The problem presented involves a sampling distribution of the sample mean, where the population is normally distributed with a standard deviation (σ) of 30. The task is to find the sample size (n), using the graph and the given information.

Key Formula

For the sampling distribution of the sample mean, the standard deviation of the sample mean (also known as the standard error, SE) is given by:

$SE = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma = 30$ is the population standard deviation.
• $SE$ is the standard error.
• $n$ is the sample size, which we are solving for.

Step-by-Step Solution:

1. From the graph, the values on the x-axis represent the sample mean's range. The central value is 123, and one standard deviation away from the mean seems to be 128 (right) and 118 (left).

2. The difference between the mean (123) and one standard deviation on either side (128 or 118) is 5. Hence, the standard error (SE) is:

$SE = 128 - 123 = 5$

3. Now, use the formula for the standard error:

$SE = \frac{\sigma}{\sqrt{n}} \quad \Rightarrow \quad 5 = \frac{30}{\sqrt{n}}$

4. Solve for $n$:

$\sqrt{n} = \frac{30}{5} = 6 \quad \Rightarrow \quad n = 6^2 = 36$

Final Answer:

The sample size $n$ is 36.

Would you like more details on any of the steps or the formulas involved?

Here are five relative questions to expand on this information:

1. What is the significance of the standard error in a sampling distribution?
2. How does increasing the sample size affect the standard error?
3. What assumptions are made about the population distribution in this problem?
4. How is the concept of confidence intervals related to the standard error?
5. Can you explain how the central limit theorem justifies the use of the normal distribution here?

Tip: In problems involving sampling distributions, always check if you're working with the standard deviation of the population or the standard error, as this can change the approach for finding the sample size.