Suppose that X1, X2, . . . , Xn are i.i.d. N(0, 2 0 ) and
Chapter , Problem 34(choose chapter or problem)
Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) are i.i.d. \(N\left(\mu_{0}, \sigma_{0}^{2}\right)\) and \(\mu \text { and } \sigma^{2}\) are estimated by the method of maximum likelihood, with resulting estimates \(\hat{\mu} \text { and } \hat{\sigma}^{2}\). Suppose the bootstrap is used to estimate the sampling distribution of \(\hat{\mu}\).
a. Explain why the bootstrap estimate of the distribution of is \(\hat{\mu} \text { is } N\left(\hat{\mu}, \frac{\hat{\sigma}^{2}}{n}\right)\).
b. Explain why the bootstrap estimate of the distribution of \(\hat{\mu}-\mu_{0} \text { is } N\left(0, \frac{\hat{\sigma}^{2}}{n}\right)\).
c. According to the result of the previous part, what is the form of the bootstrap confidence interval for \(\mu\), and how does it compare to the exact confidence interval based on the t distribution?
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