Let X1, . . . , Xn be i.i.d. with cdf F, and let Y1, . . . , Ym be i.i.d. with cdf G. Th...

Let X₁, . . . , X_n be i.i.d. with cdf F, and let Y₁, . . . , Y_m be i.i.d. with cdf G. The hypothesis to be tested is that F = G. Suppose for simplicity that m + n is even so that in the combined sample of X’s and Y ’s, (m + n)/2 observations are less than the median and (m + n)/2 are greater.

a. As a test statistic, consider T , the number of X’s less than the median of the combined sample. Show that T follows a hypergeometric distribution under the null hypothesis:

Explain how to form a rejection region for this test.

b. Show how to find a confidence interval for the difference between the median of F and the median of G under the shift model, G(x) = F(x −). (Hint:

Use the order statistics.)

c. Apply the results (a) and (b) to the data of Problem 21.

Reference

A study was done to compare the performances of engine bearings made of different compounds (McCool 1979). Ten bearings of each type were tested. The following table gives the times until failure (in units of millions of cycles):

a. Use normal theory to test the hypothesis that there is no difference between the two types of bearings.

b. Test the same hypothesis using a nonparametric method.

c. Which of the methods—that of part (a) or that of part (b)—do you think is better in this case?

d. Estimate π, the probability that a type I bearing will outlast a type II bearing.

e. Use the bootstrap to estimate the sampling distribution of and its standard error.

f. Use the bootstrap to find an approximate 90% confidence interval for π.