Let's break down and solve the problem in the image.

Given Information:

• The population has a normal distribution with:
  • Mean (μ) = 53.6
  • Standard deviation (σ) = 15
• A sample of size n = 158 is drawn.

Part (a): Mean of the distribution of sample means

For the distribution of sample means (also known as the sampling distribution of the sample mean), the mean (μₓ̄) is the same as the population mean:

$\mu_{x̄} = \mu = 53.6$

Part (b): Standard deviation of the distribution of sample means

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

$\sigma_{x̄} = \frac{\sigma}{\sqrt{n}}$

Substituting the given values:

$\sigma_{x̄} = \frac{15}{\sqrt{158}} = \frac{15}{12.57} \approx 1.19$

Final Answers:

• (a) The mean of the distribution of sample means is 53.6.
• (b) The standard deviation of the distribution of sample means is approximately 1.19.

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

Here are 5 related questions:

1. What is the Central Limit Theorem and how does it apply to this problem?
2. How does increasing the sample size affect the standard deviation of the sample means?
3. Can the sample means be normally distributed even if the population distribution is not normal?
4. How would the standard error change if the sample size were doubled?
5. Why is the standard deviation of the sample means smaller than the population standard deviation?

Tip: The standard error decreases as the sample size increases, making your estimate of the population mean more precise!