In this problem, we are tasked with finding the mean and standard deviation of the sampling distribution of sample means, given the population mean ($\mu$) and standard deviation ($\sigma$).

Given information:

• Population mean, $\mu = 156$
• Population standard deviation, $\sigma = 25$
• Sample size, $n = 44$

Solution:

1. Mean of the sampling distribution: The mean of the sampling distribution of sample means is equal to the population mean: $\mu_{\overline{x}} = \mu = 156$

2. Standard deviation of the sampling distribution: The standard deviation of the sampling distribution (also called the standard error) is calculated using the formula: $\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} = \frac{25}{\sqrt{44}} \approx \frac{25}{6.633} \approx 3.772$

Final Answers:

• The mean is $\mu_{\overline{x}} = 156$.
• The standard deviation is $\sigma_{\overline{x}} \approx 3.772$.

Would you like more details on this solution or have any questions? Here are some related topics:

1. How does sample size affect the standard deviation of the sampling distribution?
2. What is the difference between the population standard deviation and the standard deviation of the sample means?
3. How can we use the Central Limit Theorem to solve sampling distribution problems?
4. How would the sampling distribution change if the population mean and standard deviation were different?
5. What is the importance of the standard error in hypothesis testing?

Tip: The standard error decreases as the sample size increases, making sample means more closely centered around the population mean.