Probabilities

Question

Heights of Women   In a survey of U.S. women, the heights in the 20 –29 age group were normally distributed, with a mean of 64.2 inches and a standard deviation  of  2.9  inches.  Find  the  probability  that  a  randomly  selected  study participant has a height that is (a) less than 56.5 inches, (b) between 61 and 67 inches, and (c) more than 70.5 inches, and (d) identify any unusual events. Explain your reasoning.

Answer 1

Hi Jeffrey.

 

To find the probability of a specific occurrence with a normal distribution we use the following formula

        X - μ

Z = -------

         σ

 

a) height < 56.5"

       56.5 - 64.2

Z = -------------  = -2.6552

            2.9

 

Look up this value in a z-table

Probability (z<-2.6552) = 0.003965 = .4%

 

b) Height between 61 and 67 inches

We need to calculate 2 z values, one for each endpoint.

at 61

       61 - 64.2
Z = ----------- = -1.1034
          2.9 

at 67

       67 - 64.2
Z = ----------- = 0.9655
          2.9

 

 

Probability (z<-1.1034) = 0.1284

 

Probability (z<0.9655) = 0.8329

0.8329 - 0.1284 = .7048 = 70.48%

 

c) Height > 70.5"

       70.5 - 64.2
Z = -------------- = 2.1724
          2.9

 

Probability (z>2.1724) = 1-0.9851 = 0.0149 = 1.49%