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. Round to four decimal places.
A. 0.4276
B. 0.9605
C. 0.9756
D. 0.9589

Step 1 ：We are given a problem of probability involving normal distribution. The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution. This theorem applies here as we are dealing with a sample size of 40 mechanics which is sufficiently large.

Step 2 ：We are given the population mean (\(\mu = 8.4\) hours), the population standard deviation (\(\sigma = 1.8\) hours), the sample size (\(n = 40\) mechanics), and we are asked to find the probability that the sample mean is less than 8.9 hours.

Step 3 ：To solve this, we need to standardize the sample mean using the Z-score formula for sample means, which is \(Z = \frac{X - \mu}{\sigma/\sqrt{n}}\), where X is the sample mean. The Z-score tells us how many standard deviations an element is from the mean.

Step 4 ：After finding the Z-score, we can use a Z-table or a function from a software package to find the probability that a Z-score is less than the calculated value.

Step 5 ：Substituting the given values into the Z-score formula, we get \(Z = \frac{8.9 - 8.4}{1.8/\sqrt{40}} = 1.7568209223157665\).

Step 6 ：Using a Z-table or a function from a software package, we find that the probability that a Z-score is less than 1.7568209223157665 is 0.9605.

Step 7 ：Final Answer: The probability that the mean rebuild time is less than 8.9 hours for a sample of 40 mechanics is \(\boxed{0.9605}\).