Assume that the heights of women are normally distributed wi

Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 75 women are randomly selected, find the probability that they have a mean height between 63 and 65 inches.

Solution

For any normal random variable X with mean and standard deviation , X ~ Normal( , ), (note that in most textbooks and literature the notaYou can translate into standard normal units by:
Z = ( X - ) / tion is with the variance, i.e., X ~ Normal( , ² ). Most software denotes the normal with just the standard deviation.)

Now we need to find.

Find P( 63 < Xbar < 65 )
= P( ( 63 - 63.6 ) / 0.2886751 < ( Xbar - ) / < ( 65 - 63.6 ) / 0.2886751 )
= P( -2.078461 < Z < 4.849742 )
= P( Z < 4.849742 ) - P( Z < -2.078461 )
= 0.9999994 - 0.01883346
= 0.981166