For a particular brand of batteries the distribution of the

For a particular brand of batteries the distribution of the amount of time from its installation until it fails is a normal distribution with a mean of 3.2 years and a standard deviation of 0.5 years. If a installs batteries of this type in 10 of its company cars what is the that exactly 6 out of 10 batteries will be operating after 3 years?

Solution

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    3      
u = mean =    3.2      
          
s = standard deviation =    0.5      
          
Thus,          
          
z = (x - u) / s =    -0.4      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.4   ) =    0.655421742

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 =    0.655421742      
x = the number of successes =    6      
          
Thus, the probability is          
          
P (    6   ) =    0.23469068 [ANSWER]