The mileage that car owners get with a certain kind of radial tire is normally d

Question

The mileage that car owners get with a certain kind of radial tire is normally distributed with a mean of 60,000 miles and standard deviation of 5,000 miles.

a. Calculate the probability that a randomly selected tire of this type will last more than 60,000 miles. Show the necessary steps. [ 2points]

b. Calculate the probability that a randomly selected tire of this type will last less than 43,000 miles. Show the necessary steps. [2 points]

c. Calculate the probability that a randomly selected tire of this type will last between 55,000 and 60,000 miles. Show the necessary steps. [2 points]

d. What is the tire mileage above which 80 percent of these tires will last? Show the necessary steps. [2 points]

e. What is the tire mileage below which 60 percent of these tires will last? Show the necessary steps. [2 points]

f. If the worst performing 5% of these tires are eligible for free-replacement under the company’s product warrantee policy, what is the cutoff level of tire duration for free-replacement? Please show how you arrived at your answer.

Explanation / Answer

a) P(X>60000)=0.5 as 60000 lies on mean and normal distribution have equal probabilty on both side of the curve.

b)P(X<43000)=P(Z<(43000-60000)/5000)=P(Z<-3.4)=0.0003

c)P(55000<X<60000)=P(-1<Z<1)=0.5-0.1587=0.3413

d)for 80%, tires to last, zscore at 20 percentile =-0.8416

hence corresponding mileage =mean+z*std deviation =55791.9

e)for 60% below., z score at 60 percentile=0.2533

hence corresponding mileage =mean+z*std deviation =61266.7

f)for 5%, z=-1.6449

hence corresponding mileage =mean+z*std deviation =51775.7

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