Question

assume that females have pulse rates that are normally distributed with a mean of μ = 73.0 beats per minute and a standard deviation of σ = 12.5 beats per minute. complete parts (a) through (c) below.
b. if 4 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 77 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 individuals, not sample means, the distribution is a normal distribution for any sample size.
b. since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
c. since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size.
d. since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size.

Explanation:

When dealing with the distribution of sample means, if the original population is normally distributed, then the sampling distribution of the sample mean \(\bar{X}\) is also normally distributed for any sample size \(n\). In this case, the population of female pulse rates is normally distributed. So, we can use the normal distribution to find probabilities for the sample mean (even when \(n = 4<30\)).

Answer:

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