Answer to Question #4252 in Statistics and Probability for Jorda Goll

Question

If a shipment of 30,000 batteries is assumed to have a lifetimes which are normally distributed with mean 360 days and standard deviation 25 days what percentage could be expected to last more than 365 days?

Answer 1

Let X be the random variable equal to the lifetime of a battery. Then by
assumption it has normal distribution with
mean 360 days and standard
deviation 25 days.
We have to find the probability P(X>365).

Notice
that the random variable Y=(X-360)/25 has normal distribution with mean 0 days
and standard deviation 1, and the
values of its probabilities is known from
tables.
Therefore
P(X>365) = P( (X-360)/25 > (365-360)/25
)
= P( Y > 5/25 )
= P(Y>0.2)
= 1 -
P(Y<0.2) =
= 1 - F^{-1}(0.2) =
= 1 - 0,5792597094
=
= 0,4207402906

Thus about 42% of all batteries, that is

& 30000 * 0.42 = 12622 batteries
have lifetime > 365 days.