Discussion
Let \((X_{i}, Y_{i})\) \(i = 1, \ldots, 5\) be the bivariate pair of observations and \(\mathbf{Z}_{i} = X_{i} - Y_{i}\) \(i = 1, \ldots, 5\). For a given sample the number of \(Z_{i} \gt 0\) is 2. Hence the \(p\)-value to test the null hypothesis that \(H_{0}\) : median (Z) = 0 against \(H_{1}\) : median (Z) \(\lt\) 0 is :
Answer: A
The p-value is 0.5.
This is the sign test. Under \(H_{0}\) the number S of positive \(Z_{i}\) is binomial with \(n = 5\), \(p = \frac{1}{2}\). Since \(H_{1}\) says the median is negative, small values of S are evidence against \(H_{0}\), so the p-value is
\[ \begin{aligned} P(S \le 2) &= \frac{\binom{5}{0} + \binom{5}{1} + \binom{5}{2}}{32} \\ &= \frac{1 + 5 + 10}{32} = 0.5 \end{aligned} \]
The data give no evidence at all against \(H_{0}\). 0.25, 0.10 and 0.05 would require fewer positive differences.
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