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For a new type of​ tire, a racing car team found the average distance a set of tires would run during a race is 165 ​miles, with a standard deviation of 15 miles. Assume that tire mileage is independent and follows a Normal model. ​a) If the team plans to change tires twice during a​ 500-mile race, what is the expected value and standard deviation of miles remaining after two​ changes? ​b) What is the probability they​ won't have to change tires a third time​ (and use a fourth set of​ tires) before the end of a 500 mile​ race?

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Answer:

(A) 170, (B) 0.667

Step-by-step explanation:

Solution

From the question given, we solve for both A and B

Let X represent  the distance  a set of tires would run during a race.

Now,

(a) E ( Miles left ) = 500 - (2μ)

= 500 -  ( 2* 165) = 170

Standard deviation (SD) (Miles remaining) = 2σ/√2 = 2 * 15/√2

= 21.215

(b) P [X≥ E (Miles remaining)] = P (X≥ 170)

= 1 -P (X< 170)

= 1- P = (X  -μ /σ < 170 - μ/σ)

= 1- P ( Z < 170 - 165/15)

=1 - P (Z< 0.3333)

= 0.667

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