$(1$ point$)$ The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean $=531\mathrm{.}4$ and standard deviation $=29\mathrm{.}9.$

$($a$)$ What is the probability that a single student randomly chosen from all those taking the test scores $538$ or higher?

ANSWER:

For parts $($b$)$ through $($d$),$ consider a simple random sample $($SRS$)$ of $25$ students who took the test.

$($b$)$ What are the mean and standard deviation of the sample mean score $\stackrel{}{x},$ of $25$ students?

The mean of the sampling distribution for $\stackrel{}{x}$ is:

The standard deviation of the sampling distribution for $\stackrel{}{x}$ is:

$($c$)$ What z$-$score corresponds to the mean score $\stackrel{}{x}$ of $538?$

ANSWER:

$($d$)$ What is the probability that the mean score $\stackrel{}{x}$ of these students is $538$ or higher?

ANSWER: