president of Salas Products, is reviewing the warranty policy for her company's new model of automobile batteries. Life tests performed on a sample of 100 batteries indicated:
(1) an average life of 75 months,
(2) a standard deviation of 5 months,
(3) a bell shaped battery life distribution. What percentage of the batteries will fail within the first 65 months of use?

It is stated that battery life distribution is normal. First we need to find out where 65 hour battery life in this distribution falls by finding its z-score. We can use the following formula

z(x)=[tex]\frac{X-M}{s}[/tex] where

X the point we are looking to find its z-score, which is 65
M is the mean of the sample, which is 75
s is the standard deviation of the sample, which is 5

Replacing these numbers in the formula, z-score of the 65 hour battery life :

z(x)=[tex]\frac{65-75}{5}[/tex]=-2

All the batteries fail within the first 65 months if their z-score is smaller than -2. Or we can state this also as: P(z≤-2) which is 0.0228

Thus 2.28 % of the batteries will fail within the first 65 months of use.