The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a population mean of 60,000 miles and a population standard deviation of 4,000 miles. Suppose you bought a set of four tires; what is the likelihood the mean tire life of these four tires is more than 66,000 miles?
A)0.0013
B)0.9987
C)0.4987
D)0.9544

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate the distribution of the sample means. The mean of the sample means is equal to the population mean, 60,000 miles. The standard deviation of the sample means, also known as the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size, or 4,000/sqrt(4) = 2,000 miles.

We want to find the probability that the mean tire life of these four tires is more than 66,000 miles. We can standardize this value using the formula:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Plugging in the values, we get:

z = (66,000 - 60,000) / (4,000 / sqrt(4))
z = 3

Using a standard normal distribution table or calculator, we can find that the probability of getting a z-score of 3 or higher is approximately 0.0013. Therefore, the likelihood that the mean tire life of these four tires is more than 66,000 miles is 0.0013 or 0.13%.

The Tread Life of Tires Mounted on Light-Duty Trucks Follows