Could you please solve the problem below? I need detailed mathematical process steps and explanation. The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean $\backslash$mu $=548\mathrm{.}7$ and standard deviation $\backslash$sigma $=25\mathrm{.}9.$

$($a$)$ What is the probability that a single student randomly chosen from all those taking the test scores $555$ or higher?

ANSWER: $?$

For parts $($b$)$ through $($d$),$ consider a simple random sample $($SRS$)$ of $35$ students who took the test.

$($b$)$ What are the mean and standard deviation of the sample mean score $\backslash$bar x$,$ of $35$ students?

The mean of the sampling distribution for $\backslash$bar x is: $?$

The standard deviation of the sampling distribution for $\backslash$bar x is: $?$

$($c$)$ What z$-$score corresponds to the mean score $\backslash$bar x of $555?$

ANSWER: $?$

$($d$)$ What is the probability that the mean score $\backslash$bar x of these students is $555$ or higher?

ANSWER: $?$