In a large population of collegeeducated adults the mean IQ

In a large population of college-educated adults, the mean IQ is 115 with a standard deviation of 20. Suppose 100 adults from this population are randomly selected for a market research campaign.

The probability that the sample mean IQ is between 110 and 120 is:

A) 0.0062.

B) 0.9876.

C) 0.9938.

D) 0.9207.

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    110      
x2 = upper bound =    120      
u = mean =    115      
n = sample size =    100      
s = standard deviation =    20      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -2.5      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    2.5      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.006209665      
P(z < z2) =    0.993790335      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.987580669   [ANSWER, B]