# Answer to Question #4251 in Statistics and Probability for John Dewer

Question #4251

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?

b) what percentage could be expected to last more than 180 days?

b) what percentage could be expected to last more than 180 days?

Expert's answer

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 probabilities P(X>365) and

P(X>180)

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

Similarly,

P(X>180) = P( (X-360)/25 > (180-360)/25 )&

= P( Y >

-180/25 )

= P(Y>-7.2)

= 1 - P(Y<-7.2) =

= 1 - F^{-1}(-7.2) =

= 1 - 3,01062798111744E-013

~

= 0.9999999999997

Thus the probability is very close to 1,

so it is expected that all 30,000 batteries will have lifetime >

180

assumption it has normal distribution with

mean 360 days and standard

deviation 25 days.

We have to find the probabilities P(X>365) and

P(X>180)

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

Similarly,

P(X>180) = P( (X-360)/25 > (180-360)/25 )&

= P( Y >

-180/25 )

= P(Y>-7.2)

= 1 - P(Y<-7.2) =

= 1 - F^{-1}(-7.2) =

= 1 - 3,01062798111744E-013

~

= 0.9999999999997

Thus the probability is very close to 1,

so it is expected that all 30,000 batteries will have lifetime >

180

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