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:

~
25.0 versus Ha:

~ 25.0. The test statistic is Y
the number of observations that exceed 25.

a. Consider rejecting H0 if Y  15. What is the value of 

(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

~
25.0?

b. What rejection region of the form Y  c specifies a test with a significance 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 that have positive signs, hence the name sign test.]