Question

assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of μ = 1.2 kg and a standard deviation of σ = 5.7 kg. complete parts (a) through (c) below.
a. if 1 male college student is randomly selected, find the probability that he gains between 0 kg and 3 kg during freshman year.
the probability is 0.2087
(round to four decimal places as needed.)
b. if 16 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg.
the probability is 0.6957
(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 weight gain exceeds 30, the distribution of sample means 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 original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size
d. since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size

Explanation:

Step1: Recall the Central - Limit - Theorem for normal populations

If the original population is normally distributed with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample mean $\bar{X}$ for a sample of size $n$ has mean $\mu_{\bar{X}}=\mu$ and standard deviation $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$, and is normally distributed for any sample size $n$.

Step2: Analyze the given situation

In this case, the amounts of weight that male college students gain during their freshman year are normally distributed (original population is normal) with mean $\mu = 1.2$ kg and standard deviation $\sigma=5.7$ kg. When we consider a sample of $n = 16$ male college students (part b), the distribution of the sample mean $\bar{X}$ is also normally distributed because the original population is normal.

Answer:

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