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 we select a sample of 40 tires and use a simulator to determine the tread life. What is the likelihood of finding that the sample mean is between 59,050 and 60,950?
A)0.4332
B)0.8664
C)1.00
D)Very likely

Question

Quizplus · Accepted Answer

To solve this problem, we will first standardize the sample mean using the formula:

z = (x̄ - μ) / (σ / √n)

where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Substituting the given values, we get:

z = (59,050 - 60,000) / (4,000 / √40) = -2.1213
z = (60,950 - 60,000) / (4,000 / √40) = 2.1213

Next, we will use a standard normal distribution table or calculator to find the area between these two z-scores. Alternatively, we can subtract the cumulative normal probabilities from the z-score of 2.1213 and the z-score of -2.1213 (since the normal distribution is symmetric) to get the area between them.

Using either method, we get:

Area = 0.8664

Therefore, the likelihood of finding that the sample mean is between 59,050 and 60,950 is 0.8664 or 86.64%. The best choice is B, as it matches this value.

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