Questions: treated as being from a normal distribution because the sample size is too small? Explain. Choose the correct answer below. A. Yes; the sample size must be over 30 for the sample means to be normally distributed. B. No; as long as more than 30 samples are collected, the sample means will be normally distributed. C. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size. D. No; the original population is normally distributed, so the sample means will be normally distributed for any sample size.

treated as being from a normal distribution because the sample size is too small? Explain.

Choose the correct answer below.
A. Yes; the sample size must be over 30 for the sample means to be normally distributed.
B. No; as long as more than 30 samples are collected, the sample means will be normally distributed.
C. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size.
D. No; the original population is normally distributed, so the sample means will be normally distributed for any sample size.

Transcript text: treated as being from a normal distribution because the sample size is too small? Explain. Choose the correct answer below. A. Yes; the sample size must be over 30 for the sample means to be normally distributed. B. No; as long as more than 30 samples are collected, the sample means will be normally distributed. C. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size. D. No; the original population is normally distributed, so the sample means will be normally distributed for any sample size.

Solution

Solution Steps

Solution Approach

To determine if a sample can be treated as being from a normal distribution, we need to consider the Central Limit Theorem (CLT). The CLT states that the distribution of the sample means will be approximately normal if the sample size is sufficiently large, typically n > 30, or if the original population is normally distributed. Therefore, the correct answer is based on whether the original population is normally distributed or if the sample size is large enough.

Step 1: Evaluate Sample Size Condition

The Central Limit Theorem (CLT) suggests that the distribution of sample means will be approximately normal if the sample size \( n \) is sufficiently large, typically \( n > 30 \). In this case, the sample size is \( n = 30 \), which does not satisfy the condition \( n > 30 \). Therefore, option A is true.

Step 2: Evaluate Original Population Normality

If the original population is normally distributed, then the sample means will be normally distributed regardless of the sample size. Here, it is given that the original population is normally distributed. Therefore, option D is true.

Step 3: Determine the Correct Answer

Since both options A and D are true, we need to determine which one directly answers the question. The question asks if the sample can be treated as being from a normal distribution. Given that the original population is normally distributed, the sample means will be normally distributed for any sample size. Therefore, the correct answer is option D.