Question

assume that females have pulse rates that are normally distributed with a mean of μ = 75.0 beats per minute and a standard deviation of σ = 12.5 beats per minute. complete parts (a) through (c) below.
a. if 1 adult female is randomly selected, find the probability that her pulse rate is less than 79 beats per minute. the probability is 0.6255 (round to four decimal places as needed.)
b. if 4 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 79 beats per minute. the probability is 0.7389 (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 sample means, not individuals, the distribution is a normal distribution for any sample size
b. since the mean pulse rate exceeds 30, the distribution of sample means 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 part (c)

The Central Limit Theorem states that if the original population is normally - distributed, the sampling distribution of the sample mean $\bar{X}$ is normally distributed for any sample size $n$. Here, the original population of female pulse - rates is normally distributed.

Step2: Analyze each option

• Option A: The key is the normality of the original population, not just that it's a distribution of sample means. So this is incorrect.
• Option B: The mean of the population has nothing to do with the normality of the sampling distribution of the sample mean. So this is incorrect.
• Option C: Since the original population of female pulse - rates is normally distributed, the distribution of sample means is a normal distribution for any sample size. This is correct.
• Option D: The distribution in part (b) is of sample means, not individuals. So this is incorrect.

Answer:

c. C. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size
a. 0.6255
b. 0.7389