Question

A tire manufacturer warranties its tires to last at least 20,000 miles orâ "you get a new set ofâ tires." In itsâ experience, a set of these tires last on average 26,000 miles with SD 5,000 miles. Assume that the wear is normally distributed. The manufacturer profits â$200 on each setâ sold, and replacing a set costs the manufacturer â$400. Complete parts a through c.

(a) What is the probability that a set of tires wears out before 20,000 miles?

Answer 1

Probability that a set of tires wears out before 20,000 miles is 0.1151.

Explanation:

We are given that a tire manufacturer warranties its tires to last at least 20,000 miles or "you get a new set of tires." In its past experience, a set of these tires last on average 26,000 miles with S.D. 5,000 miles. Assume that the wear is normally distributed.

Let X = wearing of tires

So, X ~ N(
`\mu=26,000,\sigma^(2)=5,000^(2)`)

Now, the z score probability distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = average lasting of tires = 26,000 miles

`\sigma` = standard deviation = 5,000 miles

So, probability that a set of tires wears out before 20,000 miles is given by = P(X < 20,000 miles)

P(X < 20,000) = P(
`(X-\mu)/(\sigma)` <
`(20,000-26,000)/(5,000)` ) = P(Z < -1.2) = 1 - P(Z
`\leq` 1.2)

= 1 - 0.88493 = 0.1151

Therefore, probability that a set of tires wears out before 20,000 miles is 0.1151.