The lifetimes (in miles) of a certain brand of automobile tires is a normally di

Question

The lifetimes (in miles) of a certain brand of automobile tires is a normally distributed random variable, X, with a mean lifetime of =40,000 miles and standard deviation =2000 miles. The manufacturer would like to offer a guarantee for free replacement of any tire that does not last a specified minimum number of miles. If the manufacturer desires to have a replacement policy that they will need to honor for only 1% of all tires they sell, what number of miles should be included in the following guarantee: "We will replace any tire free of charge if the lifetime of the tire is less than ____ miles." (That is, what is the largest value for a lifetime a tire can have and still be among the shortest 1% of all tires' lifetimes?)

Explanation / Answer

Mean =40,000 miles

Standard deviation =2000 miles

Let A denote the largest value for a lifetime a tire can have and still be among the shortest 1% of all tires' lifetimes

Then P(X < A) = 0.01

P(Z < (A - 40,000)/2000) = 0.01

from standard normal distribution table, Z value can be obtained

(A - 40,000)/2000 = -2.33

A = 35340

"We will replace any tire free of charge if the lifetime of the tire is less than 35,340 miles."

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