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?

Question

Mathematics

Jason Burton

14 August, 09:42

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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Answers (1)

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Rocky · Answer 1

(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