Let's break down the information provided and clarify the points step by step:

• Population Mean ($\mu$): This is the average income of all employees in the population. $\mu = 97284$

• Population Standard Deviation ($\sigma$): This is the standard deviation of the income distribution for the entire population. $\sigma = 15868$

• Sample Size ($n$): The number of randomly selected employees in the sample. $n = 70$

Central Limit Theorem (CLT) Application:

Since the sample size is relatively large (greater than 30), the Central Limit Theorem applies. According to the CLT, the distribution of the sample means will be approximately normal, even if the population distribution is skewed (in this case, skewed to the right). So, we can conclude that the sampling distribution of the sample averages should be approximately normal.

• The Sampling Distribution of the Sample Means is centered at the population mean ($\mu$), which is $97284$.

Standard Deviation of the Sampling Distribution (Standard Error):

The standard deviation of the sample means, also known as the standard error of the mean, is calculated by the following formula: $\text{Standard Error} = \frac{\sigma}{\sqrt{n}}$ Substitute the values for $\sigma$ and $n$:

$\text{Standard Error} = \frac{15868}{\sqrt{70}} = \frac{15868}{8.3666} \approx 1896.17$

Thus, the standard deviation of the sampling distribution is approximately 1896.17.

Summary of Results:

1. Population Mean = $97284$
2. Population Standard Deviation = $15868$
3. Sample Size = $70$
4. The Sampling Distribution is approximately normal (due to the Central Limit Theorem).
5. The Sampling Distribution's Standard Deviation (Standard Error) = 1896.17 (rounded to two decimal places).

Would you like more details or explanations on any part? Here are five related questions to extend this information:

1. How does increasing the sample size affect the standard error?
2. What would happen to the sampling distribution if the sample size were smaller than 30?
3. How does skewness affect the sampling distribution when the sample size is small?
4. What is the probability that a sample mean is within $2000 of the population mean?
5. How does the Central Limit Theorem help when dealing with non-normal population distributions?

Tip: The larger the sample size, the smaller the standard error, meaning the sample mean will more closely approximate the population mean.