Refer to Exercise 13. Assume the distribution from which the sample was drawn is unknown. We want to estimate the population mean \(\mu\) with the sample mean \(\bar{X}\). This exercise shows how to use the nonparametric bootstrap to estimate the bias and uncertainty in \(\widehat{\mu}=\bar{X}\).

a. Compute \(\bar{X}\) for the given sample.

b. Generate 1000 bootstrap samples from the given sample.

c. Compute \(\widehat{\mu}_{i}^{*}=\bar{X}_{i}^{*}\) for each of the 1000 bootstrap samples.

d. Compute the sample mean \(\overline{\hat{\mu}}^{*}\) and the sample standard deviation \(s_{\widehat{\mu}^{*}}\) of \(\widehat{\mu}_{1}^{*}, \ldots, \widehat{\mu}_{1000}^{*}\).

e. Estimate the bias and uncertainty \(\left(\sigma_{\widehat{\mu}}ight)\) in \(\widehat{\mu}\).

Data From Exercise 13:

A random sample of size 8 is taken from an \(\operatorname{Exp}(\lambda)\) distribution, where \(\lambda\) is unknown. The sample values are \(2.74,6.41,4.96,1.65,6.38,0.19,0.52\), and 8.38. This exercise shows how to use the bootstrap to estimate the bias and uncertainty \(\left(\sigma_{\hat{\lambda}}ight)\) in \(\hat{\lambda}=1 / \bar{X}\).