Question: In (a) — (e) below, you may assume that the data are a simple random sample (or samples) from a normal distribution (or distributions).

1. Consider the setting of a single sample of size n=15. Suppose that the variance is NOT known. Suppose further that it is of interest to construct a 90% confidence interval estimate of the population mean µ. What are the values of the confidence coefficients?
2. For a single sample size n=35 and the estimation of a variance parameter 62 using a 95% confidence interval, what are the values of the confidence coefficients?
3. For a single sample size of n=25 and the estimation of the population mean u when the variance is known using a 80% confidence interval, what are the values of the confidence coefficients?
4. Next, consider the setting of two independent samples, one with sample size \(\mathrm{n}_{1}=13\) and the other with sample size \(\mathrm{n}_{2}=22\), and estimation of the difference between the two population means, \(\left[\mu_{1}-\mu_{2}\right]\). Assume that the two population variances are both unknown and unequal. Suppose further that the two sample variances are \(\mathrm{S}_{1}^{2}=2.20\) and \(\mathrm{S}_{2}^{2}=0.70 .\) If it is of interest to construct a \(95 \%\) confidence interval estimate of \(\left[\mu_{1}-\mu_{2}\right]\), what are the values of the confidence coefficients? Hint: This question is asking you to provide the correct percentiles from the correct distribution.
5. Finally, consider again the setting of two independent samples, one with sample size \(n_{1}=\) 13 and the other with sample size \(n_{2}=22\). Here, however, it is of interest to construct a \(90 \%\) confidence interval estimate of the ratio of the two population variances, \(\left[\sigma_{1}^{2} / \sigma_{2}^{2}\right]\). Assume again that the two sample variances are \(S_{1}^{2}=2.20\) and \(S_{2}^{2}=0.70\). What are the values of the confidence coefficients? Hint: This question is asking you to provide the correct percentiles from the correct distribution.

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