I'm Not Good At Statistics

Question

A survey found that women's heights are normally distributed with mean 63.3 in. and standard deviation 2.6 in. The survey also found that men's heights are normally distributed with a mean 67.9 in. and standard deviation 2.9.
Complete parts a through c below.

a. Most of the live characters at an amusement park have height requirements with a minimum of 4 ft 9 in. and a maximum of 6 ft 3 in. Find the percentage of women meeting the height requirement.

The percentage of women who meet the height requirement is ? %. (Round to two decimal places as needed.)

b. Find the percentage of men meeting the height requirement.

The percentage of men who meet the height requirement is ? %. (Round to two decimal places as needed.)

c. If the height requirements are changed to exclude only the tallest 5% of men and the shortest 5% of women, what are the new height requirements?

The new height requirements are at least ? in. and at most ? in.

 

Answer 1

Diane,

Two things are needed to complete this problem:  Z-score formula and Normal Distribution table

 

Z = (x - μ)/σ where x is the value being assessed, μ is the mean, and σ is the standard deviation.

 

The table should be in the back of your text book. If you don't have one, it can be found on line.

 

a)   convert women's height to inches:   57"   -   75"

 

     calculate Z-scores:  (57 - 63.3)/2.6 = -2.42          (75 - 63.3)/2.6 = 4.5

 

     The distribution table gives you the percent between the mean and the value plugged in:

 

     -2.42 ⇒ 49.22%     4.5 ⇒ 50%

 

     Adding the two tells you that the height range given statistically includes 99.22% of all women.

 

b)   Repeat all steps with the men's data:

 

      (57 - 67.9)/2.9 = -3.76          (75 - 67.9)/2.9 = 2.55

 

      -3.76 ⇒ 49.99%           2.55 ⇒ 49.46%

 

    Adding the two tells you that the height range given statistically includes 99.45% of all men.

 

c)   For part c, we must work backward from the table. Look for the Z-scores that give you 45% above the mean and 45% below the mean.  These turn out to be: +1.64 and -1.64

 

     Now we convert these back to inches away from the mean and add or subtract:

 

     men --      67.9 + 1.64(2.9) = 72.66"

 

     women --  63.3 - 1.64(2.6) = 59.04"

 

    New range:  59.04"  to  72.66"