if the car last 30 months or more, the battery will eventually need to be repaired or replaced, but not under warranty.
if the car lasts less than 30 months, the battery has to be repaired or replaced under warranty.
the average life of the battery is 48 months.
that's the mean.
the standard deviation is 8 months.
the battery life is normally distributed so we can use the normal distribution tables to determine what percent of the batteries will need to be repaired or replaced under warranty.
the formula for z-score is z = (x-m)/s
x is the raw score compared to the mean.
m is the mean.
s is the standard deviation.
with a battery life of less than 30, the z-score formula is:
z = (30 - 48) / 8.
this results in a z-score of -2.25.
this means the life of a battery that lives less than 30 months has a life that is more than 2.25 standard deviations below the mean.
you would be looking in the z-score table for a z-score of -2.25.
that will tell you the area under the normal distribution curve that is to the left of that z-score.
that area is the probability of finding a z-score of that is less than a z-score of -2.25, which corresponds to the probability of finding a battery that lives less than 30 months.
the z-score table i used is found at http://users.stat.ufl.edu/~athienit/Tables/Ztable.pdf
in that table, i found that a z-score of -2.25 has an area of .0122 to the left of it.
that means that the probability of selling a battery that has a life of less than 30 months is equal to .0122, or 1.22 percent.
that's the percent of batteries that are expected to be repaired or replaced under warranty.
visually, this would look like this:
the shaded area is the area under the normal distribution curve that represents batteries with lives less than 30 months.
i used the z-score calculator from the TI-84 Plus.
it told me that the area to the left of a z-score -2.25 is equal to .0122244334.
different calculators will give you different answers depending on the algorithms they use and how much they round the results.
rounded to 4 decimal places the answer is .0122.