Business Weekly conducted a survey of graduates from $30$ top MBA programs. On the basis of the survey, assume the mean annual salary for graduates $10$ years after graduation is $$137,000.$ Assume the standard deviation is $$36,000.$ Suppose you take a simple random sample of $42$ graduates. Round all answers to four decimal places if necessary.

a$.$ What is the distribution of $x?xN(,)$

b$.$ What is the distribution of $,$

c$.$ For a single randomly selected graduate, find the probability that her salary is between $$129,968$ and $$138,245.$

d$.$ For a simple random sample of $42$ graduates, find the probability that the average salary is between $$129,968$ and $$138,245.$

e$.$ For part d$),$ is the assumption of normal necessary? No Yes