ࡱ> `'` dubjbjLULU g.?.?ll0dddd8LD\$h@ܱܱܱ@ddUrrrܱdRrܱrr "F j޳V<k0 <4xFFZdr@@jܱܱܱܱ$jp pdddddd Statistics Review Chapters 1-2 In the paper Reproduction in Laboratory colonies of Bank Vole, the authors presented the results of a study of litter size. (A vole is a small rodent with a stout body, blunt nose, and short ears.) As each new litter was born, the number of babies was recorded, and the accompanying results were obtained. 1 4 4 5 5 6 6 7 7 8 2 4 5 5 5 6 6 7 7 8 2 4 5 5 6 6 6 7 7 8 3 4 5 5 6 6 6 7 8 8 3 4 5 5 6 6 7 7 8 9 3 4 5 5 6 6 7 7 8 9 3 4 5 5 6 6 7 7 8 9 3 4 5 5 6 6 7 7 8 10 3 4 5 5 6 6 7 7 8 10 4 4 5 5 6 6 7 7 8 11 The authors also kept track of the color of the first born in each litter. (B = brown, G = gray, W = white, and T = tan) B B T W T G G G B B W B W B T T G B T B B T B B B G W B B G G G G B B T B W T T B T B B T W W B G B B B B G T B B T T G G B B B B G W G T G B B B B G G T T W G G W T G T B B G B B B G T W B G T W G W Which variable is categorical, and which variable is quantitative? Color is categorical; litter size is quantitative Make a bar chart of the colors. Vole Color  EMBED Excel.Chart.8 \s  Make a histogram of the litter sizes.   Make a dotplot of the litter sizes. Are there any outliers in the histogram or dotplot? No, there are no outliers. Describe the shape of the histogram (symmetric or skewed). approximately symmetric Find the mean of the litter sizes. Is the mean resistant to outliers?  EMBED Equation.DSMT4  No, the mean is NOT resistant to outliers. Find the median of the litter sizes. Is the median resistant to outliers? M = 6 Yes, the median is resistant to outliers. Find the range of the litter sizes. 10 Find the 5-number summary of the litter sizes. min = 1 Q1 = 5 median = 6 Q3 = 7 max = 11 What is the interquartile range? 2 Make a boxplot of the litter sizes.   Find the standard deviation of the litter sizes. Is standard deviation resistant to outliers? sx = 1.8127 No, standard deviation is not resistant to outliers. What is the area under a density curve? 1 The (mean or median) of a density curve is the equal-areas point, the point that divides the area under the curve in half. median The (mean or median) of a density curve is the balance point, at which the curve would balance if made of solid material. mean If a density curve is skewed to the right, the (mean or median) will be further to the right than the (mean or median). mean What is the difference between x-bar and (? X-bar is the sample mean and ( is the population mean What is the difference between s and (? s is the standard deviation of the sample and ( is the standard deviation of the population How do you find the inflection points on a normal curve? They are located one standard deviation on either side of the mean,  EMBED Equation.DSMT4  Sketch the graph of N(266, 16), the distribution of pregnancy length from conception to birth for humans.  218 234 250 266 282 298 314 What is the 68-95-99.7 rule? About 68% of the observations will fall within one standard deviation of the mean. About 95% of the observations will fall within two standard deviations of the mean. About 99.7% of the observations will fall within three standard deviations of the mean. Using the empirical rule (the 68-95-99.7 rule), find the length of the longest 16% of all pregnancies. Sketch and shade a normal curve for this situation.  The longest 16% of all pregnancies are ( 282 days 218 234 250 266 282 298 314 Find the length of the middle 99.7% of all pregnancies. between 218 and 314 days Find the length of the shortest 2.5% of all pregnancies. ( 234 days What percentile rank is a pregnancy of 218 days? 0.15th percentile What percentile rank is a pregnancy of 298 days? 97.5th percentile What is the percentile of a pregnancy of 266 days? 50th percentile What z-score does a pregnancy of 257 days have? -0.5625 What percent of humans have a pregnancy lasting less than 257 days? Normalcdf (-10E99, 257, 266, 16) = 0.2869 What percent of humans have a pregnancy lasting longer than 280 days? Normalcdf (280, 10E99, 266, 16) = 0.1908 What percent of humans have a pregnancy lasting between 260 and 270 days? Normalcdf (260, 270, 266, 16) = 0.2449 Would you say pregnancy length is a continuous or discrete variable? Justify. continuous, since the possible number of days can be any value in a given interval How long would a pregnancy have to last to be in the longest 10% of all pregnancies? invNorm (0.9, 266, 16) = 286.5 days How short would a pregnancy be to be in the shortest 25% of all pregnancies? invNorm (0.25, 266, 16) = 255.2 days How long would a pregnancy be to be in the middle 20% of all pregnancies? Between invNorm (0.6, 266, 16) and invNorm (0.4, 266, 16) Between 261.9 and 270.1 days Does the vole information from the beginning of this review seem to be normal? Justify by checking actual percentages within 1, 2, and 3 standard deviations of the mean.  EMBED Equation.DSMT4  There are 62 out of 100 observations between 4.1 and 7.7  EMBED Equation.DSMT4  There are 94 out of 100 observations between 2.2 and 9.5  EMBED Equation.DSMT4  There are 100 out of 100 observations between 0.4 and 11.3 Yes, the distribution of vole litter size is approximately normal. Make a back-to-back split stemplot of the following data: Reading Scores 4th Graders 12 15 18 20 20 22 25 26 28 29 31 32 35 35 35 36 37 39 40 42 7th Graders 1 12 15 18 18 20 23 23 24 25 27 28 30 30 31 33 33 33 35 36 4th graders 7th graders0102128 515 8 8 2 0 020 3 3 49 8 6 525 7 82 130 0 1 3 3 39 7 6 5 5 535 62 044 Key 1|2 = 12 Make a comparison between 4th grade and 7th grade reading scores based on your stemplot. The distribution between 4th grade and 7th grade reading scores are similar. They are both slightly skewed left. The range is lower for the 4th grade scores and both the minimum and maximum values are higher for the 4th grade scores. The 4th grade scores peak in the upper 30s, while the 7th grade scores peak in the lower 30s. What is the mode of each set of scores? The mode for 4th grade scores is 35; the mode for 7th grade scores is 33. Is the score of 1 for one of the 7th graders an outlier? Test using the 1.5 IQR rule. 1.5(IQR) = 1.5(13) = 19.5 19 19.5 = -0.5 No, 1 is not an outlier. What is the difference between a modified boxplot and a regular boxplot? Why is a modified boxplot usually considered better? A modified boxplot is usually better because it shows all outliers. Statistics Review Chapter 3 Graph the following hot dog data: SodiumCalories149108350130345132360135360138375140380144390145400150415163400167420172450176500180505184500195515200 What is the response variable? calories What is the explanatory variable? sodium What is the direction of this scatterplot? (positive, negative) positive What is the form of this scatterplot? (linear, exponential) linear What is the strength of this scatterplot? (strong, weak) strong Are there outliers? (Outliers in a scatterplot have large residuals.) Yes, (108, 149) is an outlier. If there are outliers, are they influential? No, because it doesnt significantly change r or LSRL Calculate the correlation. r = 0.9195 Calculate the correlation without the point (108, 149). r = 0.9587 What two things does correlation tell us about a scatterplot? the strength and direction of a linear relationship If I change the units on sodium to grams instead of milligrams, what happens to the correlation? it remains the same since r is a standardized value What is the highest correlation possible? 1 or -1 What is the lowest correlation possible? 0 Correlation only applies to what type(s) of relationship(s)? linear Is correlation resistant to outliers? no, it is not resistant to outliers Does a high correlation indicate a strong cause-effect relationship? no, correlation does not necessarily imply causation Sketch a scatterplot with a correlation of about 0.8.    Sketch a scatterplot with a correlation of about 0.5.    Find the least-squares regression line (LSRL) for the calories-sodium data. y = 47.39 + 0.272X What is the slope of this line, and what does it tell you in this context? As the amount of sodium increases by 1 mg, the calories increase by 0.272 Predict the amount of calories in a hot dog with 155 mg. of sodium . 89.55 calories Predict the amount of calories in a hot dog with 600 mg. of sodium. 208.59 calories Why is the prediction in problem 64 acceptable but the prediction in problem 65 not? 155 mg is within the domain of the data; 600 mg is not Find the error in prediction (residual) for a hot dog with 505 mg of sodium. 87.65 The point (x-bar, y-bar) is always on the LSRL. Find this point, and verify that it is on your scatterplot. (x-bar, y-bar) = (156.4118, 400.8235) -85.4072 + 3.1087(156.4118) = 400.8301 Find the standard deviation of the calories. 25.6395 Find the standard deviation of the sodium. 86.6799 Find the coefficient of determination for this data. r2 = 0.8455 What does r2 tell you about this data? Approximately 84.55% of the variation in calories can be explained by the linear relationship between calories and sodium. How can you use a residual plot to tell if a line is a good model for data? The residuals should be randomly scattered and relatively close to zero. Statistics Review Chapters 4-5 If you know a scatterplot has a curved shape, how can you decide whether to use a power model or an exponential model to fit data? If (x, logy) linearized the data, an exponential model is appropriate. If (logx, logy) linearizes the data, a power model is appropriate. You can also look at the ratios between consecutive values of the response variable. Graph the following data: Time (days) Mice 6 19 60 195 597 Perform the appropriate logarithmic transformation (power or exponential) on the above data to get an equation. log y = 0.779 + 0.017X Make a residual plot to support your choice for problem 76.    Graph the following data: Diameter (inches) Cost (dollars) 3.50 8.00 14.50 22.50 39.50 Perform the appropriate logarithmic transformation (power or exponential) on the above data to get an equation. logy = -1.02 + 2.02(log x) Make a residual plot to support your choice for problem 79.    What is the correlation for the equation you found in problem 79? r = 0.9999 What is extrapolation, and why shouldnt we trust predictions using extrapolation? Extrapolation is making a prediction outside the domain of the data. It is not reliable. What is interpolation? Interpolation is making a prediction within the domain of the data. What is a lurking variable? A lurking variable is a variable that may influence the value of the variables in a study, although it is not part of the study. Why should we avoid using averaged data for regression and correlation? Averaged data has less variability, which results in a higher correlation. What is causation? Give an example. Changes in the explanatory variable cause changes in the response variable. Example: the amount of time since a pie was removed from the oven and the temperature of the pie. What is common response? Give an example. Changes in the explanatory do not cause changes in the response variable; a lurking variable does cause changes in the response variable. Example: ice cream sales at Virginia Beach and the number of drownings at Virginia Beach. What is confounding? Give an example. Changes in the explanatory cause changes in the response variable, but a lurking variable also causes changes in the response variable. Example: smoking during pregnancy may cause low birth weight, but there are other lurking variables such as poor nutrition that may also cause low birth weight. Why is a two-way table called a two-way table? There are two variables. Use this table for questions 90-95: Smoking StatusEducationNever smokedSmoked, but quitSmokesTOTALDid not complete high school8219113214Completed high school97251032251 to 3 years of college9249592004 or more years of college866337186TOTAL357156312825 Fill in the totals for this table. What percent of these people smoke? 312/825 = 37.82% What percent of never-smokers completed high school? 97/357 = 27.17% What percent of those with 4 or more years of college have quit smoking? 63/186 = 33.87% What percent of smokers did not finish high school? 113/312 = 36.22% What conclusion can be drawn about smoking and education from this table? The more education a person has completed, the less likely they are to smoke: 53% of those who did not complete high school smoke, 45% of those who completed high school smoke, 30% of those with 1 to 3 years of college smoke, and 20% of those with 4 or more years of college smoke. What is Simpsons Paradox? When data from several groups are combined to form a single group, the association may be reversed. What is the difference between an observational study and an experiment? In an experiment, a treatment is imposed on the units or subjects. What is a voluntary response sample? The subjects select themselves to be in the sample. How are a population and a sample related but different? A sample is a part of the population. Why is convenience sampling biased? A convenience sample does not accurately represent the population; some groups are inevitably underrepresented. SRS stands for what kind of sample? Name and define. In a simple random sample, each member of the population is equally likely to be selected, and each possible sample is equally likely to be selected. Discuss how to choose a SRS of 4 towns from this list: Allendale Bangor Chelsea Detour Edmonton Fennville Gratiot Hillsdale Ionia Joliet Kentwood Ludington Assign each town a number 1-12. Randomly select four numbers using randInt(1, 12, 4). You could also label each with a number from 01-12 and look on the random number table for the first four numbers in that range. What is a stratified random sample? The population is divided into groups, and may be further divided into subgroups. A SRS is chosen from each subgroup. What is a cluster sample? The population is divided into groups, and may be further divided into subgroups. All members of one or more subgroups are chosen. What is undercoverage? One or more groups with similar characteristics do not have a chance to be chosen for the sample. What is nonresponse? A subject chosen to be in the sample does not respond or refused to participate. What is response bias? A subject is not truthful or is in some way influenced to respond differently than they normally would. Why is the wording of questions important? Give an example. Wording containing bias may influence answers. Example: Prohibiting children from praying in school is a violation of the Bill of Rights. Do you think children should be allowed to pray in school? This question contains bias, since it only presents one side of the issue. How are experimental units and subjects similar but different? Subjects are human. Explanatory variables in experiments are often called _____. Factors If I test a drug at 100 mg, 200 mg, and 300 mg, I am testing one variable at three _____. Levels What is the placebo effect? Some individuals respond to any form of treatment, regardless of whether it is a real or fake treatment. What is the purpose of a control group? To reduce or eliminate the effects of lurking variables. What are the two types of matched pairs used in experiments? Either each unit/subject received both treatments, or one of each pair of units/subjects receives treatment A and the other receives treatment B. What are the three principles of experimental design? I. Control the effects of lurking variables by comparing several treatments. II. Randomly assign subjects to treatment groups. III. Replicate the experiment on many subjects to reduce chance variation. What does double-blind mean, and why would we want an experiment to be double-blind? In a double-blind experiment, neither the subjects nor the people recording the observations know which subject received which treatment. This reduces bias. What is block design? A block design divides the sample into groups of similar characteristics to reduce the effects of lurking variables. Within each group, units/subjects are randomly assigned to each of the treatment groups. I want to test the effects of aerobic exercise on resting heart rate. I want to test two different levels of exercise, 30 minutes 3 times per week and 30 minutes 5 times per week. I have a group of 20 people to test, 10 men and 10 women. I will take heart rates before and after the experiment. Draw a chart for this experimental design.  randomTreatment A 5 men Compare10 menallocationTreatment B 5 menHeart Rates randomTreatment A 5 women Compare10 womenallocationTreatment B 5 womenHeart Rates  What are the five steps of a simulation? I. State the problem II. State the assumptions III. Assign digits to represent outcomes IV. Simulate many repetitions V. State your conclusions Design and perform a simulation of how many children a couple must have to get two sons. (A simulation involves many trials. For this simulation, perform 10 trials.) Assign 0 to girl and 1 to boy. Use the command randInt(0, 1). For one trial, press enter until you have two 1s. Count the number of tries to get two 1s. Record in table. Statistics Review Chapters 6-7 What is independence? Two events are independent if knowing that one occurs does not change the probability that the other occurs. P(A|B) = P(A) and P(B|A) = P(B). You are going to flip a coin three times. What is the sample space for each flip? S = {H T} You are going to flip a coin three times and note how many heads and tails you get. What is the sample space? S = { 0 1 2 3 } You are going to flip a coin three times and note what you get on each flip. What is the sample space? S = { HHH HHT HTH HTT THH THT THT TTT } Make a tree diagram for the three flips. There are three ways I can drive from Fremont to Grand Rapids and four ways I can drive from Grand Rapids to my home. How many different ways can I drive from Fremont to my home through Grand Rapids? 12 How many different four-digit numbers can you make if digits can repeat? 104 = 10,000 How many different four-digit numbers can you make without repeating digits? 10*9*8*7 = 5,040 What is an event in probability? An outcome or a set of outcomes from a sample space Any probability is a number between (and including) __0__ and __1__. All possible outcomes together must have probability of __1__. If S is the sample space, P(S) = __1__. What are complements? Give an example and draw a Venn diagram. If A is the event that something occurs, then A complement is the event that it does not occur. Example: rolling a die and landing on an even number is the complement of rolling a die and landing on an odd number.  SHAPE \* MERGEFORMAT   SHAPE \* MERGEFORMAT  What are disjoint events? Give two examples and draw a Venn diagram. Disjoint events have no outcomes in common. Knowing that one event occurs imply that the other event will not occur.  SHAPE \* MERGEFORMAT  Use the following chart for questions 135-137: M&M ColorBrownRedYellowGreenOrangeBlueProbability0.30.20.20.10.1? What is the probability that an M & M is blue? 0.1 What is the probability that an M & M is red or green? 0.3 What is the probability that an M & M is yellow and orange? 0 Bre can beat Erica in tennis 9% of the time. Erica can swim faster than Bre 8% of the time. What is the probability that Bre would beat Erica in a tennis match and in a swimming race? (0.09)(0.92) = 0.0828 What assumption are you making in problem 138? Do you think this assumption is valid? Independence Using two dice, what is the probability that you would roll a sum of seven or eleven? 0.2222 Using two dice, what is the probability that you would roll doubles? 0.1667 Using two dice, what is the probability that you would roll a sum of 7 or 11 on the first roll and doubles on the second roll? 0.0370 What assumption are you making in problem 142? Do you think this assumption is valid? Independence. Yes, because what you get on the first roll does not change the probability of what you get on the second roll. Using two dice, what is the probability that you would roll a sum of 7 or 11 that is also doubles? 0 What is the union of two events? The event that either one or both occurs. What is an intersection of two events? The event that both occur. How can we test independence? If P(A|B) = P(A) then A and B are independent. Perform an independence test on the smoking/education chart from problem 90 to show that smoking status and education are not independent. Let A = smokes and B = 4 or more years of college P(A|B) = 0.1989 P(A) = 0.3782 Make a Venn diagram for the following situation: 45% of kids like Barney 25% of kids like Blue 55% of kids like Pooh 15% of kids like Blue and Pooh 25% of kids like Barney and Pooh 5% of kids Barney, Blue, and Pooh 5% of kids like Blue but not Barney or Pooh  Barney 0.20 0.15 0.05 Blue 0.05 0.20 0.05 0.10 0.20 Pooh A dartboard has a circle with a 20-inch diameter drawn inside a 2-foot square. What is the probability that a dart lands inside the circle given that it at least lands inside the square? (Assume a random trial here.) 0.5454 For problems 151-154 consider the process of a drawing a card from a standard deck and replacing it. Let A be drawing a heart, B be drawing a king, and C be drawing a spade. Are the events A and B disjoint? Explain. No, the king of hearts is a member of A and B. Are the events A and B independent? Explain. Yes, whether a king is drawn or not, the probability of getting a heart remains 0.25 Are the events A and C disjoint? Explain. No, no card can be both a heart and a spade. Are the events A and C independent? Explain. No, knowing the card is a heart changes the probability that it will be a spade. What does the symbol ( mean? Union means or What does the symbol ( mean? Intersection means and Give an example of a discrete random variable. The number of students absent per week. Give an example of a continuous random variable. The height of students in a class. Make a probability histogram of the following grades on a four-point scale: Grade 0 1 2 3 4Probability0.050.280.190.320.16   Using problem 159, what is P(X > 2)? 0.48 Using problem 159, what is P(X > 2)? 0.67 What is a uniform distribution? Draw a picture. A uniform distribution has constant height.  In a uniform distribution with 0 < X < 1, what is P(0.2 < X < 0.6)? 0.4 In a uniform distribution with 0 < X < 1, what is P(0.2 ( X ( 0.6)? 0.4 How do your answers to problems 160, 161, 163, and 164 demonstrate a difference between continuous and discrete random variables? For a discrete random variable, the probability that X = k, where k is a constant, could be any number between 0 and 1, inclusive. For a continuous random variable, the probability that X = k is equal to 0. Normal distributions are (continuous or discrete). continuous Expected value is another name for _____. mean Find the expected value of the grades in problem 159. 2.26 Find the variance of the grades in problem 159. 1.3724 Find the standard deviation of the grades in problem 159. 1.1715 What is the law of large numbers? As the number of observations increases, the sample mean x-bar approaches the population mean ( and the expected value of X approaches the population mean (. If I sell an average of 5 books per day and 7 CDs per day, what is the average number of items I sell per day? 12 If I charge $2 per book and $1.50 per CD in problem 192, what is my average amount of income per day? $20.50 Before you can use the rules for variances you must make sure the variables are _____. independent For problems 175-183, use the following situation: For Test 1, the class average was 80 with a standard deviation of 10. For Test 2, the class average was 70 with a standard deviation of 12. What is the average for the two tests added together? 150 What is the standard deviation for the two tests added together? 15.6205 What is the difference in the test averages? 10 What is the standard deviation for the difference in the test averages? 15.6205 If I cut the test scores on Test 2 in half and add 50, what is the new average? 85 What is the new standard deviation for Test 2 in problem 179? 6 If I add 7 points to every Test 1, what is the new standard deviation? 10 If I multiply every Test 1 by 2 and subtract 80, what is the new mean? 80 If I multiply every Test 1 by 2 and subtract 80, what is the new standard deviation? 20 Statistics Review Chapter 8 and Miscellaneous What are the four conditions of a binomial distribution? I. Each outcome is either considered success or failure II. There is a fixed number of n observations III. The n observations are independent IV. The probability of success p is the same for each observation What are the four conditions of a geometric distribution? I. Each outcome is either considered success or failure II. The variable of interest is the number trials required to obtain the first success III. The n observations are independent IV. The probability of success p is the same for each observation Use the following situation for questions 186-200: The probability that a child born to a certain set of parents will have blood type AB is 25%. The parents have four children. X is the number of those children with blood type AB. Is this binomial or geometric? binomial Using the situation in problem 186, find P(X = 2). Binompdf(4, 0.25, 2) = 0.2109 Using the situation in problem 186, find P(X < 3). Binomcdf(4, 0.25, 2) = 0.9492 Using the situation in problem 186, find P(X > 1). 1 Binompdf(4, 0.25, 0) = 0.6836 Using the situation in problem 186, find P(1 < X < 3). Binompdf(4, 0.25, 1) + Binompdf(4, 0.25, 2) + Binompdf(4, 0.25, 3) = 0.6797 Using the situation in problem 186, find P(2 < X < 4). Binompdf(4, 0.25, 3) = 0.0469 What is the mean of the situation in problem 186? 1 What is the standard deviation of the situation in problem 186? 0.8660 A set of parents continue having children until they have a child with type AB blood. X is the number of children they have to give birth to in order to have one child with type AB blood. Is this binomial or geometric? geometric Using the situation in problem 194, find P(X = 1). geometpdf(0.25, 1) = 0.25 Using the situation in problem 194, find P(X < 2). geometcdf(0.25, 2) = 0.4375 Using the situation in problem 194, find P(X > 5). 1 geometcdf(0.25, 5) = 0.2373 Using the situation in problem 194, find P(2 < X < 4). geometpdf(0.25, 2) + geometpdf(0.25, 3) = 0.3281 Using the situation in problem 194, find P(2 < X < 5). geometpdf(0.25, 3) + geometpdf(0.25, 4) + geometpdf(0.25, 5) = 0.3252 What is the mean of the situation in problem 194? 4     Answer Key A.P. 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