SUPPLEMENTARY EXERCISES for ... - WH Freeman

Chapter 45S6.15 Use the result of Exercise S6.15 to give the probabilities of Type I and Type II errors forthe test discussed there. Take the alternative hypothesis to be µ = 295.S6.16 Use the result of Exercise S6.14 to give the probability of a Type I error and the probabilityof a Type II error for the test in that exercise when the alternative is µ = 460.ChapterS6.17 A study compares two groups of mothers with young children who were on welfare twoyears ago. One group attended a voluntary training program offered free of charge at a localvocational school and advertised in the local news media. The other group did not choose toattend the training program. The study finds a significant difference (P < 0.01) between theproportions of the mothers in the two groups who are still on welfare. The difference is notonly significant but quite large. The report says that with 95% confidence the percent of thenonattending group still on welfare is 21% ± 4% higher than that of the group who attended theprogram. You are on the staff of a member of Congress who is interested in the plight of welfaremothers and who asks you about the report.(a) Explain briefly and in nontechnical language what “a significant difference (P < 0.01)”means.(b) Explain clearly and briefly what “95% confidence” means.(c) Is this study good evidence that requiring job training of all welfare mothers would greatlyreduce the percent who remain on welfare for several years?S6.18 Use a computer to generate n = 5 observations from a normal distribution with mean20 and standard deviation 5—N(20, 5). Find the 95% confidence interval for µ. Repeat thisprocess 100 times and then count the number of times that the confidence interval includes thevalue µ = 20. Explain your results.S6.19 Use a computer to generate n = 5 observations from a normal distribution with mean20 and standard deviation 5—N(20, 5). Test the null hypothesis that µ = 20 using a two-sidedsignificance test. Repeat this process 100 times and then count the number of times that youreject H 0 . Explain your results.S6.20 Use the same procedure for generating data as in the previous exercise. Now test the nullhypothesis that µ = 22.5. Explain your results.S6.21 Figure 6.2 demonstrates the behavior of a confidence interval in repeated sampling byshowing the results of 25 samples from the same population. Now you will do a similar demonstration.Suppose that (unknown to the researcher) the mean SAT-M score of all California highschool seniors is µ = 460, and that the standard deviation is known to be σ = 100. The scoresvary normally.(a) Simulate the drawing of 25 SRSs of size n = 100 from this population.(b) The 95% confidence interval for the population mean µ has the form x ± m. What is themargin of error m? (Remember that we know σ = 100.)(c) Use your software to calculate the 95% confidence interval for µ when σ = 100 for each of