The weights of people in a certain population are normally distributed with a mean of 153 lb and a standard deviation of 23 lb. Suppose that we select samples of size 5.

Determine whether the sampling distribution is normal (or approximately normal) and give its mean and standard deviation.

Choose the correct answer:

A. Normal distribution, mean = 153 lb, standard deviation = 10.29 lb
B. Normal distribution, mean = 153 lb, standard deviation = 23 lb
C. Approximately normal, mean = 153 lb, standard deviation = 10.29 lb
D. Approximately normal, mean = 153 lb, standard deviation = 4.6 lb

The sampling distribution of the sample means for a normally distributed population follows a normal distribution when the sample size is sufficiently large. In this case, the sample size is 5, which is relatively small. When dealing with small sample sizes, the Central Limit Theorem states that the sampling distribution of the sample mean tends to become approximately normal, even if the original population distribution is not normal.

Given a population with a mean of 153 lb and a standard deviation of 23 lb, when we take samples of size 5, the mean of the sampling distribution remains the same as the population mean (153 lb). However, the standard deviation of the sampling distribution is calculated by dividing the population standard deviation by the square root of the sample size[tex](5^(1/2[/tex]) ≈ 2.24), resulting in a value of approximately 10.29 lb.

This is why the answer is "Approximately normal, mean = 153 lb, standard deviation = 10.29 lb."

The Central Limit Theorem explains how the sampling distribution of the sample mean becomes approximately normal regardless of the population distribution, as long as the sample size is large enough. It is a fundamental concept in statistics, enabling reliable inference in various situations where normality assumptions might not hold for the population distribution.

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