The breaking strength $x$ of a certain rivet used in a machine engine has a mean $5000$ psi and standard deviation $250$

psi. A random sample of $64$ rivets is taken. Consider the distribution of $\stackrel{}{x},$ the sample mean breaking strength.

$($a$)$ What is the probability that the sample mean falls between $4910$ psi and $5090$ psi?

$($b$)$ What sample $n$ would be necessary in order to have

Click here to view page $1$ of the standard normal distribution table.

Click here to view page $2$ of the standard normal distribution table.

$($a$)$ The probability is

$($Round to four decimal places as needed.$)$