Question: The data for a random sample of six paired observations are shown in the table.

1. Calculate the difference between each pair of observations by subtracting observation 2 from observation 1. Use the differences to calculate \(\bar{d}\) and \(s_{\mathrm{d}}^{2}\)
2. If \(\mu_{1}\) and \(\mu_{2}\) are the means of populations 1 and 2 , respectively, express \(\mu_{\mathrm{d}}\) in terms of \(\mu_{1}\) and \(\mu_{2}\).
3. Form a \(95 \%\) confidence interval for \(\mu_{\mathrm{D}}\).
4. Test the null hypothesis \(H_{0}: \mu_{\mathrm{d}}=0\) against the alternative hypothesis \(H_{a}: \mu_{d} \neq 0 .\) Use \(\alpha=.05\).

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