Question

Assume that females have pulse rates that are normally distributed with a mean of mu equals 73.0μ=73.0 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 79 beats per minute.

The probability is ......

​(Round to four decimal places as​ needed.)

b. If 25 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 ....

​(Round to four decimal places as​ needed.)

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

D.

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

Answer 1

Solution-A:

Xbar~N(73,12.5/sqrt(1)

P(xbar<79)

use pnormgC in R studio to get the prob

Rcode:

library(tigerstats)
pnormGC(bound= 79,region="below",mean=73,sd= 12.5,graph=TRUE)

0.6843863

ANSWER:

0.6844

Solution-b:

Xbar~N(73,12.5/sqrt(25))

xbar=73

s=sigma/sqrt(n)=12.5/sqrt(25)=12.5/5= 2.5

P(xbar<79)

use pnormgC in R studio to get the prob

Rcode:

library(tigerstats)
pnormGC(bound= 79,region="below",mean=73,sd= 2.5,graph=TRUE)

0.9918025

ANSWER:

0.9918

Solution-c:

From central limit theorem,as original popaltion follows normal distribution

sample follows normal distribution

OPTION D

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