Question

if the population distribution is approximately normal, what has to be true by the central limit theorem?

incorrect. in general, this is a true statement if the population distribution is unknown. however, it is stated that the population distribution is already approximately normal so it is not necessary to have a sample size at least 30 under this special circumstance.

• the sample size for the sampling distribution must be at least 30 for the sampling distribution to have a standard deviation similar to the population deviation
• the sample size does not have to be at least 30 because the population is approximately normal
• the sample size for the sampling distribution must be at least 30 for the sampling distribution to have a mean similar to the population mean
• the sample size for the sampling distribution must be at least 30 for the sampling distribution to be approximately normal

Explanation:

Analyze the given condition

The problem states that the population distribution is already approximately normal.

Apply the Central Limit Theorem

Using the Central Limit Theorem and Sampling Distribution concepts:

• If the underlying population is normally distributed, the sampling distribution of the sample mean will be approximately normal for any sample size \(n\).
• The requirement of a sample size \(n \ge 30\) is only necessary to guarantee normality when the population distribution is non-normal or unknown.

Evaluate the options

• Option 1: Incorrect. The standard deviation of the sampling distribution (standard error) is \(\sigma/\sqrt{n}\), which depends on \(n\), not just being "at least 30".
• Option 2: Correct. Since the population is already approximately normal, the sample size does not need to be at least 30 for the sampling distribution to be approximately normal.
• Option 3: Incorrect. The mean of the sampling distribution \(\mu_{\bar{x}}\) is always equal to the population mean \(\mu\), regardless of sample size.
• Option 4: Incorrect. This is the selected incorrect answer in the image. It is incorrect because \(n \ge 30\) is not required when the population is already normal.

Answer:

• (A) The sample size for the sampling distribution must be at least 30 for the sampling distribution to have a standard deviation similar to the population deviation
• (B) The sample size does not have to be at least 30 because the population is approximately normal (Correct answer)
• (C) The sample size for the sampling distribution must be at least 30 for the sampling distribution to have a mean similar to the population mean
• (D) The sample size for the sampling distribution must be at least 30 for the sampling distribution to be approximately normal