Question

Assume that females have pulse rates that are normally distributed with a mean of mu equals 72.0μ=72.0 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 less than 76 beats per minute.

The probability is ?.

​(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 76 beats per minute.

The probability is ?.

​(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 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 sample​ means, not​ individuals, the distribution is a normal distribution for any sample size.

Answer 1

a)
Here, μ = 72, σ = 12.5 and x = 76. We need to compute P(X <= 76). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (76 - 72)/12.5 = 0.32

Therefore,
P(X <= 76) = P(z <= (76 - 72)/12.5)
= P(z <= 0.32)
= 0.6255

b)
Here, μ = 72, σ = 12.5/sqrt(4) = 6.25 and x = 76. We need to compute P(X <= 76). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (76 - 72)/6.25 = 0.64

Therefore,
P(X <= 76) = P(z <= (76 - 72)/6.25)
= P(z <= 0.64)
= 0.7389

c)
C.

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