The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 2800 miles. What warranty
Should the company use if they want 96% of the tires to outlast the warranty?
A)64,900 miles
B)62,800 miles
C)55,100 miles
D)57,200 miles

Question

Quizplus · Accepted Answer

Given that the tread life of the tires follows a normal distribution with a mean (μ) of 60,000 miles and a standard deviation (σ) of 2800 miles, and the company wants 96% of the tires to outlast the warranty, we are essentially looking for a point below which 96% of the distribution falls. This corresponds to the 4th percentile of the distribution (100% - 96% = 4%), because we want only 4% of the tires to fail before the warranty expires. Using the Z-score formula, where Z = (X - μ) / σ, and knowing that the Z-score for the 4th percentile is approximately -1.75 (from standard Z-tables), we can solve for X, the warranty mileage. X = Z*σ + μ = -1.75*2800 + 60,000 ≈ 55,100 miles. This means the company should set the warranty at a point where approximately 96% of the tires will outlast it, which is around 55,100 miles.

The Tread Life of a Particular Brand of Tire Is