The Distribution of W. In this exercise, you are to obtain the distribution of the variable W for samples of size 3 so that you can see how the Wilcoxon signed-rank table is constructed.

a. The rows of the following table give all possible signs for the signed ranks in a Wilcoxon signed-rank test with n = 3. For instance, the first row covers the possibility that all three observations are greater than μ0 and thus have positive sign ranks. Fill in the empty column with values of W. (Hint: The first entry is 6, and the last is 0.)

Rank 123 W

+++

++−

+−+

+−−

−++

−+−

−−+

−−−

b. If the null hypothesis H0:μ = μ0 is true, what percentages of samples will match any particular row of the table? (Hint: The answer is the same for all rows.)

c. Use the answer from part

(b) to obtain the distribution of W for samples of size 3.

d. Draw a relative-frequency histogram of the distribution obtained in part (c).

e. Use your histogram from part

(d) to find W0.125 for a sample size of 3.

One-Median Sign Test. Recall that the Wilcoxon signed-rank test, which can be used to perform a hypothesis test for a population median, η, requires that the variable under consideration has a symmetric distribution. If that is not the case, the one-median sign test (or simply the sign test) can be used instead. The one-median sign test is also known as the one-sample sign test and the one-variable sign test. Technically, like the Wilcoxon signed-rank test, use of the sign test requires that the variable under consideration has a continuous distribution. In practice, however, that restriction is usually ignored.

If the null hypothesis H0: η = η0 is true, the probability is 0.5 of an observation exceeding η0. Therefore, in a simple random sample of size n, the number of observations, s, that exceed η0 has a binomial distribution with parameters n and 0.5.

To perform a sign test, first assign a “+” sign to each observation in the sample that exceeds η0 and then obtain the number of

“+” signs, which we denote s0. The P-value for the hypothesis test can be found by applying Exercise 9.71 on page 409 and obtaining the required binomial probability.