Statistical Analysis Homework: Comparing Two Groups and Hypothesis Testing - Prof. Peter H, Assignments of Statistics

Download Statistical Analysis Homework: Comparing Two Groups and Hypothesis Testing - Prof. Peter H and more Assignments Statistics in PDF only on Docsity! Stat 502 Homework 2 Assigned 10/9/08 Due 10/16/08 1. (Voles): Researchers studying 24 wild prairie voles are interested in the effects of food sup- plements on reproductive success. The researchers randomly assigned 12 voles to receive supplements (group B), with the remaining 12 voles needing to forage to obtain all of their food (group A). At the end of the mating season the number of pups for each of the 24 females was recorded. (a) Make a histogram and boxplots for each of the two groups. Comment on the differences. (b) Compute the means and medians of each group. Comment on the differences. (c) Plot the density of the appropriate t-distribution if one were to use the ordinary two- sample t-test to evaluate differences between the groups. Obtain the corresponding p- value. Write down the assumptions which validate the use of this p-value, and comment on whether or not they are met for these data. (d) Make a histogram of the randomization distribution of the t-statistic, and compute the corresponding p-value. Write down the assumptions which validate the use of this p- value, and comment on whether or not they are met for these data. 2. (Null distribution of p-values): Recall that a p-value is a function of the data, and so before the experiment is run it is a random variable. (a) Consider the one-sample t-test for evaluating evidence against H0 : µ = µ0. Derive the distribution of the p-value under the null hypothesis. Using the result, show that the type I error of a level-α test is α. (b) Now consider the hypothesis H0 : µA = µB. Via simulation, compute the null distri- bution of the p-value based on the two-sample t-test under each of the following six experimental scenarios. i. Y1,A, . . . , Y10,A ∼ i.i.d. normal(1,1), Y1,B, . . . , Y10,B ∼ i.i.d. normal(1,1) ii. Y1,A, . . . , Y10,A ∼ i.i.d. normal(1,1), Y1,B, . . . , Y10,B ∼ i.i.d. normal(1,3) iii. Y1,A, . . . , Y10,A ∼ i.i.d. Poisson(1), Y1,B, . . . , Y10,B ∼ i.i.d. Poisson(1) iv. Y1,A, . . . , Y10,A ∼ i.i.d. normal(1,1), Y1,B, . . . , Y10,B ∼ i.i.d. normal(3,1) v. Y1,A, . . . , Y10,A ∼ i.i.d. normal(1,1), Y1,B, . . . , Y10,B ∼ i.i.d. normal(3,3) vi. Y1,A, . . . , Y10,A ∼ i.i.d. Poisson(1), Y1,B, . . . , Y10,B ∼ i.i.d. Poisson(3) More specifically, for each of the above scenarios, • simulate 1000 (or more) runs of the 10-sample experiment; • compute the p-value for each of the 1000 runs; • plot the empirical distribution of the 1000 p-values (using, for example, a histogram) and compare to the distribution from part (a). 1