The delivery times for all food orders at a fast food restau

The delivery times for all food orders at a fast food restaurant during lunch hour are normally distributed with a mean of 6.7 minutes and a standard deviation of 2.1 minutes. Find the probability that the mean delivery time for a random sample of 16 such orders at this restaurant is between 7 and 8 minutes.

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    7      
x2 = upper bound =    8      
u = mean =    6.7      
n = sample size =    16      
s = standard deviation =    2.1      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    0.571428571      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    2.476190476      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.716145417      
P(z < z2) =    0.993360364      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.277214948   [ANSWER]