The tread life of a particular brand of tire is a random variable best described by a normal distribution with amean of 60,000 miles and a standard deviation of 3000 miles. What warranty should the company use if theywant 96% of the tires to outlast the warranty?
A) 54,750 mi
B) 65,250 mi
C) 57,000 mi
D) 63,000 mi

Question

Quizplus · Accepted Answer

To ensure 96% of the tires outlast the warranty, the warranty should be set at a point where only 4% of the tires fail before reaching it. This corresponds to the lower 4% tail of the normal distribution. Using the Z-score formula, $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value we seek, $\mu = 60,000$ miles is the mean, and $\sigma = 3,000$ miles is the standard deviation. The Z-score for 96% in a normal distribution is approximately -1.75 (since we are looking at the lower tail). Solving for $X$ gives us $X = \mu + Z\sigma = 60,000 + (-1.75)(3,000) = 60,000 - 5,250 = 54,750$ miles.

The Tread Life of a Particular Brand of Tire Is