The average age of doctors in a certain hospital is 450 year

The average age of doctors in a certain hospital is 45.0 years old. Suppose the distribution of ages has a standard deviation of 4.0 years. If 16 doctors are chosen at random for a committee, find the probability that the average age of those doctors is less than 45.5 years. Select one: a. 19.1% b. 51.2% c. 55.0% d. 69.1% e. Cannot be determined

Solution

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    45.5      
u = mean =    45      
n = sample size =    16      
s = standard deviation =    4      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.5      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   0.5   ) =    0.691462461 = 69.1% [ANSWER, OPTION D]