Question: Two independent random samples have been selected \(-100\) observations from population 1 and 100 from population $2 .$ Sample means \(\bar{x}_{1}=15.5\) and \(\bar{x}_{2}=26.6\) were obtained.

From previous experience with these populations, it is known that the variances are \(\sigma_{1}^{2}=9\) and \(\sigma_{2}^{2}=16\)

1. Find \(\sigma_{\left(\bar{x}_{1}-\bar{x}_{2}\right)}\).
2. Sketch the approximate sampling distribution for \(\left(\bar{x}_{1}-\bar{x}_{2}\right)\) assuming \(\left(\mu_{1}-\mu_{2}\right)=10\)
3. Locate the observed value of \(\left(\bar{x}_{1}-\bar{x}_{2}\right)\) on the graph you drew in part b. Does it appear that this value contradicts the null hypothesis \(H_{0}:\left(\mu_{1}-\mu_{2}\right)=10 ?\)
4. Use the \(z\) -table on the inside of the front cover to determine the rejection region for the test of \(H_{0}:\left(\mu_{1}-\mu_{2}\right)=10\) against \(H_{0}:\left(\mu_{1}-\mu_{2}\right) \neq 10 .\) Use
   \(\alpha=.05\).
5. Conduct the hypothesis test of part \(d\) and interpret your result.
6. Construct a \(95 \%\) confidence interval for \(\left(\mu_{1}-\mu_{2}\right)\). Interpret the interval.
7. Which inference provides more information about the value of \(\left(\mu_{1}-\mu_{2}\right)-\) the test of hypothesis in part \(\mathbf{e}\) or the confidence interval in part f?

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