ࡱ> []Z} rbjbjVV 1<<\LI[[[[[ooo8totu(xxx=??????b?i[xxxxx?[[x^[[=x=yw5HGZoazP)0zp&5[5xxxxxxx??xxxxxxxxxxxxxxxx !: Math 217C, Fall 2010 Final Exam Information Our final exam is scheduled for 2:00 pm on Friday, 12-17-10, in CFA 107. If you need/want to take the exam earlier than this, the alternative exam time/place is 7:00 pm on Thursday, 12-16-10, in CFA 138. Bring your calculator and two 3-by-5 index cards of formulas etc. to the exam. I will provide copies of Table A and Table D, as well as the handout, "Commonly Used Statistics in Math 217." To prepare for the exam, you should review the following sections from your textbook and work the practice problems in this study guide. An answer key for the practice problems is posted on our class website. For additional practice, make use of the many exercises in your textbook. Daily practice is the most important factor for success in math! Key topics to review: Section 6.1: Introduction to Confidence Intervals for a Mean What is the purpose of a confidence interval? What is the exact meaning of the confidence level? What is the basic form of a confidence interval? How is the margin of error of a confidence interval affected by the confidence level? by the sample size? by the population standard deviation? What is the minimum sample size needed to achieve a specified margin of error? Know how and when to use the Z interval dialog on your calculator, and how to interpret the results (STAT > TESTS > Z Interval). See cautions p.426-427. Section 6.2: Introduction to Significance Testing for a Mean What is the purpose of a test of significance? What is the exact meaning of the P-value? Know how and when to use the z test dialog on your calculator, and how to interpret the results (STAT > TESTS > Z-Test). What should you conclude from a significance test? Section 6.3: Use and Abuse of Statistical Tests Under what circumstances are the Z procedures in chapter 6 valid and appropriate? Consider the context when choosing a level of significance. Note that .05 is not a magical or sacred cut-off for significance: P = .0501 is about as significant as P = .0499 and should yield the same final conclusion. Formal statistical inference cannot correct basic flaws in experimental design and data collection. For example, the margin of error in a confidence interval only takes into account the sampling variability due to random sampling; it does not correct for poor sampling design, poorly worded questions, etc. Statistical significance is different than practical significance (importance). In addition to a significance test, use a confidence interval to estimate the size of an effect. If you perform repeated testing and occasionally find significance (say, P < .05 about 5% of the time or less) then those tests probably show significance just due to sampling variability! We expect P to come out small now and then just due to random sampling error, even when the null hypothesis is true. This is not good evidence that the alternative hypothesis is actually true. Section 7.1: Inference for the Mean of a Population Standard error of the sample mean:  EMBED Equation.3 ; it estimates the standard deviation of the sampling distribution of the sample mean  EMBED Equation.3 . The t distributions: How do you determine the degrees of freedom? How do the t distributions compare with the standard normal? How do you use Table D to find critical values (t*) and P values? When is it correct to use the one-sample t confidence interval for a population mean? What is the margin of error? When do you use the Z interval instead of the t interval? The one-sample t test: How does it compare with the Z test from 6.2? When is it correct to use this procedure? When do you use the Z test instead of the t test? Know how and when to use the t interval dialog on your calculator, and how to interpret the results (STAT > TESTS > T Interval). Know how and when to use the t test dialog on your calculator, and how to interpret the results (STAT > TESTS > T-Test). How are the t procedures used to analyze data from matched pairs? Section 7.2: Comparing the Means of Two Populations What is the two-sample t statistic? What needs to be true about the two samples, and their populations, in order for this statistic to give good statistical results? What's the best way to find the "degrees of freedom" for a two-sample t statistic? Know how and when to use the two-sample t interval dialog on your calculator, and how to interpret the results (STAT > TESTS > 2-SampTInterval). Always choose "Pooled = No" unless there is good reason to believe the population standard deviations are equal. Know how and when to use the two-sample t test dialog on your calculator, and how to interpret the results (STAT > TESTS > 2-SampTTest). Always choose "Pooled = No" unless there is good reason to believe the population standard deviations are equal. Section 8.1: Inference for the Proportion of a Population Know how and when to use the 1-proportion Z interval dialog on your calculator, and how to interpret the results (STAT > TESTS > 1-Prop Z Int). Same for significance testing for a single proportion (STAT > TESTS > 1-Prop Z Test). Note: These procedures require a large sample size n. Know how and when to use the "plus four" ( EMBED Equation.3 ) methods for confidence intervals and significance testing for a single proportion, and how to interpret the results. Section 8.2: Comparing the Proportions of Two Populations Know how and when to use the 2-proportion Z interval dialog on your calculator, and how to interpret the results (STAT > TESTS > 2-Prop Z Int). Same for significance testing for two proportions (STAT > TESTS > 2-Prop Z Test). Note: These procedures require a large sample size n. Know how and when to use the "plus four" ( EMBED Equation.3 ) methods for confidence intervals and significance testing for two proportions, and how to interpret the results. The formulas are not in the Table of Common Statistics, but you can put them on your index card -- see the bottom of p.565. Practice problems Suppose that a random sample of 50 Hanover College students finds that 11 of the students in the sample are smokers. Which confidence interval procedure is appropriate for these data? Be specific. Give a 95% confidence interval to estimate the corresponding population proportion. Write a sentence to interpret your findings. The Registrar knows every current HC students GPA. He wants to know the mean current HC student GPA. Is it reasonable for him to use the GPA data to calculate a 95% confidence interval for the mean current HC student GPA? Explain. Which is better for detecting practical significance (in addition to statistical significance): a confidence interval, or a significance test? Explain. In a study of memory recall, 8 students from a large psychology class were selected at random and given 10 minutes to memorize a list of 20 nonsense words. Each was asked to list as many of the words as he or she could remember both 1 hour and 24 hours later, as shown in the following table. Subject #12345678After 1 hour14121871191615After 24 hours104146961212 Perform an appropriate test of significance and answer the question: Do these data provide convincing evidence that the mean number of words recalled after 1 hour will, in general, exceed the mean number of words recalled after 24 hours? (Hint: These are paired data; analyze the differences.) Write a clear, complete sentence to summarize your findings. What do you conclude? What assumption did you have to make about the population distribution of differences? McDonalds' happy meals come with either a "girl toy," currently a Hello Kitty toy, or a "boy toy," currently a Transformer. A random sample of 200 kindergarten students (100 boys and 100 girls) in Indiana finds that 83 percent of the girls prefer Hello Kitty toys to Transformers. 28 percent of the boys prefer Hello Kitty toys to Transformers. We are interested in estimating the difference in the two population proportions, p1 - p2, where p1 is the proportion of all female kindergarten students in Indiana who would prefer the Hello Kitty toy, and p2 is the proportion of all male kindergarten students in Indiana who would prefer the Hello Kitty toy. Which confidence interval procedure is appropriate for estimating p1 - p2? Be specific. Give a 95% confidence interval for p1 - p2. Write a sentence to interpret your findings. Based on the confidence interval, do we have convincing evidence that p1 > p2? Explain. Find the test statistic and P-value for testing whether p1 > p2. Write a sentence to interpret your findings. Suppose you are testing H0:  = 95 against Ha:   EMBED Equation.3  95 based on an SRS of 12 observations from a normal population. What values of the t statistic are statistically significant at the  = 0.01 level? At the  = 0.05 level? What is the exact meaning of the P-value found in a test of significance? State the null hypothesis H0 and the alternative hypothesis Ha for a significance test in the following situation: The diameter of a spindle in a small motor is supposed to be 8 mm. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target. H0 (in English and in symbols): Ha (in English and in symbols): A student reads that a 95% confidence interval for the mean SAT math score of California high school seniors is 452 to 470. Asked to explain the meaning of this interval, the student says, 95% of California high school seniors have SAT math scores between 452 and 470. Is the student correct? ____________ Justify your answer by discussing the meaning of the confidence level for a confidence interval. Chicken nuggets are a popular item in the HC dining room. Suppose that one day when chicken nuggets are served, a random sample of 12 students in the CC dining room reveals that 100% of the 12 find the CC chicken nuggets to be either "very good" or "excellent." Do these data give convincing evidence that among the population of all students who eat in the CC dining room (and are not vegetarian), more than 75% of them would find the CC chicken nuggets to be either "very good" or "excellent"? Exactly which significance test is appropriate here? Give the test statistic and P-value for testing whether the stated population proportion exceeds 0.75. Write a sentence to interpret your findings. Because sulfur compounds cause off-odors in wine, oenologists (wine experts) have determined the odor threshold, the lowest concentration of a compound that the human nose can detect. For example, the odor threshold for dimethyl sulfide (DMS) is given in the oenology literature as 25 micrograms per liter of wine (g/l). Untrained noses may be less sensitive (have a higher odor threshold). Here are the DMS odor thresholds for 10 beginning students of oenology. 31314336233432302024 Treating these data as an SRS of size 10 from an approximately normal population, carry out a significance test to determine whether the mean DMS odor threshold among all beginning oenology students is more than 25 g/l. Do piano lessons improve the spatial-temporal reasoning of preschool children? Neurobiological arguments suggest that this may be true. A study designed to test this hypothesis measured the spatial-temporal reasoning of 30 preschool children before and after six months of piano lessons. (The study also included children who took computer lessons, and a control group who continued their usual activities, but we are not concerned with those here.) The changes in the reasoning scores (after minus before) are as follows: 2 5 7 -2 2 7 4 1 0 7 4 3 4 9 4 5 2 9 6 0 6 -1 3 4 6 7 -2 7 -3 3 Find the sample mean for these data. _________ Find the sample standard deviation. _________ Find the standard error of the mean. __________ Calculate a 95% confidence interval for the population mean improvement in reasoning scores. Show your work clearly. A random sample of 20 female students and 20 male students at Franklin College finds that 3 of the women and 8 of the men are frequent binge drinkers. Do these data give strong evidence that the proportion of binge drinkers is different depending on the student's gender (for FC students)? Let p1 be the population proportion of all female FC students who are frequent binge drinkers, and let p2 be the population proportion of all male FC students who are binge drinkers. Which confidence interval procedure is appropriate for estimating p1 - p2? Be specific. Give a 95% confidence interval for p1 - p2. Write a sentence to interpret your findings. Based on the confidence interval, do we have convincing evidence that p1 `" p2? Explain. Find the test statistic and P-value for testing whether p1 `" p2. Write a sentence to interpret your findings. The one-sample t statistic for testing EMBED Equation.3 versus  EMBED Equation.3 from a sample of n = 22 observations from a normal population has the value t = -1.573. What are the degrees of freedom for this statistic? _____ What is the (approximate) P-value for this test? ____________________ A marine biologist has data on the lengths of 44 adult male great white sharks, which he is willing to treat as an SRS from the population of all adult male great white sharks. He uses a t test to see if the data give significant evidence that adult male great white sharks average more than 20 feet in length. After calculating t, he finds that the P value is P = .0023. What conclusion should he reach about great white sharks? Alternatively, suppose that he finds P = .2251. Now what conclusion should he reach about great white sharks? The placebo effect is particularly strong in patients with Parkinsons disease. To understand the workings of the placebo effect, scientists made chemical measurements at a key point in the brain when patients received a placebo that they thought was an active drug and also when no treatment was given. The same patients were measured both with and without the placebo, at different times. The statistician will analyze the data using matched pairs, so she analyzes the differences (placebo minus no treatment). She wants to set up the hypotheses to test whether there is significant evidence of a difference between placebo and no treatment. State the appropriate hypotheses. Does bread lose its vitamins when stored? Two loaves of bread were prepared with flour that was fortified with a fixed amount of vitamins. After baking, the vitamin C content of the two loaves was measured. The loaves were then stored for three days and the vitamin C content was measured again. The units are milligrams per hundred grams of flour (mg/100 g). Here are the data: loaf 1loaf 2Immediately after baking47.6249.79Three days after baking21.2522.34When bread is stored, does it lose vitamin C? Perform an appropriate t test for these data. Be sure to state any assumptions you need about the populations, your hypotheses, the test statistic with degrees of freedom, and the P-value. State a conclusion in a clear English sentence. Use the sample data to give a 90% confidence interval for the amount of vitamin C lost on average when bread is stored for three days. Statisticians prefer large samples. Describe briefly the likely effect of increasing the sample size (or the number of subjects in an experiment) on each of the following: The width of a 95% confidence interval. The P-value of a significance test, when the null hypothesis is false. The variability of the sampling distribution of a sample statistic such as  EMBED Equation.3 . What is the purpose of a test of significance? Fill in the blanks. The t distributions are symmetric about ___________ (a number). The t-distributions are ___________-shaped, but have thicker tails than a standard normal (z) distribution. As the degrees of freedom increase, the t distribution approaches the ____________________ distribution. To find the degrees of freedom, use d.f. = __________ (formula). This tells you which row of Table D is appropriate. To find the standard error of the mean for data from an SRS of size n, use SE = ____________ (formula). The number of pups in wolf dens of the southwestern United States is recorded below for 16 wolf dens. (Source: The Wolf in the Southwest: The Making of an Endangered Species, edited by D. E. Brown, University of Arizona Press.) 5875343958565647 Find the sample mean: __________ Find the sample standard deviation: ____________ Find the standard error of the mean: ___________ Find a 90% confidence interval for the population mean, and write your conclusion in a clear, detailed sentence. Let represent the population mean number of wolf pups per den in the southwestern United States. Carry out a significance test to determine whether the sample data give convincing evidence that is more than 5. Repeat part (e) but determine whether the sample data give convincing evidence that is less than 7. Tree-ring dating at archaeological excavation sites is used in conjunction with other chronologic evidence to estimate occupation dates of prehistoric Indian dwellings in the southwestern United States. It is thought that Burnt Mesa Pueblo was occupied around 1300 A.D. The following data give tree-ring dates (A.D.) from adjacent archaeological sites: 1189126712681275127512711272131613171230 Assuming that these 10 values are an SRS from a normal population, do the data provide convincing evidence that the population mean of tree-ring dates in the area is different from 1300 A.D.? Carry out the appropriate significance test and state your conclusion in a clear, detailed sentence. Also give a 95% confidence interval to estimate the population mean. Which of the following errors (indicate yes or no for each) are accounted for by the margin of error in a confidence interval? ________ error due to voluntary response survey ________ error due to random variation in choosing an SRS ________ error due to poorly calibrated measuring instruments ________ error due to non-response in a sample survey A school administrator needs to estimate the mean Degree of Reading Power (DRP) score for all third-graders in the district. If the population standard deviation of DRP scores is estimated equal 11 (over all third-graders in the district), find the minimum sample size needed to produce a 95% confidence interval for the mean DRP score with margin of error m = 2.   +,-M^_`tu   - . ? 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One group contained breast-fed infants, while the children in another group were fed a standard baby formula without any iron supplements. Here are summary results on blood hemoglobin levels at 12 months of age: Groupn EMBED Equation.3 sBreast-fed2313.31.7Formula1912.41.8 Is there significant evidence that the mean hemoglobin level is higher among breast-fed babies? State the hypotheses, find the appropriate test statistic, find the P-value, and state the conclusion. Give a 95% confidence interval based on the given statistics, and interpret the interval in a clear sentence. State the assumptions that your procedures in (a) and (b) require in order to be valid.      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