The problem provided shows a normal distribution curve and asks for the population standard deviation, given a sample size of 49 and a normal distribution.

Key information:

• The graph represents a sampling distribution of one mean.
• The sample size is 49.
• The mean of the distribution appears to be 61 (based on the center of the graph).
• The x-values range from 58 to 64.

We can apply the Central Limit Theorem (CLT), which states that the standard deviation of the sampling distribution (also known as the standard error of the mean, $\sigma_{\bar{x}}$) is related to the population standard deviation $\sigma$ by the formula:

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma_{\bar{x}}$ is the standard deviation of the sampling distribution (standard error),
• $\sigma$ is the population standard deviation (which we need to find),
• $n = 49$ is the sample size.

Looking at the graph, it appears that the standard deviation of the sampling distribution is about 1 (since each tick mark represents a single unit on the x-axis, and the curve has one standard deviation range between 60 and 62).

We can now use this information to solve for $\sigma$ (population standard deviation).

Calculation:

$\sigma_{\bar{x}} = 1 \quad \text{(from the graph)}$

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{49}} = \frac{\sigma}{7}$

Now, solving for $\sigma$: $1 = \frac{\sigma}{7}$ $\sigma = 7$

Answer:

The population standard deviation $\sigma$ is 7.

Would you like more details on this solution or have any questions?

Related Questions:

1. How does the Central Limit Theorem apply to larger sample sizes?
2. What is the significance of the standard error in sampling distributions?
3. Can this method be used for non-normally distributed populations?
4. How would the solution change with a different sample size?
5. Why is the sample size squared root used in the standard error formula?

Tip:

For a normally distributed population, as the sample size increases, the standard error decreases, making the sample mean a more accurate estimate of the population mean.