The problem asks for the sample size $n$, given the sampling distribution of the sample mean. Here’s the approach to solve it:

Formula for the Standard Error of the Mean:

$\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}$ where:

• $\sigma = 25$ (population standard deviation, as given)
• $SE$ can be estimated from the spread of the sample means shown on the graph.

Identifying Key Information from the Graph:

The bell curve shows a distribution with sample means ranging roughly from 116 to 146.

• The center (mean) of the distribution appears to be around 131.
• The curve shows a typical spread of around 5 units to either side of the mean, meaning one standard error seems to be 5.

So, we can estimate the standard error $SE \approx 5$ from the graph.

Using the Formula to Find $n$:

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

Substitute the known values:

$5 = \frac{25}{\sqrt{n}}$

Solve for $\sqrt{n}$:

$\sqrt{n} = \frac{25}{5} = 5$

Square both sides:

$n = 5^2 = 25$

Answer:

The sample size $n$ is:

$n = 25$

Do you have any questions or need further clarification?

Related Questions:

1. What is the difference between population standard deviation and standard error?
2. How does increasing the sample size affect the standard error?
3. What role does the Central Limit Theorem play in sampling distributions?
4. How would the answer change if the standard error was 10 instead of 5?
5. Why is it important to have a large sample size in statistical analysis?

Tip: Larger sample sizes reduce the standard error, giving more precise estimates of the population mean.