Part 1: Mean and Standard Deviation of the Sampling Distribution

For the given scenario, you need to find the mean $\mu_{\bar{x}}$ and the standard deviation $\sigma_{\bar{x}}$ of the sampling distribution of the sample means.

Mean of the Sampling Distribution ($\mu_{\bar{x}}$):

The mean of the sampling distribution of the sample means is the same as the population mean. Therefore: $\mu_{\bar{x}} = \mu = 126,000$

Standard Deviation of the Sampling Distribution ($\sigma_{\bar{x}}$):

The standard deviation of the sampling distribution (also called the standard error) is calculated using the formula: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ Where:

• $\sigma = 21,000$ (population standard deviation),
• $n = 37$ (sample size).

So, $\sigma_{\bar{x}} = \frac{21,000}{\sqrt{37}} \approx \frac{21,000}{6.08} \approx 3,454$ (Rounded to the nearest integer.)

Final answers:

• The mean $\mu_{\bar{x}} = 126,000$,
• The standard deviation $\sigma_{\bar{x}} \approx 3,454$.

Part 2: Graph of the Sampling Distribution

The graph of the sampling distribution is a normal distribution, as the population distribution is normal. The mean of the distribution is 126,000, and the standard deviation is about 3,454. The bell curve would be centered at 126,000, with most of the values falling within the range of $\mu_{\bar{x}} \pm 3\sigma_{\bar{x}}$ (approximately from 115,638 to 136,362).

Would you like more details or have any questions?

---

Here are 5 questions that could expand your understanding:

1. How does the sample size affect the standard deviation of the sampling distribution?
2. What is the probability of getting a sample mean greater than $130,000?
3. What is the significance of the Central Limit Theorem in this context?
4. How would the distribution change if the sample size were smaller, say 16?
5. Can you calculate a 95% confidence interval for the sample mean based on this distribution?

Tip: Larger sample sizes reduce the standard error, making the sampling distribution more concentrated around the population mean.