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 \(\boxed{0.9605}\).