Which of the following is not a commonly used practice? Choose the correct answer below.
A. If the original population is not normally distributed and n > 30, the distribution of the sample means can be approximated reasonably well by a normal distribution.
B. If the distribution of the sample means is normally distributed, and n > 30, then the population distribution is normally distributed.
C. If the original population is normally distributed, then for any sample size n, the sample means will be normally distributed.
The Correct Answer and Explanation is:
The correct answer is:
B. If the distribution of the sample means is normally distributed, and n > 30, then the population distribution is normally distributed.
Explanation:
Understanding how sample means behave in relation to the population distribution is a fundamental concept in inferential statistics. Let’s explore why option B is not a commonly used or valid statistical practice.
Central Limit Theorem (CLT):
The Central Limit Theorem is a foundational statistical principle that states:
Regardless of the population distribution, the sampling distribution of the sample means approaches a normal distribution as the sample size (n) becomes large (typically n > 30 is used as a rule of thumb).
This means:
- If the population is not normally distributed but n > 30, we can still assume that the sampling distribution of the sample means is approximately normal. (This supports option A.)
- If the population is normally distributed, then the sampling distribution of the sample means will also be normally distributed even for small sample sizes. (This supports option C.)
Why Option B is Incorrect:
Option B tries to reverse the logic of the Central Limit Theorem. It suggests that if the sample means are normally distributed, then the population must also be normally distributed. This is not necessarily true.
Conclusion:
The correct answer is B, because it incorrectly infers properties of the population from the sampling distribution. Options A and C are based on established statistical principles and are widely accepted in practice.