Question

Assume that females have pulse rates that are normally distributed with a mean of mu equals 76.0μ=76.0 beats per minute and a standard deviation of sigma equals 12.5σ=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 between 72 beats per minute and 80 beats per minute.

If

2525

adult females are randomly​ selected, find the probability that they have pulse rates with a mean between

72 beats per minute and 80 beats per minute.

Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

Answer 1

P(72 < Y < 80) = P(72 - mean < Y - mean < 80 - mean)
                  = P((72 - mean)/SD < (Y - mean)/SD < (80 - mean)/SD)
                  = P((72 - mean)/SD < Z < (80 - mean)/SD)
                  = P((72 - 76)/12.5< Z < (80 - 76)/12.5)
                  = P(-0.32 < Z < 0.32)
                  = P(Z < 0.32) - P(Z <-0.32)
                  = 0.251

Since the original population has a normal​ distribution, the distribution is a normal distribution for any given sample size.