Assume that women\'s heights are normally distributed with a mean given by =62.9
ID: 3178644 • Letter: A
Question
Assume that women's heights are normally distributed with a mean given by
=62.9 in,
and a standard deviation given by =2.4 in.
(a)If 1 woman is randomly selected, find the probability that her height is between
62.5in and 63.5in. The probability is approximately=
(Round to four decimal places as needed.)
(b) If 18 women are randomly selected, find the probability that they have a mean height between 62.5 in and 63.5 in.
The probability is approximately=
.(Round to four decimal places as needed.)
(c) Why can the central limit theorem be used in part (b), even though the sample size does not exceed 30?
A.The population size is greater than 30.
B.The sample size needs to be less than 30, not greater than 30.
C.The population is normally distributed.
D.The sample is normally distributed.
Explanation / Answer
a. P(62.5<x<63.5)
As it is given height is normally distributed we will convert x to z
P(62.5-62.9/2.4<z<63.5-62.9/2.4)=P(-0.167<z<0.25)=0.0987-(-0.0663)=0.165
b. For mean we need to find P(62.5<xbar<63.5)
Now as per central limit theorem mean is normally distributed with mean=62.9 and sd=2.4/sqrt(18)=0.566
so P(62.5-62.9/0.566<z<63.5-62.9/0.566)=P(-0.71<z<1.06)=0.3554+2611=0.6166
c. As population is normally distributed as per central limit theorem we have used mean as normal so answer is c.