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 $\overline{X}$. This exercise shows how to use the nonparametric bootstrap to estimate the bias and uncertainty in $\hat{\mu }=\overline{X}$.

a. Compute $\overline{X}$ for the given sample.

b. Generate 1000 bootstrap samples from the given sample.

c. Compute ${\hat{\mu }}_i^{\ast }={\overline{X}}_i^{\ast }$ for each of the 1000 bootstrap samples.

d. Compute the sample mean ${\overline{\hat{\mu }}}^{\ast }$ and the sample standard deviation $s_{{\hat{\mu }}^{\ast }}$ of ${\hat{\mu }}_1^{\ast },\dots ,{\hat{\mu }}_{1000}^{\ast }$.

e. Estimate the bias and uncertainty $\left({\sigma }_{\hat{\mu }}\right)$ in $\hat{\mu }$.

Data From Exercise 13:

A random sample of size 8 is taken from an $\mathrm{Exp}\left(\lambda \right)$ 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 $\left({\sigma }_{\hat{\lambda }}\right)$ in $\hat{\lambda }=1/\overline{X}$

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