Part a: Sketching the Sampling Distribution of $\bar{x}$ for Sample Size 60

The Central Limit Theorem (CLT) tells us that the sampling distribution of the sample mean $\bar{x}$ will be approximately normally distributed, provided the sample size is large enough, regardless of the population distribution. For large samples (generally $n \geq 30$), the distribution of the sample mean $\bar{x}$ will approximate a normal distribution with the following properties:

• Mean of the sampling distribution: This remains the same as the population mean, $\mu = 52000$.
• Standard error of the mean (SEM): This describes the spread of the sampling distribution and is calculated as $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

For sample size 60, the standard error (SE) is: $SE = \frac{\sigma}{\sqrt{n}} = \frac{4000}{\sqrt{60}} \approx 516.4$ Thus, the sampling distribution of $\bar{x}$ for $n = 60$ is normally distributed with:

• Mean $\mu = 52000$
• Standard error $SE \approx 516.4$

Sketch (Verbal Description):

1. The horizontal axis would be labeled with values of $\bar{x}$, centered around 52000.
2. A normal distribution curve would be plotted, with its peak at 52000.
3. The spread of the distribution (standard deviation) is much smaller than the population standard deviation due to the smaller standard error.

Part b: Sampling Distribution for Sample Size 120

When the sample size increases, the standard error decreases. For sample size 120, the standard error becomes: $SE = \frac{4000}{\sqrt{120}} \approx 365.1$ This leads to a narrower distribution compared to the case with $n = 60$.

Effect of Increasing Sample Size:

1. The mean of the sampling distribution remains $\mu = 52000$.
2. The standard error decreases from 516.4 to 365.1.
3. The distribution becomes narrower, indicating that with a larger sample size, the sample means will be more tightly clustered around the population mean.

This shrinking of the spread indicates that as the sample size increases, the precision of the sample mean as an estimate of the population mean improves.

Would you like me to provide further details or clarifications on this topic?

Relative Questions:

1. How does the Central Limit Theorem apply to non-normal population distributions?
2. What is the relationship between the standard error and the sample size?
3. How does increasing the sample size affect the confidence intervals for the sample mean?
4. What is the formula to calculate the Z-score for the sample mean?
5. How does sample size affect hypothesis testing for population means?

Tip: Larger sample sizes lead to smaller standard errors, improving the accuracy of the sample mean as an estimate of the population mean.