Question

birth weights in the united states are normally distributed with \\( \sigma _ { x } = 500 \\) grams. a random sample of 15 babies was taken. the average birth weight of these 15 babies was 3450 grams. which of the following is true regarding the distribution of sample means?
choose the correct answer below.
a. the distribution of sample means will be normal since birth weights of all babies are normally distributed.
b. the distribution of sample means will not be normal since the sample size is not large enough according to the central limit theorem.
c. the distribution of sample means will be approximately normal since the sample size is large enough according to the central limit theorem.
d. there is not enough information to determine the distribution of the sample means.

Explanation:

Step1: Recall the Central Limit Theorem

The Central Limit Theorem (CLT) states that if the population is normally distributed, then the sampling distribution of the sample mean is normally distributed for any sample size \(n\). If the population is not normally distributed, then the sampling distribution of the sample mean is approximately normally distributed when \(n\geq30\) (a common rule - of - thumb).

Step2: Analyze each option

• Option A: Since the population (birth weights in the US) is normally distributed (\(\sigma_{x}=500\) grams, and we assume the population is normal as stated in the problem description), for any sample size \(n\) (in this case \(n = 15\)), the sampling distribution of the sample mean \(\bar{X}\) is normal.
• Option B: The sample size \(n=15\) is not relevant for the normality of the sampling distribution of the sample mean when the population is normal. The statement in this option is incorrect.
• Option C: The sample size \(n = 15\) is not relevant for the normality of the sampling distribution of the sample mean when the population is normal. The statement in this option is incorrect.
• Option D: There is enough information. We know the population is normal.

Answer:

A. The distribution of sample means will be normal since birth weights of all babies are normally distributed.