Question

A tire company has developed a new type of steel-belted radial tire. Extensive testing indicates the population of mileages obtained by all tires of this new type is normally distributed with a mean of 39,531 miles and a standard deviation of 4,075 miles. The company wishes to offer a guarantee providing a discount on a new set of tires if the original tires purchased do not exceed the mileage stated in the guarantee. What should the guaranteed mileage be if the tire company desires that no more than 2 percent of the tires will fail to meet the guaranteed mileage? (Use a z-score to 2 decimal places in your intermediate calculations. Round your final answer to the nearest mile.)

Answer 1

Solution:

Given: The population of mileages obtained by all tires of this new type is normally distributed with a mean of 39,531 miles and a standard deviation of 4,075 miles.

We have to find value of x = the guaranteed mileage such that:

no more than 2 percent of the tires will fail to meet the guaranteed mileage

That is:

Thus find z such that:

Look in z  table for Area = 0.0200 or its closest area and find corresponding z value.

Area 0.0202 is closest to 0.0200 and it corresponds to -2.0 and 0.05

thus z =-2.05

Now use following formula to find x value:

Thus the guaranteed mileage = x = 31177 miles.