To determine the correct answer, let's analyze the problem:

Given Data:

• SAT Evidence-Based Reading scores are approximately normal.
• Mean ($\mu$) of the scores: $536$.
• Standard deviation ($\sigma$) of the scores: $102$.
• A simple random sample (SRS) of $100$ students is taken, and we calculate the mean of their scores.

Question:

If this sampling process is repeated many times, what is the expected mean of the sample means?

Solution:

According to the Central Limit Theorem, the mean of the sample means ($\bar{X}$) will be the same as the population mean. Thus, the expected mean of the sample means is also $536$, regardless of the sample size.

Answer:

The correct choice is:

• 536.

Would you like further details, or do you have any questions?

Related Questions:

1. What would the standard error of the mean be for this sample?
2. How would increasing the sample size affect the standard error?
3. Why does the Central Limit Theorem apply in this scenario?
4. What would happen to the distribution of sample means if the sample size were smaller?
5. How would you interpret the standard deviation of the sample mean in this context?

Tip:

The Central Limit Theorem is especially useful when dealing with sampling distributions, as it allows us to assume a normal distribution for sample means if the sample size is sufficiently large.