39. The sign test is a very simple procedure for testing hypotheses about a population median assuming only that the underlying distribution is continuous.

To illustrate, consider the following sample of 20 observations on component lifetime (hr):

1.7 3.3 5.1 6.9 12.6 14.4 16.4 24.6 26.0 26.5 32.1 37.4 40.1 40.5 41.5 72.4 80.1 86.4 87.5 100.2 We wish to test H0: versus Ha: .

The test statistic is Y  the number of observations that exceed 25.

a. Consider rejecting H0 if Y  15. What is the value of a (the probability of a type I error)

for this test? Hint: Think of a success as a lifetime that exceeds 25.0. Then Y is the number of successes in the sample. What kind of a distribution does Y have when ?

b. What rejection region of the form Yc speci es a test with a signi cance level as close to .05 as possible? Use this region to carry out the test for the given data.
Note: The test statistic is the number of differences Xi  25.0 that have positive signs, hence the name sign test.