Chapter 15, Problem 33SE

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):

We wish to test H₀: $\tilde{\mu }$ = 25.0 versus H_a: $\tilde{\mu }$ > 25.0. The test statistic is Y = the number of observations that exceed 25.

1. a. Determine the P-value of the test when Y = 15. [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 $\tilde{\mu }$ = 25.0?]
2. b. For the given data, should H₀ be rejected at significance level .05?

[Note: The test statistic is the number of differences X_i − 25 that have positive signs, hence the name sign test.]