https://file.aiquickdraw.com/mathbot/file/1730004369375xhofdyg7.jpg

Let's go through the question to determine the correct solution.

The problem states:
- The mean annual salary for graduates is $ \mu = 189,000 $.
- The standard deviation of salaries is $ \sigma = 42,000 $.
- We take a sample of $ n = 47 $ graduates.

### Part (b) Solution:
In part (b), we're asked to find the distribution of the sample mean salary $ \bar{X} $.

For the distribution of the sample mean $ \bar{X} $, when the population is normally distributed or the sample size is large enough (Central Limit Theorem), the distribution of $ \bar{X} $ is approximately normal with:
1. **Mean:** $ \mu_{\bar{X}} = \mu = 189,000 $
2. **Standard Error (SE):** $ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} $

#### Calculating the Standard Error:
\[
\sigma_{\bar{X}} = \frac{42,000}{\sqrt{47}}
\]

Calculating this:
\[
\sigma_{\bar{X}} \approx 6126.1714
\]

Thus, the distribution of $ \bar{X} $ is:
\[
\bar{X} \sim N(189,000, 6126.1714)
\]

### Final Answer:
For part (b), the correct answer is:
\[
\bar{X} \sim N(189,000, 6126.1714)
\]

Let me know if you want further details or have more questions on this topic. 

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Here are 5 related questions for further understanding:
1. How does the sample size affect the standard error of the sample mean?
2. Why is the normal distribution assumption justified for the sample mean in this question?
3. How would the distribution change if the sample size were smaller?
4. What is the probability of finding a sample mean salary within one standard deviation of the mean?
5. How would you interpret the results if the population distribution were skewed rather than normal?

**Tip:** Always check if the Central Limit Theorem applies when working with sample means, as it can simplify the calculation of probabilities.

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 $189,000. Assume the standard deviation is $42,000. Suppose you take a simple random sample of 47 graduates. Round all answers to four decimal places if necessary.

a. What is the distribution of X?
b. What is the distribution of X-bar?
c. For a single randomly selected graduate, find the probability that her salary is between $182,811 and $186,974.
d. For a simple random sample of 47 graduates, find the probability that the average salary is between $182,811 and $186,974.
e. For part d, is the assumption of normal necessary?

Explore the probability distribution and sample mean distribution for MBA graduates' salaries. Learn how to calculate the standard error, probability for salary ranges, and understand when the assumption of normality is necessary. This example involves a survey with a mean salary of $189,000 and a standard deviation of $42,000 for a sample size of 47.

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