The net weights of food jars filled by a certain machine hav

The net weights of food jars filled by a certain machine have a normal distribution with mean of 137.0 grams and standard deviation of 1.6 grams. A sample of 10 jars is randomly selected from the output of the machine. What is the probability that exactly 2 jars have a net fill weight that is less than 135.0 grams?

Solution

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    135      
u = mean =    137      
          
s = standard deviation =    1.6      
          
Thus,          
          
z = (x - u) / s =    -1.25      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -1.25   ) =    0.105649774

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    10      
p = the probability of a success =    0.105649774      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.20559409 [ANSWER]