In Iowa many farms have their own silos for holding corn At

In Iowa, many farms have their own silos for holding corn. At a specific farm with one silo, the capacity for the silo is 12,000 bushels of corn. During harvest, each wagon is filled and the corn from the wagon is placed in the silo. The amount of corn in each wagon is normally distributed with a mean of 300 bushels and a standard deviation of 30 bushels (the variability is due to machine operator). Wagons are independent of each other. What is the probability that when 39 wagons of corn are emptied into the silo, that the silo will be above capacity?

Solution

This is just like saying that the mean is above 12000/39 = 307.6923077.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    307.6923077      
u = mean =    300      
n = sample size =    39      
s = standard deviation =    30      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    1.60128154      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.60128154   ) =    0.054657288 [ANSWER]