Problem
Test the claim below about the mean of the differences for a population of paired data at the level of significance \( \alpha \). Assume the samples are random and dependent, and the populations are normally distributed. Claim: \( \mu_{\mathrm{d}}=0 ; \alpha=0.01 \). Sample statistics: \( \bar{d}=2.9, s_{d}=8.63, n=9 \)
Solution
Step 1 ：\(H_0: \mu_d = 0, H_1: \mu_d \neq 0\)
Step 2 ：\(t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}} = \frac{2.9 - 0}{8.63 / \sqrt{9}} \approx 1.0065\)
Step 3 ：\(\mathrm{Reject}\,H_0\,\mathrm{if}\,|t| > t_{\alpha/2, n-1} = t_{0.005, 8}\)
