All members of the New Modeling Agency must be between 66 an

All members of the New Modeling Agency must be between 66 and 70.5 inches tall. The heights of women aged 20 to 29 follow approximately the N(64.3, 2.7) distribution. What percentage of women in this age group are eligible (via their heights) to be members of the New Modeling Agency?

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    66      
x2 = upper bound =    70.5      
u = mean =    64.3      
n = sample size =    1      
s = standard deviation =    2.7      
          
Thus, the two z scores are          
          
z1 = lower z score =    0.62962963      
z2 = upper z score =    2.296296296      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.735531533      
P(z < z2) =    0.989170527      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.253638994 [ANSWER]