. 9.7 Because both income and contraception are related to mortality, we might expect that using them together would lead to a substantial increase in predictability. But note that they are correlated with each other, and therefore share some of the same variance. 9.9 Psychologists have a professional interest in infant mortality because some of the variables that contribute to infant mortality are behavioral ones, and we care about understanding, and often controlling, behavior. Psychologist have an important role to play in world health that has little to do with pills and irrigation systems. There is a great deal of data available on these issues, and you can easily find it on the Internet. If you are interested in this question, you might also be interested in searching for similar literature on HIV/AIDS. This question was partly intended to make students think about the fact that all sorts of things are of interest to psychologists. We dont just run animals in a maze or inquire into peoples dirty minds. In addition, low birthweight is a risk factor for all sorts of infant outcomes. 9.11 The relationship is extremely curvilinear, even though the linear correlation is quite high. You can see that the best fitting line misses almost all of the data points at each end of the distribution. 9.13 The relationship between test scores in Katz study and SAT scores for application purposes is a relevant question because we would not be satisfied with a set of data that used SAT questions and yet gave answers that were not in line with SAT performance. We want to know that the tests are measuring at least roughly the same thing. In addition, by knowing the correlation between SATs and performance without seeing the questions, we get a better understanding of some of what the SAT is measuring. 9.15 Correlation for the data in Exercise 9.14: SAT: mean = 598.57 "X = 16760 St. Dev. = 61.57 Test: mean = 46.21 "Y = 1294 St. Dev. = 6.73  EMBED Equation.2  With 26 df we would need a correlation of .374 to be significant. Since our value exceeds that, we can conclude that the relationship between test scores and the SAT is reliably different from 0. 9.17 When we say that two correlations are not significantly different, we mean that they are sufficiently close that they could both have come from samples from populations with exactly the same population correlation coefficient. 9.19 The answer to this question depends on the students expectations. 9.21 It is sometimes appropriate to find the correlation between two variables even if you know that the relationship is slightly curvilinear. A straight line often does a remarkably good job of fitting a curved function, provided that it is not too curved. 9.23 The amount of money that a country spends on health care may have little to do with life expectancy because to change a countrys life expectancy you have to change the health of a great many individuals. Spending a great deal of money on one person, even if it were to extend her life by dozens of years, would not change the average life expectancy in any noticeable way. Often the things that make a major change in life expectancy, like inoculations, really cost very little money. The African Red Cross estimates that there are 300-500 million cases of malaria each year, resulting in 1.5 to 2.5 million deaths. In particular, more than 90% of the deaths are in children under 5 years of age, and they occur predominantly in sub-Saharan Africa. Malaria cases could be cut by up to a third with insecticide treated bednets, which are very cheap by U.S. healthcare standards. 9.25 Extremely exaggerated data on male and female weight and height to show a negative slope within gender but a positive slope across gender: Height 68 72 66 69 70 66 60 64 65 63 Weight 185 175 190 180 180 135 155 145 140 150 Gender Male Male Male Male Male Fem. Fem. Fem. Fem. Fem.  What we are effectively plotting here is the relationship between Gender and Weight, more than between Height and Weight. 9.27 We have confounding effects here. If we want to claim that red wine consumption lowers the incidence of heart disease, we have a problem because the consumption of red wine is highest in those areas with the greatest solar radiation, which is another potential cause of the effect. We would have to look at the relationship between red wine and heart disease controlling for the effects of solar radiation. 9.29 This is an Internet search with no fixed answer. Chapter 10-Regression 10.1 Regression equation predicting low birthweight from high-risk fertility. Y = Low Birthweight Percentage X = High-risk Fertility  EMBED Equation.2  = 6.70 sY = 0.698 sY2 = 0.487  EMBED Equation.2  = 46.00 sX = 6.289 sX2 = 39.553 covXY = 2.7245  EMBED Equation.2  10.3 If the high risk fertility rate jumped to 70, we would predict that the incidence of birthweight < 2500gr would go to 8.35.  EMBED Equation.2  This assumes that there is a causal relationship, which is plausible in some ways, but not proven. It may be trivial to point this out, but here we have a real world situation where we can say something about changing trends in society and their possible effects. 10.5 I would be more comfortable speaking about the effects on Senegal because it is already at approximately the mean income level and we are not extrapolating for an extreme country. This may have little to do with a statistics course in psychology, but there have been some noticeable improvements in infant mortality in Senegal, and one device that has made a difference is a warm table on which newborn infants can be placed. This may interest students who probably think of advances in medicine in terms of MRIs.  HYPERLINK "http://www.usaid.gov/stories/senegal/pc_sn_infant.html" http://www.usaid.gov/stories/senegal/pc_sn_infant.html 10.7 Prediction of Symptoms score for a Stress score of 45: Regression equation:  EMBED Equation.2  = 0.7831X + 73.891 If X = 45: = 0.7831*45 + 73.891 Predicted Symptoms = 109.13 10.9 Subtracting 10 points from every X or Y score would not change the correlation in the slightest. The relationship between X and Y would remain the same. 10.11 Diagram to illustrate Exercise 10.10:  10.13 Adding a constant to Y:  From this figure you can see that adding 2.5 to Y simply raised the regression line by 2.5 units. The correlation would be unaffected. 10.15 Predicting GPA (Y) from ADDSC (X):  EMBED Equation.2  When Hans Huessy and I first collected these data I was somewhat disheartened by how well we were doing (and to some extent I still am). We can take a measure in elementary school that is quickly filled out by the teacher, and make an excellent prediction about how the student will do in high school. That may be nice statistically, but I dont think we like to feel that children are that locked in. 10.17 The correlation dropped to -.478 when I added and subtracted .04 from each Y value. This drop was caused by the addition of error variance. One way to solve for the point at which they become equal is to plot a few predicted values and draw regression lines. Where the lines cross is the point at which they are equal. A more exact way of to set the two equations equal to each other and solve for X.  EMBED Equation.2  10.19 Weight as a function of height for males:  The regression solution that follows is a modification of printout from SPSS. Equation Number 1 Dependent Variable.. WEIGHT Variable(s) Entered on Step Number 1.. HEIGHT Multiple R .60368 R Square .36443 Adjusted R Square .35287 Standard Error 14.99167 Analysis of Variance DF Sum of Squares Mean Square Regression 1 7087.79984 7087.79984 Residual 55 12361.25279 224.75005 F = 31.53637 Signif F = .0000 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T HEIGHT 4.355868 .775656 .603680 5.616 .0000 (Constant) -149.933617 54.916943 -2.730 .0085 The intercept is given as the constant and is -149.93, which has no interpretable meaning with these data. The slope of 4.356 tells us that a one-unit increase in height is associated with a 4.356 increase in weight. The correlation is .60, telling us that for females 36% of the variability in weight is associated with variability in height. Both the correlation and the slope are significantly different from 0, as shown by an F of 31.54 and a (equivalent) t of 5.616. 10.21 Predicting my own weight, for which I use the equation from Exercise 10.19:  EMBED Equation.2  = 4.356*height - 149.93  EMBED Equation.2  = 4.356*68 - 149.93 = 146.28 a) The residual is Y -  EMBED Equation.2  = 156 - 146.28 = 9.72. (I have gained some weight since I last used this example.) b) If the students who supplied the data gave biased responses, then, to the degree that the data are biased, the coefficients are biased and the prediction will not apply accurately to me. 10.23 Predictions for a 56 male and female For the male,  EMBED Equation.2  = 4.356*66 - 149.93 = 137.57 For a female,  EMBED Equation.2  = 2.578*66 - 44.859 = 125.29 Difference = 12.28 pounds 10.25 Plot of Reaction Time against Trials for only the Yes/5-stimuli trials:  The following regression solution is a modification of SPSS printout. Equation Number 1 Dependent Variable.. RXTIME Variable(s) Entered on Step Number 1.. TRIAL Multiple R .01640 R Square .00027 Adjusted R Square -.02056 Standard Error 12.76543 Analysis of Variance DF Sum of Squares Mean Square Regression 1 2.10363 2.10363 Residual 48 7821.89637 162.95617 F = .01291 Signif F = .9100 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T TRIAL -.014214 .125100 -.016397 -.114 .9100 (Constant) 67.805186 28.267795 2.399 .0204 The slope is only -0.014, and it is not remotely significant. For this set of data we can conclude that there is not a linear trend for reaction times to change over time. From the scatterplot above we can see no hint that there is any nonlinear pattern, either. 10.27 The evils of television:  Regression equations: Boys  EMBED Equation.2  = -4.821X + 283.61 Girls  EMBED Equation.2  = -3.460X + 268.39 b) The slopes are roughly equal, given the few data points we have, with a slightly greater decrease with increased time for boys. The difference in intercepts reflects the fact that the line for the girls is about 9 points below that for boys. c) Television can not be used as an explanation for poorer scores in girls, because we see that girls score below boys even when we control for television viewing. 10.29 Draw a scattering of 10 data points and drop your pencil on it. As you move the pencil vertically you are changing the intercept. As you rotate the pencil you are changing the slope. You can come up with a very good line simply by rotating and raising or lowering your pencil so as to make the deviations from the lines as small as possible. (We really minimize squared deviations, but I dont expect anyones eyes to be good enough to do that.) 10.31 Galtons data a) The correlation is .459 and the regression equation is  EMBED Equation.DSMT4  = .646midparent + 23.942. (Remember to weight cases by freq.) b) I reran the regression requesting that SPSS save the Unstandardized prediction and residual. c)  d) The children in the lowest quartile slightly exceed their parents mean (67.12 vs 66.66) and those in the highest quartile average slightly shorter than their parents (68.09 vs 68.31). e) It is easiest if you force both axes to have the same range and specify that the regression line is  EMBED Equation.DSMT4  = 1X + 0. (If you prefer, you can use an intercept of 0.22 to equate the means of the parents and children.)  Chapter 11-Multiple Regression 11.1 Predicting quality of life: All other variables held constant, a difference of +1 degree in Temperature is associated with a difference of -.01 in perceived Quality of Life. A difference of $1000 in median income, again with all other variables held constant, is associated with a +.05 difference in perceived Quality of Life. A similar interpretation applies of b3 and b4. Since values of 0 cannot reasonably occur for all predictors, the intercept has no meaningful interpretation.  EMBED Equation.2  = 5.37 - .01(55) + .05(12) + .003(500) - .01(200) = 4.92  EMBED Equation.2  = 5.37 - .01(55) + .05(12) + .003(100) - .01(200) = 3.72 11.3 Religious Influence and religious Hope contribute significantly to the prediction, but not religious Involvement. It is worth pointing out here that even though religion Involvement does not contribute significantly to the multiple regression, it does have a significant simple correlation with Optimism. The matrix of correlations (where N = 600) is OPTIMISM RELINVOL RELINF RELHOPE OPTIMISM 1.0000 .1667 .2725 .2663 P= . P= .000 P= .000 P= .000 RELINVOL .1667 1.0000 .4487 .5439 P= .000 P= . P= .000 P= .000 RELINF .2725 .4487 1.0000 .4187 P= .000 P= .000 P= . P= .000 RELHOPE .2663 .5439 .4187 1.0000 P= .000 P= .000 P= .000 P= . 11.5 I would have speculated that religious Involvement was not a significant predictor because of its overlap with the other predictors, but the tolerances kick a hole in that theory to some extent. Thats what happens when you ask a question before you are sure of the answer. ( 11.7 Adjusted R2 for 15 cases in Exercise 11.6:   Since a squared value cannot be negative, we will declare it undefined. This is all the more reasonable in light of the fact that we cannot reject H0:R* = 0. 11.9 The multiple correlation between the predictors and the percentage of births under 2500 grams is .855. The incidence of low birthweight increases when there are more mothers under 17, when mothers have fewer than 12 years of education, and when mothers are unmarried. All of the predictors are associated with young mothers. (As the question noted, there are too few observations for a meaningful analysis of the variables in question.) 11.11 The multiple correlation between Depression and the three predictor variables was significant, with R = .49 [F(3,131) = 14.11, p = .0000]. Thus approximately 25% of the variability in Depression can be accounted for by variability in these predictors. The results show us that depression among students who have lost a parent through death is positively associated with an elevated level of perceived vulnerability to future loss and negatively associated with the level of social support. The age at which the student lost his or her parent does not appear to play a role. 11.13 The fact that the frequency of the behavior was not a factor in reporting is an interesting finding. My first thought would be that it is highly correlated with the Offensiveness, and that Offensiveness is carrying the burden. But a look at the simple correlation shows that the two variables are correlated at less than r = .20. 11.15 Using random variables as predictors: I drew the following data directly from the random number tables in the appendix (and I didnt cheat). Y X1 X2 X3 X4 X5 5 3 7 2 7 5 2 1 6 0 9 5 3 5 2 9 1 2 6 4 1 8 7 9 9 1 0 2 9 4 2 7 6 7 1 7 6 9 2 8 8 1 3 7 3 0 4 9 9 3 3 7 9 4 8 5 6 5 6 4 The multiple correlation for these data is .739, which is astonishingly high. Fortunately, the F test on the regression is not significant. Notice that we have only twice as many subjects as predictors. This question is bound to lead to the question of how many cases we need per variable. There is no good answer to this question. Some will tell you that there should be at least 10 cases per predictor. I know of no argument in defense of such a rule. Harris (1985) has suggested a rule that says that N should exceed the number of predictors by at least 50. Cohen (1988) has argued from the point of view of power, and gives the example that a population correlation coefficient of .30 would require a sample size of 187 to have power = .80. This latter is sobering, but it is not a good argument here because we have not yet discussed power in any meaningful way. 11.17 Predicting weight:  11.19 The weighted average is 3.68, which is very close to the regression coefficient for Height when we control for Gender. 11.21 Sex is important to include in this relationship because women tend to be smaller than men, and thus probably have smaller, though not less effective, brains, but we probably dont want that contamination in our data. However, note that Sex was not significant in the previous answer, though the sample size (and hence power) is low. 11.23 I could argue that PctSAT is a nuisance variable because we are not particularly interested in the variable itself, but only in controlling it to allow us to have a clearer view of Expend, which is the variable in which we are interested. At the same time, it is an important contributor to the prediction of Combined, but we are led away from noticing that because of our predominant interest in Expend. 11.25 The scatterplot follows and shows that the squared correlation is .434, which is just what we found from the regression solution.  Chapter 12Hypothesis Tests Applied to Means: One Sample 12.1 Distribution of 100 random digits:  12.3 The mean and standard deviation of the sample are 4.1 and 2.82, respectively, which are reasonably close to the parameters of the population for which the sample was drawn (4.5 and 2.6, respectively). The mean of the distribution of means is 4.28, which is somewhat closer to the population mean, and the standard deviation is 1.22. The Central Limit theorem would predict a sampling distribution of the mean with a mean of 4.5 and a standard deviation of 2.6/(5 = 1.16. These values are close to the values that we would expect. 12.5 If you had drawn 50 samples of size 15, the mean of the sampling distribution should still approximate the mean of the population, but the standard error of that distribution would now be only 2.67/(15 = 0.689. 12.7 Why doesnt the previous question address the issue of the terrible state of North Dakotas educational system? These students are certainly not a random sample of high school students in North Dakota or elsewhere. Moreover, they scored above the mean of 500, which would certainly not be expected if North Dakotas system were inadequate. In addition, there is no definition of what is meant by a terrible state, nor any idea of whether or not the SAT measures such a concept. 12.9 Unlike the results in the two previous questions, this interval probably is a fair estimate of the confidence interval for P/T ratio across the country. It is not itself biased by the bias in the sampling of SAT scores. 12.11 Weight gain exercise: For these data the mean weight gain was 3.01 pounds, with a standard deviation of 7.3 pounds. This gives us  EMBED Equation.DSMT4  With 28 df the critical value at ( = .05, two-tailed, is 2.048, which will allow us to reject the null hypothesis and conclude that the girls gained weight at better than chance levels in this experiment. There is an important movement within statistics right now in the direction of laying a much heavier emphasis on confidence limits than on null hypothesis tests. I think this is a very good example of a place where a behavioral scientist might make good use of a confidence interval. I didnt ask the you to calculate these limits, but they are 0.227 and 5.787. You should think about what these limits mean and about why they are useful. 12.13 Effect size measure for data in Exercise 12.11: One effect size measure would simply be the mean weight gain of 3.01 pounds. That statistic has real meaning to us, especially if we keep the size of a standard deviation in mind. A dubious alternative method would be to calculate an estimate of  EMBED Equation.DSMT4 using the standard deviation of the gain scores as our base.  EMBED Equation.DSMT4  If I knew the standard deviation at baseline, that would make a good denominator. Unfortunately that information is not available, and 7.3 is the standard deviation of weight gains, and it is difficult to see how that creates a reasonable metric.. 12.15 I needed to solve for t in Exercise 12.14 because I did not know the population variance. 12.17 Testing the null hypothesis that children under stress report lower levels of anxiety:  EMBED Equation.DSMT4  With 35 df the critical value of t at ( = .05 (two-tailed) is 2.03. We can reject H0 and conclude that children under stress show significantly lower levels of anxiety than normal children. Here is another situation where the data say that children report lower levels of anxiety, but it was necessary to first verify that their reports could be relied upon. 12.19 Yes, the results in Exercise 12.18 are consistent with the t test in Exercise 12.17. The t test showed that these children showed lower levels of anxiety than the normal population, and the confidence interval did not include 14.55. Chapter 13Hypothesis Tests Applied to Means: Two Related Samples 13.1 Sexual satisfaction of married couples. (Dependent variable = 1 for never fun and 4 for always fun.) Husband Mean = 2.725 St. Dev. = 1.165 Wife Mean = 2.791 St. Dev. = 1.080 Difference Mean = -0.066 St. Dev. = 1.298 St. error diff = 0.136 N = 90  EMBED Equation.2  With 90 df the critical value of t is approximately 1.98, so we cannot reject the null hypothesis. We have no reason to conclude that wives are more or less satisfied, on the average, than their husbands. This is a matched-sample t because responses came from married couples. I would hope that there is some relationship between the sexual satisfaction of one member of the couple and the satisfaction of the otherbut perhaps that is hoping for too much. 13.3 Scatterplot of data from Exercise 13.1: (The frequencies of each combination are shown above the points.)  The correlation is .33, which is significant at ( = .05 This analysis finally addresses the degree of compatibility between couples, rather than mean differences. The correlation is significant, but it is not very large. That scatterplot is not very informative because of the discreteness of the scale and hence the overlapping of points. 13.5 The most important thing about a t test is the assumption that the mean (or difference between means) is normally distributed. Even though the individual values can only range over the integers 1 4, the mean of 91 subjects can take on a large number of possible values between 1 and 4. It is a continuous variable for all practical purposes, and can exhibit substantial variability. I drew 10,000 random samples of N = 91 from treating Husband scores as a population. The distribution of means follows.  13.7 We used a paired-t test for the data in Exercise 13.6 because the data were paired in the sense of coming from the same subject. Some subjects generally showed more beta-endorphins at any time than others, and we wanted to eliminate this subject-to-subject variability that has nothing to do with stress. In fact, there isnt much of a relationship between the two measures, but we cant fairly ignore it anyway. (Even though the correlation is not statistically significant, I think that we would look foolish if we did not treat these as paired data.) 13.9 If you look at the actual numbers given in Exercise 13.6, you would generally be led to expect that whatever was used to measure beta-endorphins was only accurate to the nearest half unit. Fair enough, but then where did values of 5.8 and 4.7 come from? If we can tell the difference to a tenth of a unit, why are most, but not all, of the scores reported to the nearest .5? Its a puzzle. 13.11 You would not want to use a repeated measures design in any situation where the first measure will tip off or sensitize a subject to what comes next. Thus if you are going to show a subject something and then ask him to recall it, the next time you show any item the subject will expect to have to recall it. Similarly we should be careful about repeated measures in drug studies because drugs often last surprisingly long in the body. 13.13 How many subjects do we need? First of all, in Exercise 13.6 we had 19 subjects, giving us 18 df. This means that for a one tailed test at a = .01 we will need a t of at least 2.552 to be significant. So we can substitute everything we know about the data except for the N, and solve for N.  EMBED Equation.2  This exercise should be a good lead in to power, because you should be able to see the logic of this without knowing a thing about power. But in the chapter on power we are really doing the same thing but disguising it behind a bunch of Greek symbols. (Well, perhaps thats a bit unfair.) Notice that we had to guess at N to get the critical value of t before we could calculate the needed N. Using the sample size they had is a reasonable approximation. 13.15 As the correlation between the two variables increases, the standard error of the difference will decrease, and the resulting t will increase. 13.17 First guess versus average guess   Notice that this is the same t as we had in Exercise 13.12. This is because there is a perfect linear relationship between first, second, and average guesses. (If you know the first guess and the average, you can compute what the second guess must have been.) 13.19 If I had subtracted the Before scores from the After scores I would simply change the sign of the mean and the sign of the t. There would be no other effect. 13.21 There is no answer I can give for this question because it asks the students to design a study. Chapter 14Hypothesis Tests Applied to Means: Two Independent Samples 14.1 Reanalysis of Exercise 13.1 as if the observations were independent: Males Mean = 2.725 s = 1.165 NM = 91 Females Mean = 2.791 s = 1.080 NF = 91  EMBED Equation.DSMT4  [t.05(180) = 1.98] Do not reject the null hypothesis. We can conclude that we have no reason to doubt the hypothesis that males and females are equal with respect to sexual satisfaction. There was no need to pool the variances here because the sample sizes were equal. If we did pool them, the pooled variance would have been 1.262. 14.3 The difference between the t in Exercises 13.1 and 14.1 is small because the relationship between the two variables was so small. 14.5 Random assignment plays the role of assuring (as much as is possible) that there were no systematic differences between the subjects assigned to the two groups. Without random assignment it might be possible that those who signed up for the family therapy condition were more motivated, or had more serious problems, than those in the control group. 14.7 You can not use random assignment to homophobic categories for a study like the study of homophobia because the group assignment is the property of the participants themselves. The lack of random assignment here will not invalidate the findings. 14.9 In Exercise 14.8 it could well have been that there was much less variability in the schizophrenic group than in the normal group because the number of TATs showing positive parent-child relationship could have had a floor effect at 0.0. This did not happen, but it is important to check for it anyway. 14.11 Experimenter bias effect: Expect Good Mean = 18.778 s = 3.930 N = 9 Expect Poor Mean = 17.625 s = 4.173 N = 8  EMBED Equation.DSMT4   EMBED Equation.DSMT4  [t.05(15) = 2.131] Do not reject the null hypothesis. We cannot conclude that our data show the experimenter bias effect. 14.13 Confidence limits for Exercise 14.8: Mean difference = 1.45 standard error = 0.545 t.05(38) = 2.02  EMBED Equation.DSMT4  Note that the answers to Exercises 14.11 and 14.12 are in line with the hypothesis tests, in that when we rejected the null hypothesis the confidence limits did not include 0, and when we did not reject the null, they did include 0. 14.15 Comparing GPA for those with low and high ADDSC scores:  EMBED Equation.DSMT4  [t.05(86) = 1.98] Reject H0 and conclude that people with high ADDSC scores in elementary school have lower grade point averages in ninth grade than people with lower scores. Here I pooled the variances even though the Ns were substantially different because the variance estimates were so similar. 14.17 The answer to 14.15 tells you that ADDSC scores have significant predictability of grade point average several years later. Moreover the answer to Exercise 14.16 tells you that this difference is substantial. This is a nice example of a situation in which it is easy to see a test of means as a test of predictability. 14.19 Anger with a reason is just fine.  EMBED Equation.DSMT4  The critical value is approximately 2.00, so we will reject the null hypothesis and conclude that when given a reason for a womans anger, she is given more status than when no reason was given for the anger. 14.21 If the two variances are equal, they will be equal to the pooled variance. If you have a problem seeing this, you can take any two equal variances and unequal Ns and try it for yourself. The answer becomes obvious when you do. Chapter 15Power 15.1 The statement on skiing is intended to point out that just because two things are different doesnt mean that the larger (better, greater, etc.) one will always come out ahead. To take a different example, one treatment might be better than another for anorexia, but I would be very surprised if the difference was statistically significant every time, or even that its mean was always greater than the other mean. I just hope that it is significant most of the time. 15.3 Power for socially desirable responses: Assume the population mean = 4.39 and the population standard deviation = 2.61 a) Effect size:  EMBED Equation.DSMT4  b) delta:  EMBED Equation.DSMT4  c) power = .22 Notice that the value of ( here is exactly the same as the value of t in that example. This is as it should be. 15.5 For Exercise 15.3 we would need ( approximately equal to 2.50, 2.80, and 3.25 for power of .70, .80, and .90, respectively.  EMBED Equation.DSMT4  Notice how quickly the required sample sizes increase, and how as p increases the N required increases faster and faster. 15.7 Diagram of Exercise 15.6:  15.9 Avoidance behavior in rabbits using a one-sample t test: a) For power = .50 we need ( = 1.95.  EMBED Equation.DSMT4  b) For power = .80 we need ( = 2.80.  EMBED Equation.DSMT4  Because subjects come in whole units, we would need 16 subjects for power = .50 and 32 subjects for power = .80 15.11 Avoidance behavior in rabbits with unequal sample sizes:  EMBED Equation.DSMT4  With ( = 1.46, power = .31 15.13 Cognitive development of LBW and normal babies at 1 yearmodified data: a) Power calculations  EMBED Equation.DSMT4  With ( = -1.19, power = .22 b) t test:  EMBED Equation.DSMT4  [t.05(38) = 2.205] Do not reject the null hypothesis. c) The t is numerically equal to (, although t is calculated from statistics and ( is calculated from parameters. In other words, ( is equal to the t we would get if the data came out with statistics equal to the parameters, 15.15 The significant t with the smaller N is more impressive, because that test had less power than the other, so the underlying difference is probably greater. The fact that a significant difference with a small N is more impressive should not lead you to conclude that small sample sizes are to be preferred. 15.17 Social awareness of ex-delinquentswhich subject pool would be better to use?  EMBED Equation.DSMT4 Normal = 38 N = 50  EMBED Equation.DSMT4 College = 35 N = 100  EMBED Equation.DSMT4 Dropout = 30 N = 25  EMBED Equation.DSMT4  Assuming equal standard deviations, the H. S. dropout group of 25 would result in a higher value of (, and therefore higher power. 15.19 Total Sample Sizes Required for Power = .60, ( = .05, Two-Tailed (( = 2.20) Effect Size(One-Sample tTwo-Sample t (per group)Two-Sample t (overall)Small0.20121242484Medium0.50203978Large0.8081632 15.21 When can power = (? The mean under H1 should fall at the critical value under H0. The question implies a one-tailed test. Thus the mean is 1.645 standard errors above 0, which is 100. = 100 + 1.645(X = 100 + 1.645(15)/(25 = 104.935 When = 104.935, power would equal (. 15.23 The power of the comparison of TATs of parents of schizophrenic and normal subjects.  EMBED Equation.DSMT4  Power = .75 15.25 Aronsons research on stereotype threat.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  From Appendix D5 the power of this experiment, if these are accurate estimates of the parameters, is .658. Chapter 16One-way Analysis of Variance I am assuming that most people would prefer to see the solutions to these problems as computer printout. (I will use SPSS for consistency.) 16.1 Analysis of Eysencks data: a) The analysis of variance: - - - - - O N E W A Y - - - - - Variable RECALL By Variable GROUP Group Membership Analysis of Variance Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 1 266.4500 266.4500 25.2294 .0001 Within Groups 18 190.1000 10.5611 Total 19 456.5500 Standard Standard Group Count Mean Deviation Error 95 Pct Conf Int for Mean Grp 1 10 19.3000 2.6687 .8439 17.3909 TO 21.2091 Grp 2 10 12.0000 3.7417 1.1832 9.3234 TO 14.6766 Total 20 15.6500 4.9019 1.0961 13.3558 TO 17.9442 b) t test t-tests for Independent Samples of GROUP Group Membership Number Variable of Cases Mean SD SE of Mean ---------------------------------------------------------------------- RECALL Young 10 19.3000 2.669 .844 Older 10 12.0000 3.742 1.183 ---------------------------------------------------------------------- Mean Difference = 7.3000 Levene's Test for Equality of Variances: F= .383 P= .544 t-test for Equality of Means 95% Variances t-value df 2-Tail Sig SE of Diff CI for Diff ---------------------------------------------------------------------- Equal 5.02 18 .000 1.453 (4.247, 10.353) Unequal 5.02 16.27 .000 1.453 (4.223, 10.377) Notice that if you square the t value of 5.02 you obtain 25.20, which is the same as the F in the analysis of variance. Notice also that the analysis of variance procedure produces confidence limits on the means, whereas the t procedure produces confidence limits on the difference of means. 16.3 Expanding on Exercise 16.2: a) Combine the Low groups together and the High groups together: Variable RECALL By Variable LOWHIGH Analysis of Variance Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 1 792.1000 792.1000 59.4505 .0000 Within Groups 38 506.3000 13.3237 Total 39 1298.4000 Standard Standard Group Count Mean Deviation Error 95 Pct Conf Int for Mean Grp 1 20 6.7500 1.6182 .3618 5.9927 TO 7.5073 Grp 2 20 15.6500 4.9019 1.0961 13.3558 TO 17.9442 Total 40 11.2000 5.7699 .9123 9.3547 TO 13.0453 Here we have compared recall under conditions of Low versus High processing, and can conclude that higher levels of processing lead to significantly better recall. b) The answer is still a bit difficult to interpret because both groups contain both younger and older subjects, and it is possible that the effect holds for one age group but not for the other. 16.5 (2 and (2 for the data in Exercise 16.1: SSgroup = 266.45 SStotal = 456.55 MSerror = 10.564 k = 2  EMBED Equation.DSMT4  16.7 Foa et al. (1991) study: GroupnMeanS.D.TotalVarianceSIT1411.073.9515515.6025PE1015.4011.12154123.6544SC1118.097.1319950.8369WL1019.507.1119550.5521Total4515.622703  EMBED Equation.DSMT4   EMBED Equation.DSMT4  From these values we can fill in the complete summary table and compute the F value. SourcedfSSMSFTreatment3507.840169.2803.04Error412279.06755.587Total442786.907 [F.05(3,41) = 2.84] We can reject the null hypothesis and conclude that there are significant differences between groups. Some treatments are more effective than others. b)  c) It would appear that the more interventionist treatments lead to fewer symptoms than the less interventionist ones, although we would have to run multiple comparisons to tell exactly which groups are different from which other groups. 16.9 If the sample sizes in Exercise 16.7 were twice as large, that would double the SStreat and MStreat. However it would have no effect on MSerror, which is simply the average of the group variances. The result would be that the F value would be doubled. 16.11 Effect size for tests in Exercise 16.10. It only makes sense to calculate an effect size for significant comparisons in this study, so we will deal with SIT vs SC.  EMBED Equation.DSMT4  The SIT group is nearly a full standard deviation lower in symptoms when compared to the SC group, which is a control group. 16.13 ANOVA on GPAs for the ADDSC data: Variable GPA By Variable Group Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 2 22.5004 11.2502 22.7362 .0000 Within Groups 85 42.0591 .4948 Total 87 64.5595 Standard Standard Group Count Mean Deviation Error 95 Pct Conf Int for Mean Grp 1 14 3.2536 .5209 .1392 2.9528 TO 3.5543 Grp 2 49 2.5920 .6936 .0991 2.3928 TO 2.7913 Grp 3 25 1.7436 .8020 .1604 1.4125 TO 2.0747 Total 88 2.4563 .8614 .0918 2.2737 TO 2.6388 There is a significant difference between the groups, telling us that there is a relationship between ADDSC score in elementary school and the GPA the student has in 9th grade. From the means it is clear that the GPA declines as the ADDSC score increases. These are real data, and they tell us that a teacher in elementary school can already pick out those students who will do well and badly in high school. I have always found these results depressing and worrisome, even though psychologists are supposed to like to be able to predict. There are some things I wish werent so predictable. 16.15 Analysis of Darley and Latan data: GroupnMeanTotal1130.8711.312260.7218.723130.516.63Total5236.66 EMBED Equation.DSMT4  From these values we can fill in the complete summary table and compute the F value. SourcedfSSMSFTreatment20.8540.4278.06Error492.5970.053Total513.451 [F.05(2,49) = 3.18] We can reject the null hypothesis and conclude that subjects are less likely to summon help quickly if there are other bystanders around. 16.17 Bonferroni test on data in Exercise 16.2: Both of these comparisons will be made using t tests. The means are given in Exercise 16.15 above.  EMBED Equation.DSMT4  For 36 df for error and for 2 comparisons at a familywise error rate of ( = .05, the critical value of t = 2.34. There is clearly not a significant difference between young and old subjects on tasks requiring little cognitive processing, but there is a significant difference for tasks requiring substantial cognitive processing. The probability that at least one of these statements represents a Type I error is at most .05. It is worth pointing out that when we are using MSerror as our variance estimate, and have equal sample sizes, the computations are very simple because we only need to calculate the denominator once. 16.19 Effect size for WL versus SIT  EMBED Equation.DSMT4  The two groups differ by over a standard deviation. 16.21 Spilich et al. data on a cognitive task: Variable ERRORS By Variable SMOKEGRP Analysis of Variance Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 2 2643.3778 1321.6889 4.7444 .0139 Within Groups 42 11700.4000 278.5810 Total 44 14343.7778 Standard Standard Group Count Mean Deviation Error 95 Pct Conf Int for Mean Grp 1 15 28.8667 14.6866 3.7921 20.7335 TO 36.9998 Grp 2 15 39.9333 20.1334 5.1984 28.7838 TO 51.0828 Grp 3 15 47.5333 14.6525 3.7833 39.4191 TO 55.6476 Total 45 38.7778 18.0553 2.6915 33.3534 TO 44.2022 Here we have a task that involves more cognitive involvement, and it does show a difference due to smoking condition. The non-smokers performed with fewer errors than the other two groups, although we will need to wait until the next exercise to see the multiple comparisons. 16.23 Spilich et al. data on driving simulation: Variable ERRORS By Variable SMOKEGRP Analysis of Variance Sum of Mean F F Source D.F. Squares Squares Ratio Prob. Between Groups 2 437.6444 218.8222 9.2584 .0005 Within Groups 42 992.6667 23.6349 Total 44 1430.3111 Standard Standard Group Count Mean Deviation Error 95 Pct Conf Int for Mean Grp 1 15 2.3333 2.2887 .5909 1.0659 TO 3.6008 Grp 2 15 6.8000 5.4406 1.4048 3.7871 TO 9.8129 Grp 3 15 9.9333 6.0056 1.5506 6.6076 TO 13.2591 Total 45 6.3556 5.7015 .8499 4.6426 TO 8.0685 Here we have a case in which the active smokers again performed worse than the non-smokers, and the differences are significant. 16.25 Attractiveness of faces a) The research hypothesis would be the hypothesis that faces averaged over more photographs would be judged more attractive than faces averaged over fewer photographs. b) Data analysis  c) Conclusions The group means are significantly different. From the descriptive statistics we can see that the means consistently rise as we increase the number of faces over which the composite was created. Chapter 17Factorial Analysis of Variance 17.1 Thomas and Wang (1996) study: a) This design can be characterized as a 3 ( 2 factorial, with 3 levels of Strategy and 2 levels of delay. b) I would expect that recall would be better when subjects generated their own key words, and worse when subjects were in the rote learning condition. I would also expect better recall for the shorter retention interval. (But what do I know?) c) Summaries of RECALL By levels of STRATEGY DELAY Variable Value Label Mean Std Dev Cases For Entire Population 11.602564 7.843170 78 STRATEGY 1.0000 9.461538 6.906407 26 DELAY 1.0000 14.923077 5.330127 13 DELAY 2.0000 4.000000 2.516611 13 STRATEGY 2.0000 11.269231 9.606488 26 DELAY 1.0000 20.538462 1.983910 13 DELAY 2.0000 2.000000 1.471960 13 STRATEGY 3.0000 14.076923 6.183352 26 DELAY 1.0000 15.384615 5.454944 13 DELAY 2.0000 12.769231 6.796492 13 17.3 Analysis of variance on data in Exercise 17.1: RECALL by STRATEGY DELAY UNIQUE sums of squares All effects entered simultaneously Sum of Mean Sig Source of Variation Squares DF Square F of F Main Effects 2510.603 3 836.868 42.992 .000 STRATEGY 281.256 2 140.628 7.224 .001 DELAY 2229.346 1 2229.346 114.526 .000 2-Way Interactions 824.538 2 412.269 21.179 .000 STRATEGY DELAY 824.538 2 412.269 21.179 .000 Explained 3335.141 5 667.028 34.267 .000 Residual 1401.538 72 19.466 Total 4736.679 77 61.515 There are significant differences due to both Strategy and Delay, but, more importantly, there is a significant interaction. This is a good example for showing all three effects. The Delay and Interaction effects are obvious, but the overall Strategy effect is harder to see. You would do well to calculate the Strategy means, which are 9.46, 11.27, and 14.08, respectively. It will help if you draw those means on the figure for Exercise 17.2. . 17.5 Bonferroni tests to clarify simple effects for data in Exercise 17.4:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  For 6 comparisons with 36 df, the critical value of t is 2.80. For the 5-minute delay, the condition with the key words provided by the experimenter is significantly better than both the condition in which the subjects generate their own key words and the rote learning condition. The latter two are not different from each other. For the 2-day delay, the rote learning condition is better than either of the other two conditions, which do not differ between themselves. We clearly see a different pattern of differences at the two delay conditions. The most surprising result (to me) in the superiority of rote learning with a 2 day interval. In running these Bonferroni tests, I had a choice. I could have thought of each simple effect as a family of comparisons, and obtained the critical value of t with 3 comparisons for each. Instead I chose to treat the whole set of 6 comparisons as a family and adjust the Bonferroni for 6 tests. There is no hard and fast rule here, and many might do it the other way. The results would not change regardless of what I decided. 17.7 The results in the last few exercises have suggested to me that if I were studying for a Spanish exam, I would fall back on rote learning, painful as it sounds and as much against common wisdom as it is. 17.9 In this experiment we have as many primiparous mothers as multiparous ones, which certainly does not reflect the population. Similarly, we have as many LBW infants as full-term ones, which is certainly not a reflection of reality. The mean for primiparous mothers is based on an equal number of LBW and full-term infants, which we know is not representative of the population of all primiparous births. Comparisons between groups are still legitimate, but it makes no sense to take the mean of all primiparous moms combined as a reflection of any meaningful population mean. Many of our experiments are run this way (with equal sample sizes across groups that are not equally represented in the population), and it is important to distinguish between the legitimacy of between group comparisons and the legitimacy of combined means. 17.11 Simple effects versus t tests for Exercise 17.10. a) If I had run a t test between those means my result would simply be the square root of the F = 1.328 that I obtained. b) If I used MSerror for my estimated error term it would give me a t that is the square root of the F that I would have had if I had used the overall MSerror, instead of the MSerror obtained in computing the simple effect. 17.13 Analysis of variance for Spilich et al. Study:  The main effect of Task and the interaction are significant. The main effect of Task is of no interest because there is no reason why different tasks should be equally difficult., We dont care about the main effect of Smoking either because it is created by large effects for two levels of Task and no effect for the third. What is important is the interaction. This is a good example of a situation in which main effects are of little interest. For example, saying that smoking harms performance is not really accurate. Smoking harms performance on some tasks, but not on others. Often main effects are still interpretable in the presence of an interaction, but not here. 17.15 Simple effects to clarify the Spilich et al. Example. We have already seen these simple effects in Chapter 16, in Exercises 16.18, 16.19, and 16.21. 17.17 Factorial analysis of the data in Exercise 16.2:  Here we see that we have a significant effect due to age, with younger subjects outperforming older subjects, and a significant effect due to the level of processing, with better recall of material processed at a higher level. Most importantly, we have a significant interaction, reflecting the fact that there is no important difference between younger and older subjects for the task with low levels of processing, but there is a big difference when the task calls for a high level of processingyounger subjects seem to benefit more from that processing (or do more of it). 17.19 Nurcombe et al study of maternal adaptation.  b) The program worked as intended and there was no interaction between groups and educational level. 17.21 Effect size for Level of Processing in Exercise 17.17  EMBED Equation.DSMT4  This is a very large effect size, but the data show an extreme difference between the two levels of processing. I used the square root of MSerror here because that was in line with what I did in the text. But a good case could be made for adding Age and the interaction sums of squares back in and calculating a new error term. That would produce  EMBED Equation.DSMT4  which is considerably smaller but still a very large effect. 17.23 Set of data for a 2 ( 2 design with no main effects but an interaction: Cell means:  8 12 12 8 17.25 Magnitude of effect for Exercise 17.1 Summary table from Exercise 17.1: SourcedfSSMSFStrategy2281.256140.6287.224Delay12229.3462229.346114.526S ( D2824.538412.26921.179Error721401.53819.466Total774736.679  EMBED Equation.DSMT4  17.27 Magnitude of effect for Exercise 17.13: Summary table from Exercise 17.13: SourcedfSSMSFTask228661.52614330.763132.895SmokeGrp21813.748906.8748.41T ( S41269.452317.3632.943Error12613587.200107.835Total13445331.926 EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  17.29 The two magnitude of effect measures ((2 and (2) will agree when the error term is small relative to the effect in question, and will disagree when there is a substantial amount of error relative to the effect. But notice that this is a comparison of MSerror and a sum of squares, and sums of squares can be large when there are many degrees of freedom for them. So to some extent, all other things equal, the two terms will be in closer agreement when there are several degrees of freedom for the treatment effect. 17.31 You should restrict the number of simple effects you examine to those in which you are particularly interested (on a priori grounds), because the familywise error rate will increase as the number of tests increases. Although we routinely talk about familywise error rates with respect to multiple comparison procedures, they really apply whenever you run more than one test, whether you consider them tests on main effects and interactions, or tests on simple effects, or tests on multiple contrasts. A test is a test as far as the error rate is concerned. SourcedfSSMSFGender1223.49223.4910.78Condition11.351.35<1G ( X10.690.69<1Error561161.4420.74 Chapter 18Repeated-Measures Analysis of Variance 18.1 Descriptive statistics on study of migraines:   18.3 I would have liked to collect data from students on the use of pain killers and other ways of dealing with migraines. I might also like to have data on stress levels over time so that I could possibly rule out the effects of stress Here again we are getting into issues of experimental design, which underlie all meaningful analyses. This design differs from the one in the suggestions section of the Resource Manual for Chapter 16. In that design we had separate groups tested at the different times. 18.5 Repeated-measures analysis of variance of data used in Exercise 18.4: SourcedfSSMSFSubjects8612.00Weeks1554.50554.5014.424Error8302.0037.75Total171159.7[F.05(1,24) = 4.26] There is a significant increase in decrease in severity over time. F = t2 = 3.7982 = 14.424. 18.7 Effect size for Exercise 18.4 We will use the square root of MSerror as our estimate of the standard deviation, because this is a standard deviation corrected for any differences due to subject effects.  EMBED Equation.DSMT4  The decrease in severity from baseline to training a reduction of approximately three and one half standard deviations. (I used the standard deviation of the baseline scores in line with what I said in the text. 18.9  EMBED Equation.DSMT4  for difference in Exercise 18.8 I would standardize the difference in means using the square root of the average of the variances of the two baseline measures. This would leave individual differences as part of the standard deviation, which seems appropriate. The average variance is 77.97, so the standard deviation is 8.83  EMBED Equation.DSMT4  On average, the severity of headaches decreased by nearly 1.50 standard deviations from baseline to training. 18.11 Exercise 18.10 tested the null hypothesis that condom use did not change over time. We would have hoped to see that the intervention worked and that condom use increased, but that was not what we found. There was an increase, but it was not significant. 18.13 It would appear that without the intervention, condom use would actually have declined. This suggests that the intervention may have prevented that decline, in which case that non-significant result is actually a positive finding. 18.15 Bonferroni t tests to compare the beginning and end of Baseline, and the beginning and end of Training for the data in Table 18.1. We can use a standard t test because the error term has been corrected by the repeated-measures analysis of variance, which has already removed between subject variability.  The Bonferroni alpha level would be .05/3 = .01667 We will reject all of the null hypotheses because each p value is less than .0167. Chapter 19Chi-Square 19.1 Popularity of Psychology professors: AndersonKlanskyKammTotalObserved25321067Expected22.322.322.367  EMBED Equation.2  We will reject the null hypothesis and conclude that students do not enroll at random. 19.3 Sorting one-sentence characteristics into piles: 12345TotalObserved810208450Expected5102010550Exp. %10%20%40%20%10%100%  EMBED Equation.2  Do not reject the null hypothesis that my daughters sorting behavior is in line with my theory. Notice that here is a case where my theory is aligned with the null hypothesis. 19.5 Racial choice in dolls (Clark & Clark, 1939): BlackWhiteTotalObserved83169252Expected126126252  EMBED Equation.2  We can reject H0 and conclude that the children did not choose dolls at random, but chose white dolls more often than black. This may be an old study, but it is a very important one. It played a role in Brown vs. Board of Education. It was used to argue the case that separate but equal was not equal, and that it was destructive to the self-esteem of black children. 19.7 Combining the two experiments: BlackWhiteTotal193983 (106.42)169 (145.58)252197061 (37.58)28 (51.42)89Totals144197341  EMBED Equation.2  Reject the null hypothesis and conclude that the distribution of choices between Black and White dolls was different in the two studies. Choice is not independent of the study, and could easily be related to the time at which the studies were run. We are no longer asking whether one color of doll is preferred over the other color, but whether the pattern of preference is constant across studies. In analysis of variance terms we are dealing with an interaction. 19.9 There are several ways this study could be modified. We could simply rerun the present analysis by defining smokers and non-smokers on the basis of the partners smoking behavior. Alternatively, we could redefine the Smoker variable as neither, mother, father, or both. 19.11 Howell and Huessy (1985) study of attention deficit disorder: ClassificationRemedial EnglishNonremedial English TotalNormal22 (28.374)187 (180.626)209ADD19 (12.626)74 (80.374) 93Total41261302  EMBED Equation.2  We can reject the null hypothesis and conclude that achievement level during high school varies as a function of performance during elementary school. 19.13 A one-way chi-square test on the data in the first column of Exercise 19.12 would be asking if the students are evenly distributed among the eight categories. What we really tested in Exercise 19.12 is whether that distribution, however it appears, is the same for those who later took remedial English as it is for those who later took non-remedial English. 19.15 Inescapable shock and implanted tumor rejection: Inescapable ShockEscapable ShockNo ShockTotalReject8 (14.52)19 (14.52)18 (15.97)45No Reject22 (15.48)11 (15.48)15 (17.03)48Total30303393  EMBED Equation.2  The ability to reject a tumor is affected by the shock condition. I like this example particularly because it makes it clear that psychological variables have very clear effects on physical health. We often say this, but here are some quite dramatic data. 19.17 This is another place where we see the important relationship between sample size and power. 19.19 Testosterone and childhood delinquency: High TestosteroneNormal TestosteroneTotalNot Delinquent366 (391.824)3554 (3528.176)3920Delinquent80 (54.176)462 (487.824)542 44640164462  EMBED Equation.3  a) These results show that there is a significant relationship between the two variablesc2 = 15.57. b) Testosterone levels in adults are related to the behavior of those individuals when they were children. c) This result shows that we can tie the two variables (delinquency and testosterone) together historically. I would assume that people who have high testosterone levels now also had high levels when they were children, but that is just an assumption. 19.21 We could ask a series of similar questions, evenly split between right and wrong answers. We could then sort the replies into positive and negative categories and ask whether faculty were more likely than students to give negative responses. . 19.23 Racial differences in desired weight gain. For white females, the odds of wishing to lose weight were 352/183 = 1.9235, meaning that while females are nearly twice as likely to wish to lose weight as to stay the same or gain weight. For African-American females, the corresponding ratio is 47/52 = .9038. The odds ratio is 1.9235/.9038 = 2.1281. This means that the odds of wishing to lose weight were more than twice as high among white females as compared to African American females. 19.25 Death penalty study:  The chi-square statistic is clearly significant. Nonwhite defendants are sentenced to death at a significantly higher rate than white defendants. Chapter 20Nonparametric and Distribution-Free Tests 20.1 Inferences in childrens story summaries (McConaughy, 1980): a) Analysis using the Mann-Whitney test (also known as Wilcoxons rank-sum test): Younger ChildrenOlder ChildrenRaw Data0103252476487Ranks1.531.564.594.57.511.5107.51311.5(R = 30 N = 7(R = 61 N = 6 Ws = (R for group with smaller N = 61  EMBED Equation.2  Ws < Ws; therefore Ws in Appendix E. Double the probability level for a 2-tailed test. W.025(6,7) = 27 > 23 b) Reject the null hypothesis and conclude that older children include more inferences in their summaries. 20.3 The analysis in Exercise 20.2 using the normal approximation:  EMBED Equation.2  We will reject the null hypothesis and come to the same conclusion we came to earlier. 20.5 Hypothesis formation in psychiatric residents (Nurcombe & Fitzhenry-Coor, 1979): Before8422483139After79363106787Diff.-1+5+1+4-1+2+3+6+5-2Rank28.52724.56108.54.5Signed Rank -28.527 -24.56108.5 -4.5 T+ = ((positive ranks) = 46.5 T- = ((negative ranks) = 8.5 T = smaller of |T+| or |T-| = 8.5 n = 10 T.05(10) = 8 < 8.5 Do not reject H0 b) We cannot conclude that we have evidence supporting the hypothesis that there is a reliable increase in hypothesis generation and testing over time. (Here is a case in which alternative methods of breaking ties could lead to different conclusions.) Here you might discuss how we could go about deciding how to break ties, putting the emphasis on a priori decisions. 20.7 Independence of first-born children: First12181317815165812Second10121513912138108Diff.26-24-133-3-24Rank417.54111888411Signed Rank417.5 -411 -188 -8 -411 Data Cont.: First13514201917251518Second898101411771312Diff.5-461056-5-226Rank141117.5201417.5144417.5Signed Rank14 -1117.5201417.5 -14 -4417.5 T+ = ((positive ranks) = 164 T- = ((negative ranks) = 46 T = smaller of |T+| or |T-| = 46 n = 20 T.05(20) =52 > 46 b) We can reject the null hypothesis and conclude that first-born children are more independent than their second-born siblings. Here is a good example of where we would use a matched sample test even though the same children do not perform in both conditions (nor could they). We are assuming that brothers and sisters are more similar to each other than they are to other children. Thus if the first-born is particularly independent, we would guess that the second-born has a higher than chance expectation of being more independent. They share a common environment. 20.9 Data in Exercise 20.7 plotted as a function of the first-borns score:  The scatterplot shows that the difference between the pairs is heavily dependent upon the score of the first-born. 20.11 The Wilcoxon matched-pairs signed-ranks test tests the null hypothesis that paired scores were drawn from identical populations or from symmetric populations with the same mean (and median). The corresponding t test tests the null hypothesis that the paired scores were drawn from populations with the same mean and assumes normality. This is an illustration of the argument that you buy things with assumptions. By making the more stringent assumptions of a t test, we buy greater specificity in our conclusions. However if those assumptions are false, we may have used an inappropriate test. 20.13 Rejection of the null hypothesis by a t test is a more specific statement than rejection using the appropriate distribution-free test because, by making assumptions about normality and homogeneity of variance, the t test refers specifically to population meansalthough it is also dependent on those assumptions. 20.15 Truancy and home situation of delinquent adolescents: Analysis using the Kruskall-Wallis one-way analysis of variance: Natural HomeFoster HomeGroup HomeScoreRankScoreRankScoreRank15181619109182214161313.51924.52026141614162227111054.51924.576.58854.5321211.51720431313.51822182276.51211.521Ri124.5170.583 N = 27 n = 9  EMBED Equation.2  We can reject the null hypothesis and conclude the placement of these adolescents has an effect on truancy rates. This analysis doesnt directly answer the question the psychologist wanted answered, because he wanted to show that the group home was better than the others. He might follow this up with Mann-Whitney tests serving in the role of multiple comparison procedures, applying a Bonferroni correction (although it might be difficult to find the necessary critical values.) Alternatively, he could just run a single Mann-Whitney between the group home and the combined data of the other two placements. 20.17 The study in Exercise 20.16 has the advantage over the one in Exercise 20.15 in that it eliminates the influence of individual differences (differences in overall level of truancy from one person to another). 20.19 For the data in Exercise 20-5: a) Analyzed by chi-square: MoreFewerTotalObserved7310Expected5510  EMBED Equation.2  We cannot reject the null hypothesis. b) Analyzed by Friedmans test: BeforeAfterScoreRankScoreRank82714192213221624231811023162117231829271Totals1317 N = 13 k= 2  EMBED Equation.2  These are exactly equivalent tests. 20.21 The mathematics of a lady tasting tea; First CupSecond CupThird CupScoreRankScoreRankScoreRank83322115314241162173141735241933462829341103314212310221Totals221610 N = 8 k= 3  EMBED Equation.2  We can reject the null hypothesis and conclude that people dont really like tea made with used tea bags. Chapter 21Choosing the Appropriate Test [N.B. Please review the disclaimer concerning these answers at the beginning of Chapter 21. There are many different ways to think about a study.] 21.1 This test involves comparing two proportions, and the easiest way to do that is to set up a 2 2 contingency table with Group on one dimension and Mastery on the other. 21.3 This is a repeated measures analysis of variance with assessment periods as the repeated measure and condition as the between subject variable. If measurements were taken on several occasions I would like to see the data plotted over time, but all we currently have is the data at the end of the treatment phase. 21.5 This is a t test for two independent groupschildren of divorced families and children of intact families. 21.7 This is a standard one-way analysis of variance. I would be most likely to conduct a planned comparison of the positive and negative conditions. 21.9 This is a two-way analysis of variance, with secure and insecure as one dimension and mortality vs. dental as the other. No multiple contrasts are called for because there are only two levels of each variable. 21.11 This could be treated as a two-way analysis of variance if we break the data down by race and by Afrocentric facial features. A problem with this is that we would presumably have more Afrocentric features for black inmates, which would lead to unequal sample sizes (i.e. an unbalanced design). 21.13 This is a regression problem where we are time is one variable and the difference in happiness between liberals and conservatives (by year) is the other variable. 21.15 The most important thing to do would be to plot the data over time looking for trends. 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I assume that she wants to know how rankings of characteristics agree across sexes or across years. She could first find the mean rank assigned to each characteristic separately for each sex and year. Because the raw data were originally ranks, I would probably be inclined to then rank these mean values. She could then calculate Spearmans rS between males and females for each year or between years within each sex. The correlations would be obtained for the ten pairs of scores (one per characteristic). 21.19 This is a 2 ( 4 analysis of variance with two levels of sex and 4 levels of occupation. The major emphasis is on the occupations, so multiple comparisons of those means would be appropriate. 21.21 There are two independent groups in this experiment. The authors should use a Mann-Whitney test to compare average locus of control scores. 21.23 This is a situation for a chi-square goodness-of-fit test. The conditions are Rotated versus Stationary, and the count is the number of subjects choosing that condition as giving stronger contours. The expected values would be 37/2 = 18.5. The data are sufficiently extreme that a test in superfluous. 21.25 This is another complex repeated-measures analysis of variance. The comparison of recall of the two lists (one learned before administration of the drug and the other learned after) is a repeated measurement because the same subjects are involved. The comparison of the Drug versus Saline groups is a between-subjects effect because the groups involve different subjects. 21.27 This is basically a correlational study, where we separately correlate the two dependent variables with amount of alcohol consumed. Given the 14 year gap, and all of the other factors that affect development, we should not expect very strong correlations even under the best of conditions.     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RMCHAP~2.DOC$RMChapter 2.docDyK F RMCHAP~3.DOC$RMChapter 3.docDyK F RMCHAP~4.DOC$RMChapter 4.docDyK F RMBC6B~1.DOC$RMChapter 5.docDyK F RMB07B~1.DOC$RMChapter 6.docDyK F RMB47B~1.DOC$RMChapter 7.docDyK F RMB87B~1.DOC$RMChapter 8.docDyK F RMBC7B~1.DOC$RMChapter 9.docDyK F RM927A~1.DOC& RMChapter 10.docDyK F RM927E~1.DOC& RMChapter 11.docDyK F RM9272~1.DOC& RMChapter 12.docDyK F RM9276~1.DOC& RMChapter 13.docDyK F RMA27A~1.DOC& RMChapter 14.docDyK F RMA27E~1.DOC& RMChapter 15.docDyK F RMA272~1.DOC& RMChapter 16.docDyK F RMA276~1.DOC& RMChapter 17.docDyK F RMB27A~1.DOC& RMChapter 18.docDyK F RMB27E~1.DOC& RMChapter 19.docDyK F RM967A~1.DOC& RMChapter 20.docDyK F RM967E~1.DOC& RMChapter 21.doc5DyK *http://www.uvm.edu/~dhowell/fundamentals/yK lhttp://www.uvm.edu/~dhowell/fundamentals/yX;H,]ą'cDd6O& 0  # A2[eS ÚHQ`![eS ÚHLh5gxڅKhagv7I  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      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Timesw@ CwCw w0-!X.PSymbolw@ ?CwCw w0-!-.Timesw@ CwCw w0-!X."#"#)Timesw@ @CwCw w0-!).+Timesw@ CwCw w0-!2(/PSymbolw@ ACwCw w0-!=.7Timesw@ CwCw w0-!(.@Timesw@ BCwCw w0-!N.EPSymbolw@ CwCw w0-!-.PTimesw@ CCwCw w0-!1.X!).^Timesw@ CwCw w0-!s.bTimesw@ DCwCw w0-!2(gPSymbolw@ CwCw w0-!=.oTimesw@ ECwCw w0-!27.x!(.!46.!..!6257.!).PSymbolw@ CwCw w0-!=.Timesw@ FCwCw w0-!1258.!..!8933.+!JAdding a score equal to the mean will not change the sum of the deviations@!(but will increase the denominator to 28.RTimesw@ CwCw w0-!smTimesw@ GCwCw w0-!new3pTimesw@ CwCw w0-!2hPSymbolw@ HCwCw w0-!=m!Sf Timesw@ CwCw w0-!(f(Timesw@ ICwCw w0-!Xf-PSymbolw@ CwCw w0-!-f7Timesw@ JCwCw w0-!XfA[A[HTimesw@ CwCw w0-!)fITimesw@ KCwCw w0-!2`NTimesw@ !CwCw w0-!Nv-PSymbolw@ LCwCw w0-!-v9Timesw@ "CwCw w0-!1vAiiRPSymbolw@ MCwCw w0-!=mVTimesw@ #CwCw w0-!1258f`!.fx!8933f{!28vti`iPSymbolw@ NCwCw w0-!=mTimesw@ $CwCw w0-!44m!.m!96m & 'xkdxpr8  xk"xk currentpoint ",Times .+s ( 2, Symbol +=( S)()X) -) X":)) (F2 (&N) -)1"3(O=( ZS)()X) -) X"z)) (2 (m27"Y3(=) 46) .)6257(.S)()X) -) X"#") ) ((/2 +=) ()N) -)1)))s ((g2 +=) 27) ()46) .)6257)))=) 1258).)8933(@JAdding a score equal to the mean will not change the sum of the deviations*(but will increase the denominator to 28.*s +new(h2 +=(f S)()X) -) X"[A)) (`N2 (v-N) -)1"i3(mV=(f`1258).)8933(vt28"i`2(m=) 44) .)96/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11616 div 3840 3 -1 roll exch div scale currentpoint translate 64 51 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1792 8 moveto 260 0 rlineto stroke 724 467 moveto 1652 0 rlineto stroke 3870 8 moveto 260 0 rlineto stroke 2802 467 moveto 1652 0 rlineto stroke 1036 1104 moveto 260 0 rlineto stroke 2021 2896 moveto 260 0 rlineto stroke 953 3355 moveto 1652 0 rlineto stroke 3031 3355 moveto 1627 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (s) -6 566 sh (X) 1156 325 sh (X) 1809 325 sh (N) 1172 859 sh (X) 3234 325 sh (X) 3887 325 sh (X) 400 1421 sh (X) 1053 1421 sh (N) 2155 1421 sh (s) 3077 1421 sh (s) -6 3454 sh (X) 1385 3213 sh (X) 2038 3213 sh (N) 1401 3747 sh 224 ns (new) 139 3550 sh 224 /Times-Roman f1 (2) 160 394 sh (2) 2206 153 sh (2) 4284 153 sh (2) 1450 1249 sh (2) 3243 1249 sh (2) 160 3282 sh (2) 2435 3041 sh 384 ns (1) 1791 859 sh (27) 3441 859 sh (46) 4876 566 sh (6257) 5356 566 sh (1) 2774 1421 sh (27) 3796 1421 sh (46) 4314 1421 sh (6257) 4794 1421 sh (1258) 6054 1421 sh (8933) 6918 1421 sh (1) 2020 3747 sh (1258) 3021 3213 sh (8933) 3885 3213 sh (28) 3658 3747 sh (44) 5080 3454 sh (96) 5560 3454 sh 384 /Symbol f1 (=) 406 566 sh (S) 755 325 sh (-) 1491 325 sh (-) 1532 859 sh (=) 2484 566 sh (S) 2833 325 sh (-) 3569 325 sh (=) 4562 566 sh (S) -1 1421 sh (-) 735 1421 sh (=) 1696 1421 sh (-) 2515 1421 sh (=) 3489 1421 sh (=) 5778 1421 sh (=) 635 3454 sh (S) 984 3213 sh (-) 1720 3213 sh (-) 1761 3747 sh (=) 2713 3454 sh (=) 4766 3454 sh 384 /Times-Roman f1 (\() 993 325 sh (\)) 2071 325 sh (\() 3071 325 sh (\)) 4149 325 sh (.) 5260 566 sh (\() 237 1421 sh (\)) 1315 1421 sh (\() 1996 1421 sh (\)) 2944 1421 sh (\() 4172 1421 sh (.) 4698 1421 sh (\)) 5561 1421 sh (.) 6822 1421 sh (\() 1222 3213 sh (\)) 2300 3213 sh (.) 3789 3213 sh (.) 5464 3454 sh 384 /Times-Roman f1 (Adding a score equal to the mean will no) -5 1997 sh (t change the sum of the deviations) show (but will increase the denominator to 28.) -3 2573 sh end MTsave restore d0MATH$i s 2 =S(X-X) 2 N-1=S(X-X) 2 27=46.6257S(X-X) 2 =(N-1)s 2 =27(46.6257)=1258.8933Adding a score equal to the mean will not change the sum of the deviationsbut will increase the denominator to 28.s new2 =S(X-X) 2 N-1=1258.893328=44.96Equation Native <@_10968857218KFpx px Ole ICompObjJLJo 2 =27(46.6257)=1258.8933Adding a score equal to the mean will not change the sum of the deviationsbut will increase the denominator to 28.s new2 =S(X-X) 2 N-1=1258.893328=44.96 FPicture (Enhanced Metafile)StaticEnhancedMetafile9q FPicture (Enhanced Metafile)StaticEnhancedMetafile9qObjInfoMLCONTENTS_1096885811PFpx px Ole Mll." EMFX @l." EMFX @    '% &%  +% % +  Rp"ArialD$f%M.xd%D$`w`}$ wwwwL0&$Zwn~wwMapInfo Cartographic-GDI-VECTORBOLDLDBOLD$"S/_@809A,w@wdv% %   %   Rp"ArialLDLDBOLD$"S/_@809A,w@wdvwL0&$6bwaw o w oP #`` ' w o@w %x@ @wdv% % TzfuAAzvLCognitive Behavior Therapy  % TdS4gAAASBLT20.0Td:ENRAA:SLT17.5Td 44AAA BLT15.0TdERAASLT12.5Td4AAABLT10.0T`ERAASLT7.5T`4AAABLT5.0T`ERAASLT2.5T`4AAABLT0.0TdrERAArSLT-2.5TdY4kAAAYBLT-5.0Td?EQRAA?SLT-7.5Tl#4;AAA#BLX-10.0TX AALP10TTDQAARLP8TT}AALP6TTAALP4TTAALP2TT#0AA1LP0To AAo LpStd. Dev = 7.31 ToAAo L`Mean = 3.0To!.AAo/ L`N = 29.00_888"% Zh=J !$!L$L!$!$!$!1$1% ZwM ^2^4E2E4,2,4242424242424|2|4c2c4J2J41214% %  +$k1&%  ZP$k1$1k1k1kk$$$1'% &%  %  [@Rk1R1R1k1kRR1R1% %  %  [@9R19191R1R99191% %  %  [@ 91 1 1919  1 1% %  %  [@ 111 1 11% %  %  [@111111% %  %  [@i1111ii11% %  %  [@L1111LL11% %  %  [@o1o1o11oo1o1% %  %  [@Vo1V1V1o1oVV1V1% %  %  [@=V1=1=1V1V==1=1% %  %  [@$=1$1$1=1=$$1$1% Z,$$1$1$Z,$1k1$1k1% ( ( % % ( % ( % ( % ( % ( % ( CompObjOQNoObjInfoRPCONTENTS_1096886464NgUFpx px  FPicture (Enhanced Metafile)StaticEnhancedMetafile9qd0ll." EMF  @l." EMF  @    '% &%  +% % +  Rp"ArialD$%V$H%D$`w`}$ wwwwL0&$Zwn~wwMapInfo Cartographic-GDI-VECTORBOLDLDBOLD$"S/_@8092,w@wdv% %   %   Rp"ArialLDLDBOLD$"S/_@8092,w@wdvwL0&$6bwaw  w x #`` ' w @w %x@ @wdv% % TfuAAvLhFamily Therapy % TdREfRAARSLT22.5Td64JAAA6BLT20.0TdE.RAASLT17.5Td4AAABLT15.0TdERAASLT12.5Td4AAABLT10.0T`ERAASLT7.5eT`4AAABLT5.0T`wERAAwSLT2.5kT`\4jAAA\BLT0.0Td>EPRAA>SLT-2.5Td"44AAA"BLT-5.0TT AALP5TTDQAARLP4TT}AALP3 TTAALP2TTAALP1 TT#0AA1LP0dTo AAo LpStd. Dev = 7.16 ToAAo L`Mean = 7.3To!.AAo/ L`N = 17.00a_888"% Zh7J LL11% ZvM ]2]4A2A4&2&4 2 42424242424c2c4H2H4,2,4% %  +k1&%  ZPk11k1k1kk1'% &%  %  [@Ok1O1O1k1kOO1O1% %  %  [@L1111LL11% %  %  [@L1111LL11% %  %  [@111111% %  %  [@111111% %  %  [@Uq1U1U1q1qUU1U1% %  %  [@:U1:1:1U1U::1:1% %  %  [@:111:1:11% Z,11Z,1k11k1% ( ( % % ( % ( % ( % ( % ( % ( Ole QCompObjTVRoObjInfoWTCONTENTSpll." EMF @l." EMF @    '% &%  +% % +  Rp"ArialD$RhG0ذRD$`wpw$ wwwwL0&$Zwn~wwMapInfo Cartographic-GDI-VECTORBOLDLDBOLD$"S/_@8222,w@wdv% %   %   Rp"ArialLDLDBOLD$"S/_@8222,w@wdvwL0&$6bwaw  w ( #`` ' w @w % @wdv% % TfuAAv LhControl Group  % TdREfRAARSLT15.0Td64JAAA6BLT12.5TdE.RAASLT10.0T`4AAABLT7.5T`ERAASLT5.0T`4AAABLT2.5T`ERAASLT0.0Td4AAABLT-2.5TduERAAuSLT-5.0TdZ4lAAAZBLT-7.5Tl;ESRAA;SLX-10.0Tl47AAABLX-12.5TT AALP6TT;HAAILP5TTjwAAxLP4TTAALP3TTAALP2TTAALP1TT#0AA1LP0To AAo LpStd. 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"CwCw w0-!=Timesw@$ CwCw w0-!225 PSymbolw@! #CwCw w0-!- Timesw@$ CwCw w0-!195 !30  PSymbolw@! $CwCw w0-!= Timesw@$ CwCw w0-!1!.!0 & '@#dxpr  #"# currentpoint ",Times .+z, Symbol)=( X) -) m(s" (4=( =165)-)195(N30" =.(o=) -PICT []|@ObjInfo^Equation Native _929457822aFpx px )1).)0) ) z)=( X) -) m(s" (=( 225)-)195(30" .( =) 1).)0/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9312 div 896 3 -1 roll exch div scale currentpoint translate 64 39 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 571 406 moveto 922 0 rlineto stroke 1919 406 moveto 1477 0 rlineto stroke 5483 406 moveto 922 0 rlineto stroke 6831 406 moveto 1508 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (z) 0 505 sh (X) 620 264 sh (z) 4912 505 sh (X) 5532 264 sh 384 /Symbol f1 (=) 253 505 sh (-) 955 264 sh (m) 1244 264 sh (=) 1601 505 sh (-) 2553 264 sh (=) 3504 505 sh (-) 3818 505 sh (=) 5165 505 sh (-) 5867 264 sh (m) 6156 264 sh (=) 6513 505 sh (-) 7496 264 sh (=) 8447 505 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) 884 798 sh (s) 5796 798 sh 384 /Times-Roman f1 (165) 1909 264 sh (195) 2812 264 sh (30) 2462 798 sh (1) 4028 505 sh (0) 4316 505 sh (225) 6852 264 sh (195) 7755 264 sh (30) 7390 798 sh (1) 8723 505 sh (0) 9011 505 sh 384 /Times-Roman f1 (.) 4220 505 sh (.) 8915 505 sh 384 /Times-Roman f1 ( ) 4528 505 sh end MTsave restore dMATH#  z=X-ms=165-19530=-1.0    z=X-ms=225-19530=1.0 z=X-ms=165-19530=-1.0    z=X-ms=225-19530=1.0d p N  , .  & Timesw@ 0CwCw w0-!Ole PIC `cdMETA PICT be1!.!28 PSymbolw@' CwCw w0-!=Timesw@ 1CwCw w0-!X $PSymbolw@' CwCw w0-!- /Timesw@ 2CwCw w0-!75 8!7,!.2!55- # C! F!1R!.28X!*h!7p!.v!5yPSymbolw@' CwCw w0-!+Timesw@ 3CwCw w0-!75PSymbolw@' CwCw w0-!=Timesw@ 4CwCw w0-!X!X* PSymbolw@' CwCw w0-!=*Timesw@ 5CwCw w0-!9*"!.*(!6*+PSymbolw@' CwCw w0-!+*4Timesw@ 6CwCw w0-!75*=PSymbolw@' CwCw w0-!=*LTimesw@ 7CwCw w0- ! 84.6*W & ',dxpr  ,", currentpoint ",Times .*1).)28, Symbol)=( $X) -) 75(,7).)5" # (F ) 1).28)*)7).)5) +) 75)=) X(* X) =) 9).)6) +) 75)=)  84.6/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5568 div 1408 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1065 414 moveto 1070 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (1) -42 513 sh (28) 246 513 sh (75) 1743 264 sh (7) 1369 814 sh (5) 1657 814 sh (1) 2571 513 sh (7) 3548 513 sh (5) 3836 513 sh (75) 4392 513 sh (9) 1054 1315 sh (6) 1342 1315 sh (75) 1909 1315 sh 384 /Times-Roman f1 (.) 150 513 sh (.) 1561 814 sh (*) 3292 513 sh (.) 3740 513 sh (.) 1246 1315 sh 384 /Symbol f1 (=) 731 513 sh (-) 1449 264 sh (+) 4097 513 sh (=) 4876 513 sh (=) 731 1315 sh (+) 1614 1315 sh (=) 2393 1315 sh 384 /Times-Italic f1 (X) 1114 264 sh (X) 5227 513 sh (X) 364 1315 sh 384 /Times-Roman f1 ( ) 2199 513 sh (.28) 2763 513 sh ( 84.6) 2727 1315 sh end MTsave restore dMATH 1.28=X-757.5    1.28*7.5+75=XX=9.6+75= 84.6FMicrosoft Equation Editor 2.0DNQE Equation.2CompObjYObjInfodfEquation Native _1096887038iFpx px  1.28=X-757.5    1.28*7.5+75=XX=9.6+75= 84.6dhp   , .  & Ole PIC hkdMETA ^PICT jm      "%&'()*+.0123456789:;<=>?@ACDEFGHIJKLMNOPQRSTUVWX[^`abcdefghijklmnopqrstuvwxz{|}~PSymbolw@ %CwCw w0-!-Timesw@ CwCw w0-!1!.!645"PSymbolw@ &CwCw w0-!=&Timesw@ CwCw w0-!X 1PSymbolw@ 'CwCw w0-!- <Timesw@ CwCw w0-!195 D!30=- 0 U! XPSymbolw@ (CwCw w0-!-gTimesw@ CwCw w0- !1.645o!*!30PSymbolw@ )CwCw w0-!+Timesw@ CwCw w0-!1954PSymbolw@ *CwCw w0-!=Timesw@ CwCw w0-!X!X*PSymbolw@ +CwCw w0-!=*&Timesw@ CwCw w0-!1954*.PSymbolw@ ,CwCw w0-!-*BTimesw@ CwCw w0-!49*L!.*X!35*[PSymbolw@ -CwCw w0-!=*iTimesw@ CwCw w0- ! 145.65*s & ',dxpr  ,", currentpoint ", Symbol .+-,Times)1).)645)=( 1X) -)195(=30" 0%(X )-)1.645)*)30)+)195)=) X(*X) =)195)-) 49) .)35)=)  145.65/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6720 div 1408 3 -1 roll exch div scale currentpoint translate 64 48 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1472 406 moveto 1227 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (-) -4 505 sh (=) 1154 505 sh (-) 1856 264 sh (-) 3239 505 sh (+) 5124 505 sh (=) 6044 505 sh (=) 1154 1299 sh (-) 2074 1299 sh (=) 3319 1299 sh 384 /Times-Roman f1 (1) 206 505 sh (645) 494 505 sh (195) 2115 264 sh (30) 1890 798 sh (30) 4657 505 sh (195) 5384 505 sh (195) 1430 1299 sh (49) 2371 1299 sh (35) 2851 1299 sh 384 /Times-Roman f1 (.) 398 505 sh (*) 4410 505 sh (.) 2755 1299 sh 384 /Times-Italic f1 (X) 1521 264 sh (X) 6379 505 sh (X) 803 1299 sh 384 /Times-Roman f1 ( ) 2763 505 sh (1.645) 3498 505 sh ( 145.65) 3637 1299 sh end MTsave restore dMATH&= -1.645=X-19530    -1.645*30+195=XX=195-49.35= 145.65 FMathType 5.0 Equation MathType EFEquation.DSMT49q DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  "-1.645CompObj!iObjInfoln#Equation Native $_929775117X}qFpx px == X"-7530    "-1.645*30++75==XX==75"-49.35== 25.65d $ 0  M  u .Ole ,PIC ps-dMETA /PICT rtB  & Timesw@b CwCw w0-!zPSymbolw@ CwCw w0-!= Timesw@b CwCw w0-!X PSymbolw@ CwCw w0-!- !m (PSymbolw@b CwCw w0-!s-  /PSymbolw@ CwCw w0-!=4Timesw@b CwCw w0-!30 >PSymbolw@ CwCw w0-!- MTimesw@b CwCw w0-!25 V!5M = aPSymbolw@ CwCw w0-!=eTimesw@b CwCw w0-!1n & 'udxpr  u"u currentpoint ",Times .+z, Symbol)=( X) -) m(s" (4=( >30)-) 25(M5" =$(e=) 1'/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3744 div 896 3 -1 roll exch div scale currentpoint translate 64 39 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 571 406 moveto 922 0 rlineto stroke 1919 406 moveto 1164 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (z) 0 505 sh (X) 620 264 sh 384 /Symbol f1 (=) 253 505 sh (-) 955 264 sh (m) 1244 264 sh (=) 1601 505 sh (-) 2401 264 sh (=) 3191 505 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) 884 798 sh 384 /Times-Roman f1 (30) 1935 264 sh (25) 2691 264 sh (5) 2411 798 sh (1) 3467 505 sh end MTsave restore dTMATHH F z=X-ms=30-255=1H z=X-ms=30-255=1d p  *  ,p .ObjInfouYEquation Native Zd_929775485xFpx px Ole \PIC wz]dMETA _\PICT y{yObjInfo|  & Timesw@ <CwCw w0-!1!.!28 PSymbolw@ CwCw w0-!=Timesw@ =CwCw w0-!X #PSymbolw@ CwCw w0-!- .!m 7PSymbolw@ >CwCw w0-!s,- " >PSymbolw@ CwCw w0-!=BTimesw@ ?CwCw w0-!X MPSymbolw@ CwCw w0-!- XTimesw@ @CwCw w0-!50 a!10V L m!12*!.* !8*PSymbolw@ CwCw w0-!+*Timesw@ ACwCw w0-!50* PSymbolw@ CwCw w0-!=*0Timesw@ BCwCw w0-!X*:PSymbolw@ CwCw w0-!=*ETimesw@ CCwCw w0-!62*N!.*Z!8*] & ',pdxpr  ,p",p currentpoint ",Times .*1).)28, Symbol)=( #X) -) m(,s" "(B=( MX) -) 50(V10" L!(*12) .)8)+) 50)=) X) =) 62) .)8/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3584 div 1408 3 -1 roll exch div scale currentpoint translate 64 48 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1033 406 moveto 922 0 rlineto stroke 2381 406 moveto 1079 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (1) -42 505 sh (28) 246 505 sh (50) 3054 264 sh (10) 2712 798 sh (12) -42 1302 sh (8) 438 1302 sh (50) 990 1302 sh (62) 2463 1302 sh (8) 2943 1302 sh 384 /Times-Roman f1 (.) 150 505 sh (.) 342 1302 sh (.) 2847 1302 sh 384 /Symbol f1 (=) 715 505 sh (-) 1417 264 sh (m) 1706 264 sh (=) 2063 505 sh (-) 2765 264 sh (+) 700 1302 sh (=) 1472 1302 sh (=) 2158 1302 sh 384 /Times-Italic f1 (X) 1082 264 sh (X) 2430 264 sh (X) 1807 1302 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) 1346 798 sh end MTsave restore dMATH{Q 1.28=X-ms=X-501012.8+50=X=62.8{ 1.28=X-ms=X-501012.8+50Equation Native _929776411vFpx px Ole PIC ~d=X=62.8d 00    \ .  & Timesw@ @CwCw w0-!zPSymbolw@A CwCw w0META DPICT ObjInfoEquation Native U-!= Timesw@ ACwCw w0-!62 PSymbolw@A CwCw w0-!- #Timesw@ BCwCw w0-!44 ,!7#-  7PSymbolw@A CwCw w0-!=<Timesw@ CCwCw w0-!2E!.K!57N & '\dxpr  \"\ currentpoint ",Times .+z, Symbol)=( 62)-) 44(#7" $(<=) 2).)57/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2944 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 571 403 moveto 1187 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (z) 0 502 sh 384 /Symbol f1 (=) 253 502 sh (-) 1056 261 sh (=) 1866 502 sh 384 /Times-Roman f1 (62) 590 261 sh (44) 1353 261 sh (7) 1075 795 sh (2) 2173 502 sh (57) 2461 502 sh 384 /Times-Roman f1 (.) 2365 502 sh end MTsave restore dEMATH9  z=62-447=2.579 z=62-447=2.57dx px t  , .  & Timesw@ %CwCw w0-!0!.!6757 PSy_1308312076GFpx px Ole PIC dMETA mbolw@ ECwCw w0-!=Timesw@ &CwCw w0-!X +PSymbolw@ FCwCw w0-!- 6Timesw@ 'CwCw w0-!489 ?!1264- * Q!0*!.*!675* !**!126*%PSymbolw@ GCwCw w0-!+*9Timesw@ (CwCw w0-!489*BPSymbolw@ HCwCw w0-!=*XTimesw@ )CwCw w0-!X*cPSymbolw@ ICwCw w0-!=*nTimesw@ *CwCw w0-!574*x & 'CompObjiObjInfoEquation Native _929780494Fpx px  FMathType 6.0 Equation MathType EFEquation.DSMT49q¯d@S DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  0.675== X"-1.626.63730.675*.6373++1.626==X==2.06d  p  {  ,{ .  & Ole PIC dMETA PICT Timesw@ ?CwCw w0-!2!.!05 PSymbolw@ CwCw w0-!=Timesw@ @CwCw w0-!X $PSymbolw@ CwCw w0-!- /Timesw@ ACwCw w0-!50 8!10-- # C!2*!.*!05* !**!10*PSymbolw@ CwCw w0-!+*-Timesw@ BCwCw w0-!50*6PSymbolw@ CwCw w0-!=*ETimesw@ CCwCw w0-!X*PPSymbolw@ CwCw w0-!=*[Timesw@ DCwCw w0-!70*d!.*p!5*s & ',{dxpr  ,{",{ currentpoint ",Times .+2).)05, Symbol)=( $X) -) 50(-10" # (*2).)05)*)10)+) 50)=) X) =) 70) .)5Z/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3936 div 1408 3 -1 roll ex      !"#$%&'()*+,-/46789:;<=>?@ABCEFGHIJKLMNOPQRSTUVWXY\_abcdefghijklmnopqrsuxyz{|ch div scale currentpoint translate 64 49 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1063 406 moveto 1079 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (2) -11 505 sh (05) 277 505 sh (50) 1736 264 sh (10) 1394 798 sh (2) -11 1299 sh (05) 277 1299 sh (10) 930 1299 sh (50) 1687 1299 sh (70) 3166 1299 sh (5) 3646 1299 sh 384 /Times-Roman f1 (.) 181 505 sh (.) 181 1299 sh (*) 709 1299 sh (.) 3550 1299 sh 384 /Symbol f1 (=) 745 505 sh (-) 1447 264 sh (+) 1397 1299 sh (=) 2169 1299 sh (=) 2855 1299 sh 384 /Times-Italic f1 (X) 1112 264 sh (X) 2504 1299 sh end MTsave restore dxMATHl 2.05=X-50102.05*10+50=X=70.5l 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Timesw@ CwCw w0-!Y  & ' dxpr   " currentpoint ",Times .+  ( Y /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 448 3 -1 roll exch div scale currentpoint translate 64 51 translate /cat { dup length 2 index length       !"#$%&'()*+,-./0123456789:;<=>?@ABDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 76 258 sh 384 /Times-Italic f1 (Y) -16 365 sh end MTsave restore dMATH. 2Y  2Y _1097483965!Fpx px Ole PIC dMETA  d<Xx<   F .  & Timesw@ XCwCw w0-!bPSymbolw@  8CwCw w0-!= Timesw@ YCwCw w0-!cov Timesw@  9CwCw w0-!XY 'Timesw@ ZCwCw w0-!sTimesw@  :CwCw w0-!X#Timesw@ [CwCw w0-!2#-  1PSymbolw@  ;CwCw w0-!=5!- ATimesw@ \CwCw w0-!6 H!. N!580+ Q!154?!.Q!431T ? dPSymbolw@  <CwCw w0-!==iTimesw@ ]CwCw w0-!0y!.!0426Timesw@  =CwCw w0-!a/PSymbolw@ ^CwCw w0-!=/ Timesw@  >CwCw w0-!Y/$$PSymbolw@ _CwCw w0-!-/Timesw@  ?CwCw w0-!b/'!X/.$.$5PSymbolw@ `CwCw w0-!=/9Timesw@  @CwCw w0-!2/C!./I!456/LPSymbolw@ aCwCw w0-!-/`Timesw@  ACwCw w0-!0/i!./o!0426/r!*/!52/!./!602/PSymbolw@ bCwCw w0-!=/Timesw@  BCwCw w0-!4/!./!699/! @Timesw@ cCwCw w0-!Y DPSymbolw@  CCwCw w0-!=D Timesw@ dCwCw w0-!0D!.D!0426DTimesw@  DCwCw w0-!XD7PSymbolw@ eCwCw w0-!+DBTimesw@  ECwCw w0-!4DK!.DQ!699DT & 'Fdxpr  F"F currentpoint ",Times .+b, Symbol) =( cov +XY (s +X(#2"  (5=( A-)6).)580(?154).)431" ?%(i==)0).)0426(/a) =PICT CCompObjiObjInfoEquation Native ) Y"$) -) b)X"$.) =) 2).)456)-) 0).)0426)*)52) .)602)=) 4).)699(@ (DY ) =) 0).)0426)X) +) 4).)6998/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7104 div 2240 3 -1 roll exch div scale currentpoint translate 64 53 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 599 406 moveto 940 0 rlineto stroke 1965 406 moveto 1229 0 rlineto stroke 603 1134 moveto 214 0 rlineto stroke 1413 1134 moveto 260 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (b) -9 505 sh (s) 900 798 sh (a) -5 1451 sh (Y) 587 1451 sh (b) 1201 1451 sh (X) 1430 1451 sh (Y) -16 2131 sh (X) 1721 2131 sh 224 ns (XY) 1205 360 sh (X) 1060 894 sh 384 /Symbol f1 (=) 281 505 sh (=) 1647 505 sh (-) 2045 264 sh (==) 3302 505 sh (=) 285 1451 sh (-) 909 1451 sh (=) 1781 1451 sh (-) 3031 1451 sh (=) 5843 1451 sh (=) 322 2131 sh (+) 2057 2131 sh 384 /Times-Roman f1 (cov) 622 264 sh (.) 2447 264 sh (.) 2531 798 sh (.) 4013 505 sh (.) 2280 1451 sh (.) 3515 1451 sh (*) 4438 1451 sh (.) 5073 1451 sh (.) 6349 1451 sh $/mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 76 2024 sh (.) 823 2131 sh (.) 2547 2131 sh 224 /MT_Times-Roman f1 (2) 1066 626 sh 384 ns (6) 2255 264 sh (580) 2543 264 sh (154) 1955 798 sh (431) 2627 798 sh (0) 3821 505 sh (0426) 4109 505 sh (2) 2088 1451 sh (456) 2376 1451 sh (0) 3323 1451 sh (0426) 3611 1451 sh (52) 4689 1451 sh (602) 5169 1451 sh (4) 6157 1451 sh (699) 6445 1451 sh (0) 631 2131 sh (0426) 919 2131 sh (4) 2355 2131 sh (699) 2643 2131 sh end MTsave restore dBMATH6>6 b=cov XY s X2 =-6.580154.431==0.0426a=Y-bX=2.456-0.0426*52.602=4.6992Y =0.0426X+4.699 FMathType 5.0 Equation MathType EFEquation.DSMT49qEµDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  b== cov XY s X2 == "-6.580154.431=="-0.0426a== Y"-b X==2.456"-0.0426*52.602==4.699Y =="-0.0426X++4.699ddg p` dg    x .  & Timesw@ CwCw w0-!_929887293Fpx px Ole PIC dMETA  0 !. !9 Timesw@" CwCw w0-!X PSymbolw@ CwCw w0-!+ Timesw@" CwCw w0-!31 %PSymbolw@ CwCw w0-!= 3Timesw@" CwCw w0-!1 ;!. A!5 DTimesw@ CwCw w0-!X KPSymbolw@" CwCw w0-!+ VTimesw@ CwCw w0-!18 ^!Collecting terms we get!31.PSymbolw@" CwCw w0-!-.Timesw@ CwCw w0-!18.PSymbolw@" CwCw w0-!=.%Timesw@ CwCw w0-!1..!..4!5.7Timesw@" CwCw w0-!X.>PSymbolw@ CwCw w0-!-.HTimesw@" CwCw w0-!0.Q!..W!9.ZTimesw@ CwCw w0-!X.aTimesw@" CwCw w0-!13@PSymbolw@ CwCw w0-!=@Timesw@" CwCw w0-!0@!.@!6@!Timesw@ CwCw w0-!X@(!XRPSymbolw@" CwCw w0-!=R Timesw@ CwCw w0-!13R!/R#!0R(!.R.!6R1PSymbolw@" CwCw w0-!=R:Timesw@ CwCw w0-!21RD!.RP!67RS !To check this,d!# substitute 21.67 in both equationsdF!0v!.v!9v !*v!21v!.v&!67v)PSymbolw@" CwCw w0-!+v7Timesw@ CwCw w0-!31v@PSymbolw@" CwCw w0-!=vNTimesw@ CwCw w0-!50vX!.vd!503vgPSymbolw@" CwCw w0-!=v{Timesw@ CwCw w0-!1v!.v!5v!*v!21v!.v!67vPSymbolw@" CwCw w0-!+vTimesw@ CwCw w0-!18v & ' xdxpr8  x"x currentpoint ",Times .+ 0).)9)X, Symbol) +) 31)=)1).)5)XPICT  ObjInfoEquation Native _929902741Fpx px ) +)18(Collecting terms we get*31)-)18)=) 1).)5)X) -) 0).)9)X(@13)=) 0).)6)X(RX) =) 13) /)0).)6) =) 21) .)67(dTo check this,)E# substitute 21.67 in both equations(v0).)9)*)21) .)67)+) 31)=) 50) .)503)=) 1).)5)*)21) .)67)+)18p/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7552 div 3840 3 -1 roll exch div scale currentpoint translate 64 53 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (0) -9 264 sh (9) 279 264 sh (31) 1124 264 sh (1) 1851 264 sh (5) 2139 264 sh (18) 2950 264 sh (31) -16 1419 sh (18) 678 1419 sh (1) 1423 1419 sh (5) 1711 1419 sh (0) 2553 1419 sh (9) 2841 1419 sh (13) -42 1995 sh (0) 732 1995 sh (6) 1020 1995 sh (13) 644 2571 sh (0) 1239 2571 sh (6) 1527 2571 sh (21) 2121 2571 sh (67) 2601 2571 sh (0) -9 3726 sh (9) 279 3726 sh (21) 779 3726 sh (67) 1259 3726 sh (31) 2002 3726 sh (50) 2759 3726 sh (503) 3239 3726 sh (1) 4172 3726 sh (5) 4460 3726 sh (21) 4952 3726 sh (67) 5432 3726 sh (18) 6149 3726 sh 384 /Times-Roman f1 (.) 183 264 sh (.) 2043 264 sh (.) 1615 1419 sh (.) 2745 1419 sh (.) 924 1995 sh (/) 1069 2571 sh (.) 1431 2571 sh (.) 2505 2571 sh (.) 183 3726 sh (*) 527 3726 sh (.) 1163 3726 sh (.) 3143 3726 sh (.) 4364 3726 sh (*) 4700 3726 sh (.) 5336 3726 sh 384 /Times-Italic f1 (X) 502 264 sh (X) 2354 264 sh (X) 1926 1419 sh (X) 3064 1419 sh (X) 1246 1995 sh (X) 17 2571 sh 384 /Symbol f1 (+) 838 264 sh (=) 1575 264 sh (+) 2690 264 sh (-) 419 1419 sh (=) 1147 1419 sh (-) 2261 1419 sh (=) 423 1995 sh (=) 368 2571 sh (=) 1814 2571 sh (+) 1716 3726 sh (=) 2453 3726 sh (=) 3896 3726 sh (+) 5889 3726 sh 384 /Times-Roman f1 (Collecting terms we get) -12 840 sh (To check this,) -6 3147 sh ( substitute 21.67 in both equations) 2200 3147 sh end MTsave restore dMATH| 0.9X+31=1.5X+18Collecting terms we get31-18=1.5X-0.9X13=0.6XX=13/0.6=21.67To check this, substitute 21.67 in both equations0.9*21.67+31=50.503=1.5*21.67+18 0.9X+31=1.5X+18Collecting ter     !"#$%&'()*+,-./01689:;=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^cefghjklmnopqrstuvwxyz{|}~ms we get31-18=1.5X-0.9X13=0.6XX=13/0.6=21.67To check this, substitute 21.67 in both equations0.9*21.67+31=50.503=1.5*21.67+18Ole PIC dMETA  @PICT d r r     .  & Timesw@ CwCw w0-! Timesw@ jCwCw w0-!Y  & ' dxpr   " currentpoint ",Times .+  ( Y /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 448 3 -1 roll exch div scale currentpoint translate 64 51 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 76 258 sh 384 /Times-Italic f1 (Y) -16 365 sh end MTsave restore dMATH. 2Y  2Y d r r     .ObjInfo2Equation Native 3+_929902785Fpx px Ole 4PIC 5dMETA 7@PICT <ObjInfo_  & Timesw@ 5CwCw w0-! Timesw@ :CwCw w0-!Y  & ' dxpr   " currentpoint ",Times .+  ( Y /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 448 3 -1 roll exch div scale currentpoint translate 64 51 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 76 258 sh 384 /Times-Italic f1 (Y) -16 365 sh end MTsave restore dMATH. 2Y Equation Native `+_929902934 Fpx px Ole aPIC bd 2Y d r r     .  & Timesw@ CwCw w0-! Timesw@ CwCw w0META d@PICT iObjInfoEquation Native +-!Y  & ' dxpr   " currentpoint ",Times .+  ( Y /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 448 3 -1 roll exch div scale currentpoint translate 64 51 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 76 258 sh 384 /Times-Italic f1 (Y) -16 365 sh end MTsave restore dMATH. 2Y  2Y d r_929903421Fpx px Ole PIC dMETA @ r     .  & Timesw@ CwCw w0-! Timesw@ CwCw w0-!Y  & ' dxpr   " currentpoint ",Times .+  ( PICT ObjInfoEquation Native +_929903469Fpx px Y /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 448 3 -1 roll exch div scale currentpoint translate 64 51 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 76 258 sh 384 /Times-Italic f1 (Y) -16 365 sh end MTsave restore dMATH. 2Y  2Y d r r     .  & Ole PIC dMETA @PICT Timesw@ wCwCw w0-! Timesw@i CwCw w0-!Y  & ' dxpr   " currentpoint ",Times .+  ( Y /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 448 3 -1 roll exch div scale currentpoint translate 64 51 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 76 258 sh 384 /Times-Italic f1 (Y) -16 365 sh end MTsave restore dMATH. 2Y  2Y ObjInfoEquation Native +_929907991Fpx px Ole   OmRU1! 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ThereforeQ~#Times !w*wgw -!N  - -PSymbol, !w*wgw, -!= Times !w*wgw -!2+!.1!5524!*H!13O!.[!52^!7@!.F!70I*iPSymbol, !w*wgw, -!=nTimes !w*wgw -!4y!.!481Times, !w*wgw, -!NPSymbol !w*wgw -!= Times, !w*wgw, -!4*!.0!4813Times !w*wgw -!2EPSymbol, !w*wgw, -!=MTimes !w*wgw -!20W!.c!078fPSymbol, !w*wgw, -!={Times !w*wgw -!21 ! subjects & ' udxprY  " currentpoint ",Times .+&t, Symbol)=(+D"+) -) 0(,3s +D *N";/#">3# "0.""*(V2).)552)=(N.7).)70(\+13) .)52(n8N"k0#"n4# "a+"R*(~# Therefore(N" #"# )=(+2).)552)*)13) .)52(@7).)70"*?(n=) 4).)481(N) =) 4).)481 (E2 +=) 20) .)078)=) 21) subjects/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6016 div 5888 3 -1 roll exch div scale currentpoint translate 64 57 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1333 584 moveto 267 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 573 377 384 1463 1962 16 sqr 8 th 1431 1526 moveto 637 0 rlineto stroke 16 th 1301 1067 moveto 897 0 rlineto stroke 573 377 384 1485 3503 16 sqr 8 th 1333 3067 moveto 876 0 rlineto stroke 16 th 1301 2605 moveto 940 0 rlineto stroke 573 377 384 270 4860 16 sqr 1301 4725 moveto 2054 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (t) 733 1166 sh (D) 1343 917 sh (s) 1577 1376 sh (N) 1736 1926 sh (N) 1758 3467 sh (N) 543 4824 sh (N) 575 5686 sh 224 ns (D) 1733 1472 sh 384 /Symbol f1 (=) 967 1166 sh (-) 1692 917 sh (=) 967 2704 sh (=) 967 4824 sh (=) 3479 4824 sh (=) 967 5686 sh (=) 2408 5686 sh (=) 3888 5686 sh 384 /Times-Roman f1 (0) 1984 917 sh (2) -11 2704 sh (552) 277 2704 sh (7) 1437 2455 sh (70) 1725 2455 sh (13) 1323 2917 sh (52) 1803 2917 sh (2) 1322 4575 sh (552) 1610 4575 sh (13) 2469 4575 sh (52) 2949 4575 sh (7) 1994 5125 sh (70) 2282 5125 sh (4) 3809 4824 sh (481) 4097 4824 sh (4) 1297 5686 sh x(481) 1585 5686 sh (20) 2731 5686 sh (078) 3211 5686 sh (21) 4211 5686 sh 224 ns (2) 2146 5514 sh 384 /Times-Roman f1 (.) 181 2704 sh (.) 1629 2455 sh (.) 1707 2917 sh (.) 1514 4575 sh (*) 2248 4575 sh (.) 2853 4575 sh (.) 2186 5125 sh (.) 4001 4824 sh (.) 1489 5686 sh (.) 3115 5686 sh 384 /Times-Roman f1 (Therefore) 1066 3996 sh ( subjects) 4584 5686 sh end MTsave restore dMATHvC; 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RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  t==  X 1 "- X 2   s 12 N 1 ++ s 22 N 2 == 2.725"-2.791  1.165 2 91++ 1.080 2 91  == "-0.066 .028  == "-0.0660.166=="-.40 FMathType 6.0 Equation MathType EFEquation.DSMT49q_1308470056NFpl{ pl{ Ole CompObjMOiObjInfoPƒd@SlDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  s p2 == N 1 "-1()s 12 ++N 2 "-1()s 22 N Equation Native _1308470059LVSFpl{ pl{ Ole CompObjRTi1 ++N 2 "-2== 8(15.44)++7(17.41)9++8"-2==16.359 FMathType 6.0 Equation MathType EFEquation.DSMT49qObjInfoUEquation Native O_1308470066XFpl{ pl{ Ole 3,@S4DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  t==  X 1 "- X 2   s p2 N 1 ++ s p2 N 2 == 18.778"-17.625  16.3599++ 16.3598  == 1.153 3.863  == 1.1531.965==0.587 FMathType 6.0 Equation MathTyCompObjWYiObjInfoZEquation Native _1308470072Q]Fpl{ pl{ pe EFEquation.DSMT49q©<@S,DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  CI(.95)==( X 1 "- X 2 )t .05 (38)s  X 1 "- X 2 2 ==1.452.02(0.545)==1.451.100.35d"md"2.55 FMathType 6.0 Equation MathTyOle CompObj\^iObjInfo_Equation Native pe EFEquation.DSMT49q€ @S<DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   X L ==2.59     s L2 ==0.658     N L ==75 X H ==1.68     s H2 ==0.560     N H ==13s p2 == N 1 "-1()s 12 ++N 2 "-1()s 22 N 1 ++N 2 "-2== 74(0.658)++12(0.560)75++13"-2==0.644t==  X 1 "- X 2   s p2 N 1 ++ s p2 N 2 == 2.59"-1.68  0.64475++ 0.64413  == 0.91 .058  == 0.910.241==3.77 FMathType 6.0 Equation MathType EFEquation.DSMT49q_1308471967bFpl{ pl{ Ole CompObjaciObjInfod     "%&'()*+.12347:;<=>?@ABCDEFGHKNOPQRSTUVWX[^_`abcdgjklmnopqrsvyz{|}~d@S DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   X NoAttrib ==3.40     s NoAttrib2 ==2.0736 Equation Native _1308465854gFpl{ pl{ Ole  CompObjfh!i    N NoAttrib ==41 X Attrib ==5.02     s Attrib2 ==2.7556      N Attrib ==41s p2 == n 1 "-1()s 12 ++n 2 "-1()s 22 n 1 ++n 2 "-2== 41"-1()2.0736 2 ()++41"-1()2.7556 2 ()41++41"-2==5.9466t==  X A "- X S   s p2 N 1 ++ s p2 N 2 == 3.40"-5.02  5.946641++ 5.946641  == "-1.620.538==3.01 FMathType 6.0 Equation MathType EFEquation.DSMT49q»l@S DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  g== m ObjInfoi#Equation Native $_1308465853oelFpl{ pl{ Ole ,1 "-m 0 s== 4.39"-3.872.61== 0.522.61==.20 FMathType 6.0 Equation MathType EFEquation.DSMT49qCompObjkm-iObjInfon/Equation Native 0,_1308465852qFpl{ pl{ 4@S\DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  d==.20 36  ==1.20 FMathType 6.0 Equation MathType EFEquation.DSMT49qOle 5CompObjpr6iObjInfos8Equation Native 9d@SlDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  d==g N  2.50==.20 N   therefore N== 2.50.20() 2 ==156.252.80==.20 N   therefore N== 2.80.20() 2 ==1963.25==.20 N   therefore N== 3.25.20() 2 ==264.06 FMathType 6.0 Equation MathType EFEquation.DSMT49q¨L@SDDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_1308465848~jvFpl{ pl{ Ole ICompObjuwJiObjInfoxLEquation Native M_1308465847{Fpl{ pl{ Ole YCompObjz|Zi_A  g== m 1 "-m 0 s== 5.8"-4.82== 1.02==.50d==g N  1.95==.5 N   therefore N== 1.95.50() 2 ==15.21 FMathType 6.0 Equation MathType EFEquation.DSMT49q @S<DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_EObjInfo}\Equation Native ]_1308465844yFpl{ pl{ Ole e_A  d==g N  2.8==.5 N   therefore N== 2.8.50() 2 ==31.36 FMathType 6.0 Equation MathType EFEquation.DSMT49qCompObjfiObjInfohEquation Native i_1308465842Fpl{ pl{ ™4@S\DSMT6WinAllBasicCodePagesTimes New 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VO3J8A"<{t C]Wd*9=Q^/r-}v4Stץ]TҴ/FQ-i:BEoWa*f}v׈o.]4~c?.G8MszܷUܶ%ȹ]U~DmZIV:|ӾU-ݨ;ۜ^k5`j>]~k'/No IZ=^^b%%ZBtYKBHw񷇕Z;}dMx:/]pm zf٫L|넑O2;%g,j',hefq:&NL^@Fz>(^\|9ӒV}55$?FvȹRBd%E3]ĦƞjH4YɵE"?LCmvL Tz2~f II2K4uhKvjVmɸZ.9\N,jGQpvR3rk&nWD)W ڰ0rʧn"mQ<L kK2\`jLrT3?mw]E azk7 .ŹeIRQr[2wI~[Pt_o&VBR}8lܦCuNq({3keY۹LcoٯE127mL]O5co+95sNTIB&Ij#(?XNfm\F0ʛ)P%6\Se'r>5;=x8B%Why7cuvn`> 2Ugmvl~1 g~ $בgy]r$Y"I[ +) :-7Ĉ7gC{1|̫ G|t2y\1]+|WҜ9Dl˙&%(5ܛwwQ`N_c^hw$3&5v,R1{`ImNy߉ڟ]N " !ua3M#|3~}" )bQT3uJD4__M䧕rrozbGxҿieU$VZ0'd !|a$"JH+ZKf(d ťѮ2kegBgg0Isb'q3NwF/ȇ!CP `T̄e ^ N6ȾPH{CYߣS=PLgNLs;@5bOhQ W!OgBdtNeaPFѶS-S0uE;{նtasL>)*T6ֲk5)\2J~IՓ<%N٤ĴY :#6"] >,D؁~Whaz/9+^|(S#>s CG7v \..w^2F̕%3?ϗ σ1'02 i9ɲޟMߊE LIUlX'dM>QkZKf)%,6Mݝy;*0 zM@X5c^j|/t>L(A\*(R.`jOI#Cq95mlBIaE Pw^JTe s_SxɘqZR{>-'p49L𖬿`;ߏw`{o_ve*IɇҖs-ʢ) Y:l1wZ@ۀ$h^ lSUд'=.+F2ÏCcJjgGs1[K\`T-9bJd4daE+}c٥xp͑&wF1Ӟd"Vt^HG2EҨ97jzڰ؅:s9W:Meޜ'MrwH܏Nɓ"l=bm_Qnh@d "aZ B"ż0x;ScSQ3wy7bƦ32D(~0m"^-/ TKUwR##oΡ^∙&XEiO-\ؖ_и|aV5OlMvÔO.QFp.hQIXڔ$wc<\d>X?ή9_gaoA>iS`J_O y:s\]1R5r8=YCo!eo5C꧋G:>IUk_1308477056Fp} p} Ole  CompObj iObjInfo Equation Native  J_1308477059Fp} p} Ole  CompObj i.4@SDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  d == X Hi "-X low  MS error == 15.65"-6.75 6.628  == 8.902.5739==3.46 FMathType 6.0 Equation MathType EFEquation.DSMT49qR<@SDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APObjInfo Equation Native  n_1308477062Fp} p} Ole  G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  d == X Hi "-X Low  MS error"-revised == 15.65"-8.90 13.323  == 8.903.65==2.44 FMathType 6.0 Equation MathType EFEquation.DSMT49q \@SDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  h StrCompObj iObjInfo Equation Native  _1308477072Fp} p}                           ! " # $ % & ' ( + . / 0 1 2 3 4 5 6 7 8 9 : ; < = > A D E F G H I J K L M N O P Q R S T W Z [ \ ] ^ _ ` a b c d e f g h i j m p q r s t u v w z } ~  ategy2 == SS Strategy SS total == 281.2564736.679==.06          w Strategy2 == SS Strategy "-(s"-1)MS error SS total ++MS error == 281.256"-(3"-1)19.4664736.679++19.466==.05h Delay2 == SS Delay SS total == 2229.3464736.679==.47          w Delay2 == SS Delay "-(d"-1)MS error SS total ++MS error == 2229.346"-(2"-1)19.4664736.679++19.466==.46h SD2 == SS SD SS total == 824.5384736.679==.17          w SD2 == SS SD "-(s"-1)(d"-1)MS error SS total ++MS error == 824.538"-(3"-1)(2"-1)19.4664736.679++19.466==.16 FMathType 6.0 Equation MathType EFEquation.DSMT49q&<@SDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_EOle ) CompObj* iObjInfo, Equation Native - B_A  h Task2 == SS Task SS total == 28661.52645331.926==.63          w Task2 == SS Task "-(t"-1)MS error SS total ++MS error == 28661.526"-(3"-1)107.83545331.926++107.835==.63_1308477075Fp} p} Ole ? CompObj@ iObjInfoB  FMathType 6.0 Equation MathType EFEquation.DSMT49q0\@SDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  h Smoke2 == SS SmEquation Native C L_1308477078Fp} p} Ole U CompObjV ioke SS total == 1813.74845331.926==.04          w Smoke2 == SS Smoke "-(s"-1)MS error SS total ++MS error == 1813.748"-(3"-1)107.83545331.926++107.835==.04 FMathType 6.0 Equation MathType EFEquation.DSMT49qObjInfo X Equation Native Y D_1308483202# Fp} p} Ole k (l@SDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  h TS2 == SS TS SS total == 1269.45245331.926==.03          w TS2 == SS TS "-(t"-1)(s"-1)MS error SS total ++MS error == 1269.452"-(3"-1)(3"-1)107.83545331.926++107.835==.02 FMathType 6.0 Equation MathType EFEquation.DSMT49ql@S DSMT6WinAllBasicCodePages                    K    ! 3 " # $ 2 % & ' ( ) * + , - . / 0 1 4 6 5 A H 8 9 : ; < = > ? @ B C D E F G I L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  x;W.|vTikЇ &n X=]bUtC3F FNzfF0{Jyr_(~JOrGAӃH&P%d{HxI/n6;sG{t)#5Xcef zTPe/8pN8@]ѤGj5m¡|G+.1aʸeb$^ο/G&=R`߄2ȝ<'ں6BiBb^ECXjXE{ͲR|ch%i(WJ~ v5iC(x^tˈ6wK 5/!̡_!\@&Zb;]-*Mz MRr59, A=)KU*Jv4̎-xnݶ *L-Ӷ᧙g.9Λo ю]-%M@q+lCLq4 ?,3=| ym[ga4auVG-u=,Α:ECnNjRll1WF ":qfElRy L Xf4V* /Ii趜b۬c 2rCPCc[ZH64R+J^Tn!ΑZX锚*dqkr!\$WO(Sf>ycuXQ raԕ**V7qa*4 rp{5._PMؾب& DuQ)_G>(,Es[;t /7iF`ˀo[k(/Oʴ4wn ch.RJ˚AY&axۨT+ ~" Ԋ z$/ ,ΡjT[AZC0J|fMLEa23}qȴ$ж h[L9Q?+&!q@7N.|+yGoxHAj_ b!~=&#<ޅqr_.T(| 5.otǩ&^h?Dd(P ::N   C *ANurcombeb?_T)S$<> n>_T)S$wIDATx^]|Eٽ; Bh"HQ@*]'E,^[5 Ү|3{w!\vC7?H{?3of^|?> r~XR$KK(vnˁrE^@9Zij%o@~#@.j=r/°+߬:?r=8ʷj]O|o-^s֛&)wFE4H U1rČV+FK"mO7`FSo6>L67$f"5% DE0$PWsP4=MQ{Ia2VP $@R@TE` 7ቐg,=ahK)Tk  ](wpM{]^qyȕ8SX>6XЪmΑ.q>ze l.%:Q,c `i;$qtCuk˯a8U5Ou:)U3FѠ kss 2 ~bn5 1Oh1D/551H 33B4@dx+f;*X"b|qoNfyy^+K@unb-3"{deTButZ[E48Tt* u*bǿ|͹)J-*E"7W,)hdkaq.ghأ&Hm(d56H䘃$ZV`VVY8f{~6<b|4`)1x;iPNkA)b#e( TL_}5I1Rfg 1L#/oՙC&XRnk|(@;K7Q2Wބc5@A~PcVRnkR|\xN$Ehv-I٫f'Ss!PNhbM2_ lQ}ܶWo 6qi5=%oYe,F,?t5lj --٫TFlG.Lr¾5vZ]c2gP ulEL4EVs|UR:f5ҵ xψkjE4FYf&xۮHB\LXѽ4U5t])wJcBKX:# ݾK/ v-|S];u``)km*rr潀?ruВ,1#C=|󟍢g$> Dk;4P$8?\BV51{fG |ZD=t:']v. 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"2I6W+گM ڙB$%J`!6W+گM ڙB$ & < &xڵVMl[EcّE'nd~i|'yF$r`HI]ԨTi R)H\P jWzKQ!0P3!جgwofvv1P@`$T 8     h .  & PSymbolm !w*wgwm -!cTimes4 !w*wgw4 -!2"PSymbolm !w*wgwm -!=+META R PICT ,/ CompObj YObjInfo.0  Times4 !w*wgw4 -!( DTimesm !w*wgwm -!O HPSymbol4 !w*wgw4 -!- STimesm !w*wgwm -!E \Times4 !w*wgw4 -!) dTimesm !w*wgwm -!2iTimes4 !w*wgw4 -!EU-CmPSymbolm !w*wgwm -!5PSymbol4 !w*wgw4 -!=6+Timesm !w*wgwm -!(.6!83.9PSymbol4 !w*wgw4 -!-.HTimesm !w*wgwm -!126.P!).bTimes4 !w*wgw4 -!2)fTimesm !w*wgwm -!126?G252jPSymbol4 !w*wgw4 -!+6nTimesm !w*wgwm -!(.x!169.{PSymbol4 !w*wgw4 -!-.Timesm !w*wgwm -!126.!).Times4 !w*wgw4 -!2)Timesm !w*wgwm -!126?2x2PSymbol4 !w*wgw4 -!=O+Timesm !w*wgwm -!29O5!.OA!35OD![cPSymbol4 !w*wgw4 -!ccTimesm !w*wgwm -!2] !.c!05cTimes4 !w*wgw4 -!(c!1c!)c#PSymbolm !w*wgwm -!=c+Times4 !w*wgw4 -!3c5!.c;!84c>!]cJ & ' hdxpr1  h"h currentpoint ", Symbol .+c,Times ("2 + =( D()O) -) E)) (i2 (UE"C* (5 (6+=(.6()83)-)126)) ()f2 (?G126"255(6n+(.x()169)-)126)) ()2 (?126"2x;(O+=) 29) .)35(c[)c (] 2+.)05 )()1)))=) 3).)84) ]/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5824 div 3328 3 -1 roll exch div scale currentpoint translate 64 57 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2107 474 moveto 1359 0 rlineto stroke 1651 1580 moveto 1735 0 rlineto stroke 3781 1580 moveto 1916 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 801 573 sh (c) 123 3120 sh 224 /Times-Roman f1 (2) 1055 401 sh (2) 3296 152 sh (2) 3216 1258 sh (2) 5527 1258 sh (2) 377 2948 sh (05) 556 3139 sh 384 ns (83) 1791 1430 sh (126) 2499 1430 sh (126) 2216 1980 sh (169) 3899 1430 sh (126) 4810 1430 sh (126) 4437 1980 sh (29) 1640 2481 sh (35) 2120 2481 sh (1) 914 3120 sh (3) 1635 3120 sh (84) 1923 3120 sh 384 /Symbol f1 (=) 1317 573 sh (-) 2602 324 sh (=) 1317 1679 sh (-) 2240 1430 sh (+) 3479 1679 sh (-) 4551 1430 sh (=) 1317 2481 sh (=) 1317 3120 sh 576 ns (\345) 1643 660 sh 384 /Times-Roman f1 (\() 2121 324 sh (\)) 3161 324 sh (\() 1665 1430 sh (\)) 3081 1430 sh (\() 3795 1430 sh (\)) 5392 1430 sh (.) 2024 2481 sh ([) -33 3120 sh (\() 810 3120 sh (\)) 1084 3120 sh (.) 1827 3120 sh (]) 2313 3120 sh 224 ns (.) 500 3139 sh 384 /Times-Italic f1 (O) 2251 324 sh (E) 2909 324 sh (E) 2673 874 sh end MTsave restore d3MATH'. c 2 =(O-E) 2 E   =(83-126) 2 126+(169-126) 2 126=29.35[c 2  .05 (1)=3.84]FMicrosoft Equation Editor 2.0DNQE Equation.2' c 2 =(O-E) 2 E   =(83-126) 2 126+(169Equation Native  C_9331651083Fp} p} Ole  PIC 25 T-126) 2 126=29.35[c 2  .05 (1)=3.84]T$  fcdxpr0  fc"fc currentpoint ", Symbol .+c,META  PICT 46 ObjInfo7 Equation Native  $  z  fc .  & PSymbol !w*wgw -!cTimes+ &!w*wgw+ & -!2 PSymbol !w*wgw -!="Times+ '!w*wgw+ ' -!( ;Times !w*wgw -!O ?PSymbol+ (!w*wgw+ ( -!- JTimes !w*wgw -!E STimes+ )!w*wgw+ ) -!) [Times !w*wgw -!2_Times+ *!w*wgw+ * -!EL-:dPSymbol !w*wgw -!,PSymbol+ +!w*wgw+ + -!=4"Times !w*wgw -!(-,!83-0PSymbol+ ,!w*wgw+ , -!-->Times !w*wgw -!106-F!.-X!42-[!)-hTimes+ -!w*wgw+ - -!2'lTimes !w*wgw -!106>>!.>P!42>S1,1pPSymbol+ .!w*wgw+ . -!+4tTimes !w*wgw -!(-~!169-PSymbol+ /!w*wgw+ / -!+-Times !w*wgw -!145-!.-!58-!)-Times+ 0!w*wgw+ 0 -!2'Times !w*wgw -!145>!.>!58>1~1PSymbol+ 1!w*wgw+ 1 -!+4Times !w*wgw -!(-!61-PSymbol+ 2!w*wgw+ 2 -!--Times !w*wgw -!37-!.-!58-!)- Times+ 3!w*wgw+ 3 -!2'Times !w*wgw -!37>!.>!58>11PSymbol+ 4!w*wgw+ 4 -!+4Times !w*wgw -!(-!!28-%PSymbol+ 5!w*wgw+ 5 -!--3Times !w*wgw -!51-<!.-H!42-K!)-WTimes+ 6!w*wgw+ 6 -!2'\Times !w*wgw -!51>3!.>?!42>B1 1`PSymbol+ 7!w*wgw+ 7 -!=M"Times !w*wgw -!34M+!.M7!184M:PSymbol+ 8!w*wgw+ 8 -!caTimes !w*wgw -!.d !05d !2\ Times+ 9!w*wgw+ 9 -!(a!1a!)aPSymbol !w*wgw -!=a"Times+ :!w*wgw+ : -!3a+!.a1!84a4 & 'Times ( 2 +=( ;()O) -) E)) (_2 (LE":* (, (4"=(-,()83)-)106).)42) ) ('l2 (>>106).)42"1,D(4t+(-~()169)+)145).)58) ) ('2 (>145).)58"1~I(4+(-()61)-) 37) .)58) ) ('2 (>37) .)58"1>(4+(-!()28)-) 51) .)42) ) ('\2 (>351) .)42"1 @(M"=) 34) .)184(ac +.)05(\ 2 + ()1)))=) 3).)84/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11360 div 3264 3 -1 roll exch div scale currentpoint translate 64 36 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1812 466 moveto 1359 0 rlineto stroke 1356 1556 moveto 2218 0 rlineto stroke 3969 1556 moveto 2388 0 rlineto stroke 6752 1556 moveto 2032 0 rlineto stroke 9179 1556 moveto 2069 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 538 565 sh (c) 12 3085 sh 224 /Times-Roman f1 (2) 792 393 sh (2) 3001 152 sh (2) 3404 1242 sh (2) 6187 1242 sh (2) 8614 1242 sh (2) 11078 1242 sh (05) 293 3183 sh (2) 266 2913 sh 384 ns (83) 1496 1414 sh (106) 2204 1414 sh (42) 2876 1414 sh (106) 1921 1948 sh (42) 2593 1948 sh (169) 4087 1414 sh (145) 5000 1414 sh (58) 5672 1414 sh (145) 4626 1948 sh (58) 5298 1948 sh (61) 6899 1414 sh (37) 7619 1414 sh (58) 8099 1414 sh (37) 7340 1948 sh (58) 7820 1948 sh (28) 9328 1414 sh (51) 10070 1414 sh (42) 10550 1414 sh (51) 9780 1948 sh (42) 10260 1948 sh (34) 1340 2446 sh (184) 1820 2446 sh (1) 651 3085 sh (3) 1340 3085 sh (84) 1628 3085 sh 384 /Symbol f1 (=) 1038 565 sh (-) 2307 324 sh (=) 1038 1655 sh (-) 1945 1414 sh (+) 3667 1655 sh (+) 4740 1414 sh (+) 6450 1655 sh (-) 7334 1414 sh (+) 8877 1655 sh (-) 9781 1414 sh (=) 1038 2446 sh (=) 1038 3085 sh 576 ns (\345) 1348 652 sh 384 /Times-Roman f1 (\() 1826 324 sh (\)) 2866 324 sh (\() 1370 1414 sh (.) 2780 1414 sh (\)) 3269 1414 sh (.) 2497 1948 sh (\() 3983 1414 sh (.) 5576 1414 sh (\)) 6052 1414 sh (.) 5202 1948 sh (\() 6766 1414 sh (.) 8003 1414 sh (\)) 8479 1414 sh  (.) 7724 1948 sh (\() 9193 1414 sh (.) 10454 1414 sh (\)) 10943 1414 sh (.) 10164 1948 sh (.) 1724 2446 sh (\() 547 3085 sh (\)) 821 3085 sh (.) 1532 3085 sh 224 ns (.) 237 3183 sh 384 /Times-Italic f1 (O) 1956 324 sh (E) 2614 324 sh (E) 2378 858 sh end MTsave restore dMATHOb c 2 =(O-E) 2 E   =(83-106.42) 2 106.42+(169+145.58) 2 145.58+(61-37.58) 2 37.58+(28-51.42) 2 51.42=34.184c .052 (1)=3.84 c 2 =(O-E) 2 E   =(83-106.42) 2 106.42+(169+145.58) 2 145.58+(61-37.58) 2 37.58+(28-51.42) 2 51.42=34.184c .052 (1)=3.84_933168568)`:Fp} p} Ole  PIC 9< TMETA  T& 8 fvdxpr0  fv"fv currentpoint ", Symbol .+c,Times ( 2 +=( ;()O) -) E)) (_2 (LE":* (, (4"=(-,()22)-) 28) .)374)) ('n2 (=?28) .)374"1,F(4v+(-()187)-)180)&    fv .  & PSymbol !w*wgw -!cTimes !w*wgw -!2 PSymbol !w*wgw -!="Times !w*wgw -!( ;Times !w*wgw -!O ?PSymbol !w*wgw -!- JTimes !w*wgw -!E STimes !w*wgw -!) [Times !w*wgw -!2_Times !w*wgw -!EL-:dPSymbol !w*wgw -!,PSymbol !w*wgw -!=4"Times !w*wgw -!(-,!22-1PSymbol !w*wgw -!--?Times !w*wgw -!28-H!.-T!374-W!)-iTimes !w*wgw -!2'nTimes !w*wgw -!28=?!.=K!374=N1,1rPSymbol !w*wgw -!+4vTimes !w*wgw -!(-!187-PSymbol !w*wgw -!--Times !w*wgw -!180-!.-!626-!)-Times !w*wgw -!2'Times !w*wgw -!180=!.=!626=11PSymbol !w*wgw -!+4Times !w*wgw -!(-!19-PSymbol !w*wgw -!--Times !w*wgw -!12-!.-!626-!)-Times !w*wgw -!2'Times !w*wgw -!12=!.=!626=11 PSymbol !w*wgw -!+4$Times !w*wgw -!(-.!74-2PSymbol !w*wgw -!--ATimes !w*wgw -!80-J!.-V!374-Y!)-kTimes !w*wgw -!2'oTimes !w*wgw -!80=@!.=L!374=O1.1sPSymbol !w*wgw -!=M"Times !w*wgw -!5M,!.M2!38M5PSymbol !w*wgw -!caTimes !w*wgw -!.d !05d !2\ Times !w*wgw -!(a!1a!)aPSymbol !w*wgw -!=a"Times !w*wgw -!3a+!.a1!84a4 & 'PICT ;= ObjInfo>2 Equation Native 3 _933170173AFp} p}                           ! " # $ % & ' ( ) * + , - . / 0 1 4 5 6 7 8 9 : = ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | ~  .)626)) ('2 (=180).)626"1P(4+(-()19)-) 12) .)626)) ('2 (=12) .)626"1D(4$+(-.()74)-) 80) .)374)) ('o2 (=@80) .)374"1.E(M"=) 5).)38(ac +.)05(\ 2 + ()1)))=) 3).)84/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11968 div 3264 3 -1 roll exch div scale currentpoint translate 64 35 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1812 466 moveto 1359 0 rlineto stroke 1356 1556 moveto 2274 0 rlineto stroke 4025 1556 moveto 2584 0 rlineto stroke 7004 1556 moveto 2204 0 rlineto stroke 9603 1556 moveto 2268 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 538 565 sh (c) 12 3088 sh 224 /Times-Roman f1 (2) 792 393 sh (2) 3001 152 sh (2) 3460 1242 sh (2) 6439 1242 sh (2) 9038 1242 sh (2) 11701 1242 sh (05) 293 3186 sh (2) 266 2916 sh 384 ns (22) 1505 1414 sh (28) 2261 1414 sh (374) 2741 1414 sh (28) 1965 1948 sh (374) 2445 1948 sh (187) 4143 1414 sh (180) 5050 1414 sh (626) 5722 1414 sh (180) 4679 1948 sh (626) 5351 1948 sh (19) 7122 1414 sh (12) 7841 1414 sh (626) 8321 1414 sh (12) 7564 1948 sh (626) 8044 1948 sh (74) 9756 1414 sh (80) 10502 1414 sh (374) 10982 1414 sh (80) 10205 1948 sh (374) 10685 1948 sh (5) 1344 2449 sh (38) 1632 2449 sh (1) 651 3088 sh (3) 1340 3088 sh (84) 1628 3088 sh 384 /Symbol f1 (=) 1038 565 sh (-) 2307 324 sh (=) 1038 1655 sh (-) 1971 1414 sh (+) 3723 1655 sh (-) 4791 1414 sh (+) 6702 1655 sh (-) 7582 1414 sh (+) 9301 1655 sh (-) 10221 1414 sh (=) 1038 2449 sh (=) 1038 3088 sh 576 ns (\345) 1348 652 sh 384 /Times-Roman f1 (\() 1826 324 sh (\)) 2866 324 sh (\() 1370 1414 sh (.) 2645 1414 sh (\)) 3325 1414 sh (.) 2349 1948 sh (\() 4039 1414 sh (.) 5626 1414 sh (\)) 6304 1414 sh (.) 5255 1948 sh (\() 7018 1414 sh (.) 8225 1414 sh (\)) 8903 1414 sh (.) 7948 1948 sh (\() 9617 1414 sh (.) 10886 1414 sh (\)) 11566 1414 sh (.) 10589 1948 sh (.) 1536 2449 sh (\() 547 3088 sh (\)) 821 3088 sh (.) 1532 3088 sh 224 ns (.) 237 3186 sh 384 /Times-Italic f1 (O) 1956 324 sh (E) 2614 324 sh (E) 2378 858 sh end MTsave restore dMATHRc c 2 =(O-E) 2 E   =(22-28.374) 2 28.374+(187-180.626) 2 180.626+(19-12.626) 2 12.626+(74-80.374) 2 80.374=5.38c .052 (1)=3.84 c 2 =(O-E) 2 E   =(22-28.374) 2 28.374+(187-180.626) 2 180.626+(19-12.626) 2 12.626+(74-80.374) 2 80.374=5.38c .052 (1)=3.84T     f .  & Ole ; PIC @C< TMETA > PICT BD} ] PMT Extra *!w*wgwR * -! fPSymbol !w*wgw -!cTimesR +!w*wgwR + -!2 PSymbol !w*wgw -!=$TimesR ,!w*wgwR , -!( =Times !w*wgw -!O APSymbolR -!w*wgwR - -!- LTimes !w*wgw -!E UTimesR .!w*wgwR . -!) ]Times !w*wgw -!2aTimesR /!w*wgwR / -!EN-<fPSymbol !w*wgw -!.PSymbolR 0!w*wgwR 0 -!=4$Times !w*wgw -!(-.!8-2PSymbolR 1!w*wgwR 1 -!--:Times !w*wgw -!14-B!.-N!52-Q!)-^TimesR 2!w*wgwR 2 -!2'bTimes !w*wgw -!14==!.=I!52=L1.1fPSymbolR 3!w*wgwR 3 -!+4jTimes !w*wgw -!(-t!19-wPSymbolR 4!w*wgwR 4 -!--Times !w*wgw -!14-!.-!52-!)-TimesR 5!w*wgwR 5 -!2'Times !w*wgw -!14=!.=!52=1t1PSymbolR 6!w*wgwR 6 -!+4PMT Extra !w*wgw -!L4PSymbolR 7!w*wgwR 7 -!+4Times !w*wgw -!(-!15-PSymbolR 8!w*wgwR 8 -!--Times !w*wgw -!17-!.-!03-!)- TimesR 9!w*wgwR 9 -!2' Times !w*wgw -!17=!.=!03=11PSymbolR :!w*wgwR : -!=M$Times !w*wgw -!8M-!.M3!852M6PSymbolR ;!w*wgwR ; -!caTimes !w*wgw -!.d !05d !2\ TimesR (4+ ) L) +(-()15)-)17) .)03) ) (' 2 (=17) .)03"1=(M$=) 8).)852(ac +.)05(\ 2 + ()2)))=) 5).)99/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8864 div 3264 3 -1 roll exch div scale currentpoint translate 64 35 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1874 466 moveto 1359 0 rlineto stroke 1418 1556 moveto 1838 0 rlineto stroke 3651 1556 moveto 2015 0 rlineto stroke 6754 1556 moveto 1990 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 600 565 sh (c) 12 3088 sh 224 /Times-Roman f1 (2) 854 393 sh (2) 3063 152 sh (2) 3086 1242 sh (2) 5496 1242 sh (2) 8574 1242 sh (05) 293 3186 sh (2) 266 2916 sh 384 ns (8) 1558 1414 sh (14) 2078 1414 sh (52) 2558 1414 sh (14) 1889 1948 sh (52) 2369 1948 sh (19) 3769 1414 sh (14) 4488 1414 sh (52) 4968 1414 sh (14) 4210 1948 sh (52) 4690 1948 sh (15) 6872 1414 sh (17) 7583 1414 sh (03) 8063 1414 sh (17) 7310 1948 sh (03) 7790 1948 sh (8) 1398 2449 sh (852) 1686 2449 sh (2) 682 3088 sh (5) 1406 3088 sh (99) 1694 3088 sh 384 /Symbol f1 (=) 1100 565 sh (-) 2369 324 sh (=) 1100 1655 sh (-) 1819 1414 sh (+) 3349 1655 sh (-) 4229 1414 sh (+) 5759 1655 sh (+) 6452 1655 sh (-) 7324 1414 sh (=) 1100 2449 sh (=) 1100 3088 sh 576 ns (\345) 1410 652 sh 384 /Times-Roman f1 (\() 1888 324 sh (\)) 2928 324 sh (\() 1432 1414 sh (.) 2462 1414 sh (\)) 2951 1414 sh (.) 2273 1948 sh (\() 3665 1414 sh (.) 4872 1414 sh (\)) 5361 1414 sh (.) 4594 1948 sh (\() 6768 1414 sh (.) 7967 1414 sh (\)) 8439 1414 sh (.) 7694 1948 sh (.) 1590 2449 sh (\() 547 3088 sh (\)) 883 3088 sh (.) 1598 3088 sh 224 ns (.) 237 3186 sh 384 /Times-Italic f1 (O) 2018 324 sh (E) 2676 324 sh (E) 2440 858 sh 5384 /MT-Extra f1 (L) 6018 1655 sh end MTsave restore dMATHv? c 2 =(O-E) 2 E   =(8-14.52) 2 14.52+(19-14.52) 2 14.52+L+(15-17.03) 2 17.03=8.852c .052 (2)=5.99ObjInfoE Equation Native  _957384770HFp} p} Ole  v c 2 =(O-E) 2 E   =(8-14.52) 2 14.52+(19-14.52) 2 14.52+L+(15-17.03) 2 17.03=8.852c .052 (2)=5.99CompObjGI fObjInfoJ Equation Native  _933174223?YMFp} p}  FMicrosoft Equation 3.0 DS Equation Equation.39q|cIgI  2 =O"E() 2 E  " =366"391.824() 2 391.824+...+462"487.824() 2 487.824=15.57 .052 1()=3.84TrT r    .  &                           ! 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" # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R U V W X Y Z ] _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  640 div 448 3 -1 roll exch div scale currentpoint translate -425 -3362 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1597 3429 moveto 323 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Ws) 476 3746 sh (W) 1584 3746 sh (Ws) 2300 3746 sh V/mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 926 3746 sh 384 /Symbol f1 (=) 1078 3746 sh (-) 2012 3746 sh (=) 2867 3746 sh (-) 3630 3746 sh (=) 4369 3746 sh 384 /MT_Times-Roman f1 (2) 1385 3746 sh (84) 3165 3746 sh (61) 3918 3746 sh (23) 4676 3746 sh end MTsave restore dRMATHF Ws=2W-Ws=84-61=23F Ws=2W-ObjInfoQ Equation Native  b_946293537gFTFp} p} Ole  Ws=84-61=23T9TX  FMicrosoft Equation 3.0 DS Equation Equation.39q'ؽ z=Ws"PIC SV TMETA  CompObjUW fObjInfoX 9T X    .  & Timesr !w*wgwr -!z PSymbol= !w*wgw= -!=Timesr !w*wgwr -!WsPSymbol= !w*wgw= -!-0Timesr !w*wgwr -!n ;Times= !w*wgw= -!1 @Timesr !w*wgwr -!( DTimes= !w*wgw= -!n HTimesr !w*wgwr -!1 MPSymbol= !w*wgw= -!+ TTimesr !w*wgwr -!n ]Times= !w*wgw= -!2 cPSymbolr !w*wgwr -!+ jTimes= !w*wgw= -!1 r!) x!2X- : {Timesr !w*wgwr -!n',Times= !w*wgw= -!1*1Timesr !w*wgwr -!n'5Times= !w*wgw= -!2*;Timesr !w*wgwr -!('?Times= !w*wgw= -!n'DTimesr !w*wgwr -!1*IPSymbol= !w*wgw= -!+'OTimesr !w*wgwr -!n'YTimes= !w*wgw= -!2*^PSymbolr !w*wgwr -!+'fTimes= !w*wgw= -!1'n!)'s!128K+++v.#.$-.%8'-8'**w|PSymbolr !w*wgwr -!=]Times= !w*wgw= -!53P!PSymbolr !w*wgwr -!-P/Times= !w*wgw= -!9H:!(H?!9HDPSymbolr !w*wgwr -!+HLTimes= !w*wgw= -!11HTPSymbolr !w*wgwr -!+HbTimes= !w*wgw= -!1Hj!)Ho!2YSL9Ls!9f(!(f.!11f1!)(f=!9fEPSymbolr !w*wgwr -!+fMTimes= !w*wgw= -!11fUPSymbolr !w*wgwr -!+fcTimes= !w*wgw= -!1fk!)fq!12wHj(jtm m!-m"w$-w$['['[uZZvPSymbolr !w*wgwr -!=Times= !w*wgw= -!3!.$!15'Timesr !w*wgwr -!pTimes= !w*wgw= -!(Timesr !w*wgwr -!z PSymbol= !w*wgw= -!!Timesr !w*wgwr -!3%!.+!15.!):PSymbol= !w*wgw= -!=@Timesr !w*wgwr -!(J!2N!(.T!0009[!)sPSymbol= !w*wgw= -!=zTimesr !w*wgwr -!.!0018PSymbol= !w*wgw= -!<Timesr !w*wgwr -!.!05 & 'Equation Native  C_933228194[Fp} p} Ole  PIC Z] Tn 1 (n 1 +n 2 +1)2 n 1 n 2 (n 1 +n 2 +1)12  =53"9(9+11+1)2 9(11)(9+11+1)12  ="3.15p(ze"3.15)=(2(.0009)=.0018<.05T^  n dxpr4  n "n currentpoint ",Times .+H, Symbol) =( =12(/N) ()N) +)1))".+( lR +i(s2 (mn +i"META  PICT \^ ObjInfo_S Equation Native T ^    n  .  & Times !w*wgw -!HPSymbol !w*wgw -!=$Times !w*wgw -!12 =Times !w*wgw -!N/Times !w*wgw -!(8Times !w*wgw -!N=PSymbol !w*wgw -!+HTimes !w*wgw -!1P!)V-.YTimes !w*wgw -!R lTimes !w*wgw -!irTimes !w*wgw -!2sTimes !w*wgw -!nmTimes !w*wgw -!isjxPSymbol !w*wgw -!-{Times !w*wgw -!3!(Times !w*wgw -!NPSymbol !w*wgw -!+Times !w*wgw -!1!)PSymbol !w*wgw -!\PSymbol !w*wgw -!=9$Times !w*wgw -!121@!27B.!(B:!27B>PSymbol !w*wgw -!+BMTimes !w*wgw -!1BU!)BZ5.5^!1241e!.1w!51zTimes !w*wgw -!2,Times !w*wgw -!9Bs5f5PSymbol !w*wgw -!+9Times !w*wgw -!1701!.1!51Times !w*wgw -!2,Times !w*wgw -!9B55PSymbol !w*wgw -!+9Times !w*wgw -!831Times !w*wgw -!2,Times !w*wgw -!9B55PSymbol !w*wgw -! 1a! Ca! ?a! 1! C! ?!-9Times !w*wgw -!39!(9!279PSymbol !w*wgw -!+9Times !w*wgw -!19!)9PSymbol !w*wgw -!=U$Times !w*wgw -!6U-!.U3!757U6PSymbol !w*wgw -!ciTimes !w*wgw -!.l !05l !2c Times !w*wgw -!(i!2i!)iPSymbol !w*wgw -!=i$Times !w*wgw -!5i-!.i3!99i6 & '                 &        ! " # $ % ' ( ) 9 3 + , - . / 0 1 2 4 5 6 7 8 : < = > ? @ A B C D E F G H I J K L M N O P Q R T U V W X Y Z [ \ ] ^ _ ` a b c d e f h i j k l m n o p q r s t u v w x y z { | } ~  j ({-) 3)()N) +)1)) (\ (9$=(1@12(B.27) ()27)+)1))"5.0(1e124).)5 (,2 (Bs9"5f(9+(1170).)5 (,2 (B9"5(9+(183 (,2 (B9"5(1a * (?a (1 * (? (9-) 3)()27)+)1))(U$=) 6).)757(ic +.)05(c 2 + ()2)))=) 5).)99/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8576 div 3520 3 -1 roll exch div scale currentpoint translate 64 47 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1418 466 moveto 1416 0 rlineto stroke 3354 466 moveto 457 0 rlineto stroke 1418 1686 moveto 1560 0 rlineto stroke 3202 1686 moveto 1026 0 rlineto stroke 4623 1686 moveto 1026 0 rlineto stroke 6044 1686 moveto 565 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (H) 708 565 sh (N) 1463 858 sh (N) 1902 858 sh (R) 3398 324 sh (n) 3443 858 sh (N) 4524 565 sh 224 ns (i) 3605 420 sh (i) 3624 954 sh 384 /Symbol f1 (=) 1100 565 sh (+) 2263 858 sh (-) 3903 565 sh (+) 4885 565 sh (=) 1100 1785 sh (+) 2407 2078 sh (+) 4321 1785 sh (+) 5742 1785 sh (\351) 3042 1543 sh (\353) 3042 2154 sh (\352) 3042 1912 sh (\371) 6632 1543 sh (\373) 6632 2154 sh (\372) 6632 1912 sh (-) 6862 1785 sh (+) 7916 1785 sh (=) 1100 2680 sh (=) 1100 3319 sh 576 ns (\345) 2890 652 sh 384 /Times-Roman f1 (12) 1918 324 sh (1) 2523 858 sh (3) 4188 565 sh (1) 5145 565 sh (12) 1990 1544 sh (27) 1439 2078 sh (27) 1950 2078 sh (1) 2667 2078 sh (124) 3192 1544 sh (5) 3864 1544 sh (9) 3622 2078 sh (170) 4613 1544 sh (5) 5285 1544 sh (9) 5043 2078 sh (83) 6056 1544 sh (9) 6233 2078 sh (3) 7147 1785 sh (27) 7459 1785 sh (1) 8176 1785 sh (6) 1405 2680 sh (757) 1693 2680 sh (2) 682 3319 sh (5) 1406 3319 sh (99) 1694 3319 sh 224 ns (2) 3641 152 sh (2) 4058 1372 sh (2) 5479 1372 sh (2) 6439 1372 sh (05) 293 3417 sh (2) 266 3147 sh 384 /Times-Roman f1 (\() 1743 858 sh (\)) 2693 858 sh (\() 4365 565 sh (\)) 5315 565 sh (\() 1815 2078 sh (\)) 2837 2078 sh (.) 3768 1544 sh (.) 5189 1544 sh (\() 7324 1785 sh (\)) 8346 1785 sh (.) 1597 2680 sh (\() 547 3319 sh (\)) 883 3319 sh (.) 1598 3319 sh 224 ns (.) 237 3417 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 12 3319 sh end MTsave restore dMATHAl H=12N(N+1)R i2 n i -3(N+1)   =1227(27+1)124.5 2 9+170.5 2 9+83 2 9[]-3(27+1)=6.757c .052 (2)=5.99 H=12N(N+1)R i2 n i -3(N+1)   =1227(27+1)124.5 2 9+170.5 2 9+83 2 9[]-3(27+1)=6.757c .052 (2)=5.99T<x<   F .  & _933233833KRbFp} p} Ole [ PIC ad\ TMETA ^ PSymbolD !w*wgwD  -!cTimes !w*wgw -!2PSymbolD !w*wgwD  -!="Times !w*wgw -!( ;TimesD !w*wgwD  -!O ?PSymbol !w*wgw -!- JTimesD !w*wgwD  -!E STimes !w*wgw -!) [TimesD !w*wgwD  -!2_Times !w*wgw -!EL-:dPSymbolD !w*wgwD  -!=hTimes !w*wgw -!( r!7 wPSymbolD !w*wgwD  -!- Times !w*wgw -!5 !) TimesD !w*wgwD -!2Times !w*wgw -!5rPSymbolD !!w*wgwD ! -!+Times !w*wgw -!3 PSymbolD "!w*wgwD " -!- Times !w*wgw -!5 PSymbolD #!w*wgwD # -!( !) Times !w*wgw -!2TimesD $!w*wgwD $ -!5PSymbol !w*wgw -!,PSymbolD %!w*wgwD % -!=-"Times !w*wgw -!1-+!.-1!60-4PSymbolD &!w*wgwD & -!c@Times !w*wgw -!.D !05D !2; TimesD '!w*wgwD ' -!(@!1@!)@PSymbol !w*wgw -!=@"TimesD (!w*wgwD ( -!3@+!.@1!84@4 & ' Fdxpr  F"F currentpoint ", Symbol .+c,PICT ce ObjInfof Equation Native  _933233928oiFp} p} Times (2 +=( ;()O) -) E)) (_2 (LE":*(h=( r()7)-) 5)) (2 (5"r$(+( 3)-) 5 ( ()) (2 (5"# (, (-"=) 1).)60(@c +.)05(; 2 + ()1)))=) 3).)84/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6464 div 2240 3 -1 roll exch div scale currentpoint translate 64 56 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1812 494 moveto 1359 0 rlineto stroke 3597 494 moveto 1198 0 rlineto stroke 5190 494 moveto 1176 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 538 593 sh (c) 12 2023 sh 224 /Times-Roman f1 (2) 792 421 sh (2) 3001 180 sh (2) 4625 180 sh (2) 6196 152 sh (05) 293 2121 sh (2) 266 1851 sh 384 ns (7) 3750 352 sh (5) 4303 352 sh (5) 4106 886 sh (3) 5331 352 sh (5) 5877 352 sh (5) 5688 886 sh (1) 1314 1384 sh (60) 1602 1384 sh (1) 651 2023 sh (3) 1340 2023 sh (84) 1628 2023 sh 384 /Symbol f1 (=) 1038 593 sh (-) 2307 352 sh (=) 3279 593 sh (-) 4014 352 sh (+) 4888 593 sh (-) 5588 352 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1190 /Symbol f3 (\() 5202 364 sh (\)) 6064 364 sh 384 /Symbol f1 (=) 1038 1384 sh (=) 1038 2023 sh 576 ns (\345) 1348 680 sh 384 /Times-Roman f1 (\() 1826 352 sh (\)) 2866 352 sh (\() 3611 352 sh (\)) 4490 352 sh (.) 1506 1384 sh (\() 547 2023 sh (\)) 821 2023 sh (.) 1532 2023 sh 224 ns (.) 237 2121 sh 384 /Times-Italic f1 (O) 1956 352 sh (E) 2614 352 sh (E) 2378 886 sh end MTsave restore dMATH# c 2 =(O-E) 2 E=(7-5) 2 5+3-5() 2 5   =1.60c .052 (1)=3.84 c 2 =(O-E) 2 E=(7-5) 2 5+3-5() 2 5   =1.60c .052 (1)=3.84T'  '    h .Ole  PIC hk TMETA  PICT jm x   & PSymbol !w*wgw -!cTimes1 !w*wgw1  -!FTimes !w*wgw -!2 PSymbol1 !w*wgw1  -!=#Times !w*wgw -!12 >Times1 !w*wgw1  -!Nk0Times !w*wgw -!(=Times1 !w*wgw1  -!kBPSymbol !w*wgw -!+JTimes1 !w*wgw1 -!1R!)W- . [PSymbol !w*wgw -!S^Times1 !w*wgw1 -!RfTimes !w*wgw -!ilTimes1 !w*wgw1 -!2 nPSymbol !w*wgw -!-uTimes1 !w*wgw1 -!3~Times !w*wgw -!NTimes1 !w*wgw1 -!(Times !w*wgw -!kPSymbol1 !w*wgw1  -!+Times !w*wgw -!1!)PSymbol1 !w*wgw1  -!=4#Times !w*wgw -!12,D!12>.!(>:!2>>!)(>D!2>MPSymbol1 !w*wgw1  -!+>UTimes !w*wgw -!1>]!)>b0.0f![4h!134lTimes1 !w*wgw1  -!2/xPSymbol !w*wgw -!+4Times1 !w*wgw1  -!174Times !w*wgw -!2/Times1 !w*wgw1  -!]4PSymbol !w*wgw -!-4Times1 !w*wgw1  -!34!(4!104!)(4!24PSymbol !w*wgw -!+4Times1 !w*wgw1  -!14!)4PSymbol !w*wgw -!=O#Times1 !w*wgw1  -!1O-!.O3!60O6PSymbol !w*wgw -!ccTimes1 !w*wgw1  -!.f !05f !2^ Times !w*wgw -!(c!1c!)cPSymbol1 !w*wgw1  -!=c#Times !w*wgw -!3c.!.c4!84c7 & ' xhdxpr1  h"h      !"#$%&'()*+,.123458:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghjmnopqstuvwxyz|}~ currentpoint ", Symbol .+c,Times +F( 2 + =( >12(0Nk) ()k)+)1))" .-(^S)R +i( n2 +-) 3)N) ()k) +)1))(4#=(,D12(>.12) ()2))() 2)+)1))"0.8(4h[)13 (/x2 ++)17 (/2 +])-) 3)()10) )()2)+)1))(O#=) 1).)60(cc +.)05(^ 2 + ()1)))=) 3).)84/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7168 div 3328 3 -1 roll exch div scale currentpoint translate 64 38 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1428 417 moveto 1457 0 rlineto stroke 1428 1540 moveto 1808 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 519 516 sh (c) 12 3146 sh 224 /Times-Italic f1 (F) 775 612 sh (i) 3421 612 sh 384 ns (Nk) 1473 824 sh (k) 2057 824 sh (R) 3201 516 sh (N) 4181 516 sh (k) 4601 516 sh 224 /Times-Roman f1 (2) 786 344 sh (2) 3457 344 sh (2) 3791 1467 sh (2) 4686 1467 sh (05) 306 3244 sh (2) 279 2974 sh 384 ns (12) 1948 261 sh (1) 2574 824 sh (3) 3973 516 sh (1) 5118 516 sh (12) 2124 1384 sh (12) 1418 1947 sh (2) 1939 1947 sh (2) 2402 1947 sh (1) 2925 1947 sh (13) 3395 1639 sh (17) 4283 1639 sh (3) 5313 1639 sh (10) 5594 1639 sh (2) 6249 1639 sh (1) 6772 1639 sh (1) 1386 2507 sh (60) 1674 2507 sh (1) 677 3146 sh (3) 1412 3146 sh (84) 1700 3146 sh 384 /Symbol f1 (=) 1087 516 sh (+) 2320 824 sh (S) 2974 516 sh (-) 3694 516 sh (+) 4864 516 sh (=) 1087 1639 sh (+) 2671 1947 sh (+) 4029 1639 sh (-) 5034 1639 sh (+) 6518 1639 sh (=) 1087 2507 sh (=) 1087 3146 sh 384 /Times-Roman f1 (\() 1917 824 sh (\)) 2744 824 sh (\() 4461 516 sh (\)) 5288 516 sh (\() 1804 1947 sh (\)\() 2140 1947 sh (\)) 3095 1947 sh ([) 3293 1639 sh (]) 4854 1639 sh (\() 5490 1639 sh (\)\() 5987 1639 sh (\)) 6942 1639 sh (.) 1578 2507 sh (\() 573 3146 sh (\)) 847 3146 sh (.) 1604 3146 sh 224 ns (.) 250 3244 sh end MTsave restore dJMATH>:e c F2 =12Nk(k+1)SR i2 -3N(k+1)=1212(2)(2+1)[13 2 +17 2 ]-3(10)(2+1)=1.60c .052 (1)=3.84FMicrosoft Equation Editor 2.0DNQE Equation.2CompObj-YObjInfoln/Equation Native 0Z_933237284qFp} p} > c F2 =12Nk(k+1)SR i2 -3N(k+1)=1212(2)(2+1)[13 2 +17 2 ]-3(10)(2+1)=1.60c .052 (1)=3.84T  Ole 6PIC ps7TMETA * PICT ru9 hdxpr1  h"h currentpoint ", Symbol .+c,Times +F( 2 +=( ?12(0Nk)()k) +)1))" /-(_S)R +i( m2 +-)3)N) ()k) +)1))(4$=(,A12(=/8)()3))()3)+)1))"0/1(4b[)22 (.r2 ++) 16 (.2   m  h .  & PSymbolo J!w*wgwo J -!cTimes !w*wgw -!FTimeso K!w*wgwo K -!2 PSymbol !w*wgw -!=$Timeso L!w*wgwo L -!12 ?Times !w*wgw -!Nk0Timeso M!w*wgwo M -!(>Times !w*wgw -!kBPSymbolo N!w*wgwo N -!+KTimes !w*wgw -!1S!)X- / \PSymbolo O!w*wgwo O -!S_Times !w*wgw -!RfTimeso P!w*wgwo P -!ilTimes !w*wgw -!2 mPSymbolo Q!w*wgwo Q -!-uTimes !w*wgw -!3}Timeso R!w*wgwo R -!NTimes !w*wgw -!(Timeso S!w*wgwo S -!kPSymbol !w*wgw -!+Timeso T!w*wgwo T -!1!)PSymbol !w*wgw -!=4$Timeso U!w*wgwo U -!12,A!8=/!(=5!3=9!)(=?!3=GPSymbol !w*wgw -!+=OTimeso V!w*wgwo V -!1=W!)=\0/0`![4b!224fTimes !w*wgw -!2.rPSymbolo W!w*wgwo W -!+4yTimes !w*wgw -!164Timeso X!w*wgwo X -!2.PSymbol !w*wgw -!+4Timeso Y!w*wgwo Y -!104Times !w*wgw -!2.Timeso Z!w*wgwo Z -!]4PSymbol !w*wgw -!-4Timeso [!w*wgwo [ -!34!(4!84!)(4!34PSymbol !w*wgw -!+4Timeso \!w*wgwo \ -!14!)4PSymbol !w*wgw -!=O$Timeso ]!w*wgwo ] -!9O.!.O4!00O7PSymbol !w*wgw -!ccTimeso ^!w*wgwo ^ -!.f !05f !2] Times !w*wgw -!(c!2c!)cPSymbolo _!w*wgwo _ -!=c$Times !w*wgw -!5c.!.c4!99c7 & ' ++)10 (.2 +])-) 3)()8))()3)+)1))(O$=) 9).)00(cc +.)05(] 2 + ()2)))=) 5).)99/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7680 div 3328 3 -1 roll exch div scale currentpoint translate 64 51 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1450 411 moveto 1469 0 rlineto stroke 1450 1521 moveto 1593 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) 581 510 sh (c) 12 3120 sh 224 /Times-Italic f1 (F) 824 606 sh (i) 3416 606 sh 384 ns (Nk) 1495 811 sh (k) 2079 811 sh (R) 3209 510 sh (N) 4175 510 sh (k) 4595 510 sh 224 /Times-Roman f1 (2) 835 338 sh (2) 3452 338 sh (2) 3607 1448 sh (2) 4495 1448 sh (2) 5386 1448 sh (05) 293 3218 sh (2) 266 2948 sh 384 ns (12) 1976 261 sh (1) 2608 811 sh (3) 3967 510 sh (1) 5124 510 sh (12) 2038 1371 sh (8) 1462 1921 sh (3) 1773 1921 sh (3) 2214 1921 sh (1) 2732 1921 sh (22) 3207 1620 sh (16) 4098 1620 sh (10) 4986 1620 sh (3) 6012 1620 sh (8) 6315 1620 sh (3) 6760 1620 sh (1) 7278 1620 sh (9) 1439 2481 sh (00) 1727 2481 sh (2) 682 3120 sh (5) 1438 3120 sh (99) 1726 3120 sh 384 /Symbol f1 (=) 1116 510 sh (+) 2348 811 sh (S) 2982 510 sh (-) 3682 510 sh (+) 4864 510 sh (=) 1116 1620 sh (+) 2472 1921 sh (+) 3838 1620 sh (+) 4726 1620 sh (-) 5727 1620 sh (+) 7018 1620 sh (=) 1116 2481 sh (=) 1116 3120 sh 384 /Times-Roman f1 (\() 1939 811 sh (\)) 2778 811 sh (\() 4455 510 sh (\)) 5294 510 sh (\() 1643 1921 sh (\)\() 1957 1921 sh (\)) 2902 1921 sh ([) 3074 1620 sh (]) 5541 1620 sh (\() 6189 1620 sh (\)\() 6503 1620 sh (\)) 7448 1620 sh (.) 1631 2481 sh (\() 547 3120 sh (\)) 883 3120 sh (.) 1630 3120 sh 224 ns (.) 237 3218 sh end MTsave restore dZMATHN<' c F2 =12Nk(k+1)SR i2 -3N(k+1)=128(3)(3+1)[22 2 +16 2 +10 2 ]-3(8)(3+1)=9.00c .052 (2)=5.99FMicrosoft Equation Editor 2.0DNQE Equation.2N c F2 =12Nk(k+1)SR i2 -3N(k+1)=128(3)(3+1)[22 2 +16 2 +10 2CompObjiYObjInfotvkEquation Native lj1TableS  ]-3(8)(3+1)=9.00c .052 (2)=5.99Oh+'0 $0<P ht    XSolutions Manual for Fundamental Statistics for 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