A standard chemistry examination administered nationally by

A standard chemistry examination administered nationally by the American Chemical Society has a mean of 500 and a standard deviation of 90. What is the probability that the average of a random sample of examination scores of 25 students will be between 450 and 500?

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    450      
x2 = upper bound =    500      
u = mean =    500      
n = sample size =    25      
s = standard deviation =    90      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -2.777777778      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    0      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.002736602      
P(z < z2) =    0.5      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.497263398   [answer]