6.66. Samples of a population are used to estimate characteristics of a population, e.g., the voting characteristics in state or national elections. Estimates obtained from sampling a population are not exact and are subject to error that depends on $n$, the size of the sample. Suppose $X$ is $N[\mu, \sigma^2]$, and we sample a population of independent, identically distributed random variables $X_i$, $i = 1, 2, \dots, n$. What is the distribution of the sample mean \begin{equation*} \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i ? \end{equation*} Suppose we choose $n$ to obtain a .95 probability that $\bar{X}$ will be within two standard deviations of $\mu$. How close will $\bar{X}$ be to $\mu$ in actual numbers? How large should $n$ be so that $\mu = \bar{X} \pm .1$ for $\sigma = 20$?