Answer:This problem involves the sampling distribution of the sample mean. We are given that the population mean is μ = 8.4 hours and the population standard deviation is σ = 1.8 hours. We want to find the probability that the mean rebuild time for a sample of 40 mechanics exceeds 8.7 hours.
The sampling distribution of the sample mean is approximately normal by the central limit theorem, as long as the sample size is sufficiently large (n >= 30 in most cases).
The mean and standard deviation of the sampling distribution of the sample mean can be calculated as follows:
μ_xbar = μ = 8.4 hours
σ_xbar = σ / sqrt(n) = 1.8 / sqrt(40) = 0.2843 hours
To find the probability that the sample mean exceeds 8.7 hours, we need to standardize the sample mean using the formula:
z = (xbar - μ_xbar) / σ_xbar
where xbar is the sample mean. Substituting the values, we get:
z = (8.7 - 8.4) / 0.2843 = 1.054
Using a standard normal distribution table or a calculator, we can find that the probability of a standard normal variable exceeding z = 1.054 is 0.1486.
Therefore, the probability that the mean rebuild time for a sample of 40 mechanics exceeds 8.7 hours is approximately 0.1486 or 14.86%.
Step-by-step explanation: