(Solved): Problem 9 (10 points): Let \( X_{1}, \ldots, X_{8} \) be a sample from normal population with popu ...

Problem 9 (10 points): Let \( X_{1}, \ldots, X_{8} \) be a sample from normal population with population mean \( \mu=60 \) and population variance \( \sigma^{2}=15 \). Let \( \bar{X} \) and \( S^{2} \) denote the sample mean and sample variance. (a) Find \( P\{58<\bar{X}<63\} \). (b) Express \( P\left\{S^{2} \geq 15\right\} \) in terms of a chi-square random variable. (c) Find the conditional probability \( P\left(\{58<\bar{X}<63\} \mid\left\{S^{2} \geq 15\right\}\right) \).