Question

SupposeSuppose the heights of 18-year-old men are approximately normally distributed, with mean 66 inches and standard deviation 1 inches.

(a) What is the probability that an 18-year-old man selected at random is between 65 and 67 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of nine 18-year-old men is selected, what is the probability that the mean height x is between 65 and 67 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

The probability in part (b) is much higher because the standard deviation is larger for the x distribution.    

The probability in part (b) is much higher because the mean is smaller for the x distribution.

The probability in part (b) is much higher because the mean is larger for the x distribution.

The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.

Answer 1

(a) What is the probability that an 18-year-old man selected at random is between 65 and 67 inches tall? (Round your answer to four decimal places.)
P(65<X<67)

mean=66 sd=1

using R software nad pnorm function we get the probability

pnorm(67,mean=66,sd=1)-pnorm(65,mean=66,sd=1)

Output:

0.6826895

ANSWER:0.6827

(b) If a random sample of nine 18-year-old men is selected, what is the probability that the mean height x is between 65 and 67 inches? (Round your answer to four decimal places.)
sample stddev=popsd/sqrt(n)

=1/sqrt(9)

=1/3

=0.333333

P(67<xbar<65)

use below R code:

pnorm(67,mean=66,sd=0.333333333)-pnorm(65,mean=66,sd=0.333333333)

output;

0.9973002

ANSWER:0.9973

c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

yes higher

The probability in part (b) is much higher because the standard deviation is smaller for the xdistribution.