Suppose a pharmacy fills its customers prescriptions each da

Suppose a pharmacy fills its customer\'s prescriptions each day based on the mean of 403 and the standard deviation of 42. What is the probability that the average number of customers in the random sample of 48 business days is between the values 300 (lower boundary) and 425 (upper boundary)?

Please show work.

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    300      
x2 = upper bound =    425      
u = mean =    403      
n = sample size =    48      
s = standard deviation =    42      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -16.99059364      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    3.629058835      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    4.82042E-65      
P(z < z2) =    0.999857772      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.999857772   [ANSWER]