Statistical applications

Question

Based on data from the national health survey, the heights of adult woman in the US are normally distributed with mean 63.6 inches and standard deviation 2.5 inches. This semester there are 2460 women enrolled in classes. Based on this survey, how many of these women are expected to have heights between 65 and 70 inches?

Answer 1

 

When X represents a random variable which follows a normal distribution

of which mean, μ, is 63.6 and standard deviation, σ, is 2.5.

 

The probability we need to find out is

 

     Prob(65 ≤ X ≤ 70) = Prob((65-63.6)/2.5 ≤ Z ≤ (70-63.6)/2.5)

 

                                 = Prob(0.56 ≤ Z ≤ 2.56)

 

                                 = Prob(Z ≤ 2.56) - Prob(Z ≤ 0.56)

 

where Z is a random variable which follows the standard normal distribution.

 

From the table for the standard normal distribution,

Prob(Z ≤ 2.56) reads as 0.9948, and Prob(Z ≤ 0.56) does as 0.7123.

 

     Prob(65 ≤ X ≤ 70) = Prob(Z ≤ 2.56) - Prob(Z ≤ 0.56)

 

                                 = 0.9948 - 0.7123

 

                                 = 0.2825

 

As 2460·0.2825 = 694.95,

694 women out of 2460 are expected to be between 65 and 70 inches tall.