Suppose that a random sample of size $100$ is to be selected from a population with mean $50$ and standard deviation $6.$

$($a$)$ What are the mean and standard deviation of the sampling distribution of $\stackrel{}{x}?$

${}_x^{-}=$

${}_x=$

Describe the shape of the sampling distribution of $\stackrel{}{x}.$

Since $n$ is greater than or equal to $30$ the distribution of $\stackrel{}{x}$ will be approximately normal.

Since $n$ is greater than or equal to $30$ we can't say what the shape of the distribution of $\stackrel{}{x}$ will be$.$

Since $n$ is less than $30$ we can't say what the shape of the distribution of $\stackrel{}{x}$ will be$.$

Since $n$ is less than $30$ the distribution of $\stackrel{}{x}$ will be approximately normal.

$($b$)$ Whas is the approximate probability that $\stackrel{}{x}$ will be within $0\mathrm{.}3$ of the population mean $?($Round your answer to four decimal places.$)$

$($c$)$ What is the approximate probability that $\stackrel{}{x}$ will differ from $$ by more than $0\mathrm{.}9?($Round your answer to four decimal places.$)$