Statistical Inference: Confidence Intervals and Hypothesis Testing, Lab Reports of Probability and Statistics

Download Statistical Inference: Confidence Intervals and Hypothesis Testing and more Lab Reports Probability and Statistics in PDF only on Docsity! Lab 12. Confidence intervals and hypothesis testing www.nmt.edu/~olegm/382/labs/Lab12.pdf Note: the menus and other things you will read or type on the computer are in italics. Attach the printouts whenever needed. Confidence intervals (C.I.’s) and hypothesis tests are cornerstones of statistical inference. In this Lab, we will discuss C.I.’s for one-sample problems. Also, we will take a look at hypothesis tests, and a general discussion of the p-value. 1 C.I. for the mean The confidence interval is usually of the format point estimate± margin of error The actual calculation depends on the problem at hand, but Minitab can take care of it for you. In the menu Stat → Basic Statistics there is a battery of options for 1- and 2-sample tests. Central Limit Theorem tell us that the mean X for a sample of size n has approximately Normal distribution with the mean µ and variance σ2/n. For example, when σ is known, and n is large, the following is a (100)(1−α)%- C.I. for the mean µ X ± zα/2 σ√ n where zα/2 is a (1− α/2) quantile of the standard Normal (Z) distribution. The C.I. for the mean gives us a range of “plausible” values for µ. The in- terpretation of, say, 95% C.I. is that in 95% cases it will contain the “true” (unknown) population mean µ. Of course we could opt for higher confidence, say 99%, but we’ll pay the price with a wider interval. Another way to increase the precision of C.I. is, of course, increase the sample size n for your study. Problem 1 (a) In the file sample1.txt, based on the sample in C1, compute 95% and 99% C.I.’s for the mean. (Assume σ = 1.) Which one is wider and why? (b) Based on the sample in C2, compute 95% C.I. for the mean. Compare with the part (a) 95% C.I. Which one is wider and why? (c) Can you compute a 100% C.I.? Why or why not? 1 (d) Do a simulation study of generating 100 90% C.I.’s from a Normal pop- ulation, mean 0 and variance 1. First, obtain 100 samples (rows) of size 5. Then, use Row statistics to find 100 sample means and compute the 90% C.I.’s lower and upper limits using Calculator, assume σ = 1 and zα/2 = 1.645. How many of 90% C.I.’s out of a 100 would you expect to cover your true mean? Count how many of them among your samples contained the true mean 0. 1.1 Case of unknown σ With σ unknown, we would use the t-distribution confidence interval; however, when n is small we need to also take care that the underlying distribution is close to Normal. Problem 2 The data below are the survival times (in hours) of 72 guinea pigs after they were injected with a given dose of tubercule bacilli in a medical experiment. The data are from the article “Acquisition of resistance of guinea pigs in- jected with different doses of virulent tubercule bacilli,” by T. Bjerkedal in the American Journal of Hygiene, (1960), pp. 130-148. 43 45 53 56 56 57 58 66 67 73 74 79 80 80 81 81 81 82 83 83 84 88 89 91 91 92 .... (see file bacilli.txt) (a) Obtain a 95% t-C.I. for the mean survival time. (b) Check the normality of the data using Normal probability plot. (c) Take the log of survival times and find a 95% t-C.I. for mean log survival time. Check the normality of log survival times. (d) Translate your C.I. from part (c) back into unlogged time scale and compare with part (a). Which of the C.I.’s do you believe more? (e) Nonparametric alternative When normality is violated, we might use a nonparametric procedure, i.e. the one that does not rely on a particular distribution assumption for it to work. Compute a 95% Wilcoxon C.I. for the median survival time using Stat → Nonparametric → 1-sample Wilcoxon. Compare it to the one from the log transformation, part (d). 2