This problem describes a situation where the annual salary is normally distributed with a mean of $127,000 and a standard deviation of $23,000. Random samples of 30 individuals are taken, and you're asked to find the mean and standard deviation of the sampling distribution of these sample means.

Step 1: Mean of the Sampling Distribution (μ_x)

The mean of the sampling distribution is the same as the population mean. Therefore: $\mu_x = 127,000$

Step 2: Standard Deviation of the Sampling Distribution (σ_x)

The standard deviation of the sampling distribution (also called the standard error) is given by the formula: $\sigma_x = \frac{\sigma}{\sqrt{n}}$ where:

• $\sigma = 23,000$ (the population standard deviation)
• $n = 30$ (the sample size)

Plugging the values into the formula: $\sigma_x = \frac{23,000}{\sqrt{30}} \approx \frac{23,000}{5.477} \approx 4,203.16$ Rounding to the nearest integer: $\sigma_x \approx 4,203$

Step 3: Sketching the Graph

The graph of the sampling distribution will be a normal distribution with:

• The mean at 127,000
• The standard deviation at approximately 4,203

From the options in the image, option A is the most appropriate because it shows a normal distribution centered around 127,000, with the spread of the graph reflecting the typical spread of a sampling distribution, not the entire population's spread.

Would you like more detailed steps for any part of the solution? Here are some follow-up questions that might interest you:

1. What is the purpose of the Central Limit Theorem in this context?
2. How does the sample size affect the standard error in a sampling distribution?
3. What would happen if we increased the sample size from 30 to 100?
4. How do you interpret the area under a normal distribution curve?
5. How can you use the standard deviation of the sampling distribution to calculate probabilities?

Tip: When calculating the standard deviation of a sampling distribution, always use the formula $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.