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 $\$ 156,000$. Assume the standard deviation is $\$ 40,000$. Suppose you take a simple random sample of 8 graduates. Round all answers to four decimal places if necessary. a. What is the distribution of c. For a single randomly selected graduate, find the probability that her salary is between $\$ 161,487$ and $\$ 173,358$. d. For a simple random sample of 8 graduates, find the probability that the average salary is between $\$ 161,487$ and $\$ 173,358$. e. For part d), is the assumption of normal necessary? $\bigcirc$ Yes No

Step 1 ：First, we need to understand the problem. We are given the mean annual salary of graduates 10 years after graduation, the standard deviation, and the sample size. We are asked to find the probability that the salary of a single randomly selected graduate is between $161,487 and $173,358.

Step 2 ：Next, we need to calculate the z-scores for the given salary range. The z-score is calculated by subtracting the mean from the value and dividing by the standard deviation. So, for $161,487, the z-score is \((161487 - 156000) / 40000 = 0.137175\). For $173,358, the z-score is \((173358 - 156000) / 40000 = 0.43395\).

Step 3 ：Then, we need to find the probability that the z-score is between 0.137175 and 0.43395. This can be done by looking up these values in a standard normal distribution table or using a calculator with a normal distribution function. The probability is the value at 0.43395 minus the value at 0.137175.

Step 4 ：Finally, we need to round the answer to four decimal places. This is the probability that the salary of a single randomly selected graduate is between $161,487 and $173,358.