Question

Assume that females have pulse rates that are normally distributed with a mean of μ=76.0 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 between 70 beats per minute and

82 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 between

70 beats per minute and 82 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 original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.

C.Since the distribution is of sample​ means, not​ individuals, the distribution is a normal distribution for any sample size.

D.Since the mean pulse rate exceeds​ 30, the distribution of sample means is a normal distribution for any sample size.

Answer 1

Solution:

Given:

Population mean = μ=76.0

Population standard deviation = σ=12.5

a)

We have to find P(70 X82) = ...?

Using Z-score,

Therefore,

P(−0.48≤Z≤0.48) =P(Z≤0.48)−P(Z≤−0.48)

=0.6844−0.3156=0.3688 ...Using z-table

b)

Here, n = 4

We have to find P(70 82) = ...?

Using z-score,

Therefore,

P(−0.96≤Z≤0.96) = P(Z≤0.96)−P(Z≤−0.96)

=0.8315−0.1685=0.6629 ...Using z-score,

C)

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

Hence, option B is correct

Done