The image you uploaded contains a problem related to statistics involving a population of numbers and sampling. It asks for the following:

1. The mean of the population.
2. The standard deviation of the population.
3. The mean of the sampling distribution of sample means.
4. The standard deviation of the sampling distribution of sample means.

Let me break down the given information and solve each part:

Given Population:

The population consists of the following 5 numbers: 2, 3, 6, 8, and 11.

Steps to Solve:

1. Mean of the Population:

The population mean (μ) is the sum of all values in the population divided by the number of values.

$\mu = \frac{2 + 3 + 6 + 8 + 11}{5} = \frac{30}{5} = 6$

2. Standard Deviation of the Population:

The population standard deviation (σ) is calculated using the formula:

$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$

Where $x_i$ are the population values, $\mu$ is the mean, and $N$ is the number of values.

Let's compute it:

$\sigma = \sqrt{\frac{(2-6)^2 + (3-6)^2 + (6-6)^2 + (8-6)^2 + (11-6)^2}{5}}$

$\sigma = \sqrt{\frac{16 + 9 + 0 + 4 + 25}{5}} = \sqrt{\frac{54}{5}} \approx \sqrt{10.8} \approx 3.29$

3. Mean of the Sampling Distribution of Sample Means:

The mean of the sampling distribution of the sample means is equal to the population mean. Therefore, it is:

$\mu_{\text{sample means}} = \mu = 6$

4. Standard Deviation of the Sampling Distribution of Sample Means:

The standard deviation of the sampling distribution of the sample means (also known as the standard error) is calculated as:

$\sigma_{\text{sample means}} = \frac{\sigma}{\sqrt{n}}$

Where $\sigma$ is the population standard deviation and $n$ is the sample size.

Since the sample size is 2:

$\sigma_{\text{sample means}} = \frac{3.29}{\sqrt{2}} \approx \frac{3.29}{1.414} \approx 2.33$

Summary of Results:

1. Population Mean: 6
2. Population Standard Deviation: 3.29
3. Mean of Sampling Distribution of Sample Means: 6
4. Standard Deviation of Sampling Distribution of Sample Means: 2.33

Would you like to discuss any part of this solution in more detail, or is there another question you need help with?

Here are some related questions you might find interesting:

1. How does the sample size affect the standard deviation of the sampling distribution?
2. What happens if you increase the population size in this scenario?
3. How do you calculate the standard error for a sample size of 3?
4. What would the mean of the sampling distribution of sample means be for a larger population?
5. How does drawing samples with replacement differ from drawing without replacement?

Tip: When calculating standard deviations, remember that sample size and population size both influence the spread of the data, and adjusting one can reduce or increase variability.