A study of the amount of time it takes a mechanic to rebuild the transmission for a 2010 Chevrolet Colorado shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected,
Find the probability that their mean rebuild time is less than 8.9 hours.
A) 0.4276
B) 0.9589
C) 0.9756
D) 0.9608

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate the distribution of the sample means. Since the sample size is sufficiently large (n = 40), we can assume that the sample means are normally distributed with mean μ = 8.4 and standard deviation σ/√n = 1.8/√40 = 0.284.

To find the probability that the mean rebuild time is less than 8.9 hours, we need to standardize the sample mean:

z = (x̄ - μ) / (σ/√n) = (8.9 - 8.4) / 0.284 = 1.76

Using a standard normal distribution table or calculator, we find that the probability of getting a z-score less than 1.76 is approximately 0.9608. Therefore, the probability that the mean rebuild time is less than 8.9 hours is 0.9608. The best choice is D.

A Study of the Amount of Time It Takes a Mechanic