Heights X of adult women are normally distributed with mean

Heights X of adult women are normally distributed with mean 63.7 inches and standard deviation 2.71 inches. Romeo, who is 69.25 inches tall, wishes to date only women who are shorter than he but within 4 inches of his height. Find the probability that the next woman he meets will have such a height.

Solution

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound = 69.25 - 4 =   65.25      
x2 = upper bound =    69.25      
u = mean =    63.7      
          
s = standard deviation =    2.71      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    0.57195572      
z2 = upper z score = (x2 - u) / s =    2.04797048      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.716324013      
P(z < z2) =    0.979718554      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.263394541   [ANSWER]