Question

assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of \\( \mu = 1.1 \mathrm{kg} \\) and a standard deviation of \\( \sigma = 4.5 \mathrm{kg} \\). complete parts (a) through (c) below. the probability is.4874. (round to four decimal places as needed.) c. why can the normal distribution be used in part (b), even though the sample size does not exceed 30? a. since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size. b. since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size. c. since the weight gain exceeds 30, the distribution of sample means is a normal distribution for any sample size. d. since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.

Explanation:

The Central Limit Theorem states that if the original population is normally distributed, then the sampling distribution of the sample means is also normally distributed, regardless of the sample size. In this case, the population (amounts of weight male college students gain during freshman year) is normally distributed. So, for any sample size, the distribution of sample means will be normal.

Answer:

D. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.