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  1. Generate 100 samples of size 7 from a normal population with mean 129 and standard deviation of 66.8. For each sample, perform a bootstrap hypothesis test of 𝜇𝑥 = 129.0 using the test statistics 𝑥̅-129 and using as the estimated null distribution

a. The empirical distribution

b. The empirical distribution translated to have mean 129.0

Compute the average ASL for each test, averaged over 100 simulations.

  1. According to the U.S. National Transportation Safety Board, the number of airline accidents by year from 1983 to 2006 were 23, 16, 21, 24, 34, 30, 28, 24, 26, 18, 23, 23, 36, 37, 49, 50, 51, 56, 46, 41, 54, 30, 40, and 31.

a. Find a 95% confidence interval on the mean using the standard Student ’ s t – distribution.

b. Find a 95% confidence interval on the mean using Efron ’ s percentile method.

c. Find a 95% confidence interval on the mean using the BCa method.

d. Find a 95% confidence interval on the mean using the percentile – t method

e. How do the intervals compare? Which intervals do you trust? What does this tell you about the benefits of parametric methods on small ( n < 30) samples and the problems of using bootstrap on such samples? What does it tell you about the percentile – t method compared with the other bootstrap methods, at least when a formula for the standard error is known?

  1. Dataset catsM contains a set of data on the heart weights and body weights of 97 male cats. We investigate the dependence of heart weight (g) on body weight (kg). The data set is available in the boot package.

Sample Solution

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