Final answer:
To compute probabilities regarding the sample mean using the normal model, a sample size of at least 30 is typically considered sufficient. To find the probability that a sample mean time is less than 10 minutes, use the Z-score formula. To find the mean oil-change time with a 10% probability, find the corresponding Z-score.
Step-by-step explanation:
(a) To compute probabilities regarding the sample mean using the normal model, you would need a sample size that is large enough to satisfy the Central Limit Theorem. According to the Central Limit Theorem, a sample size of at least 30 is typically considered sufficient for the normal distribution approximation.
(b) To find the probability that a random sample of n = 40 oil changes results in a sample mean time of less than 10 minutes, you can use the Z-score formula. First, calculate the Z-score using the formula Z = (X - μ) / (σ / √n), where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Once you have the Z-score, you can use a Z-table or a calculator to find the corresponding probability.
(c) To find the mean oil-change time that would have a 10% chance of being at or below, you need to find the Z-score that corresponds to a 10% probability in the left tail of the standard normal distribution. Once you have the Z-score, you can use the formula Z = (X - μ) / (σ / √n) to solve for X, the mean oil-change time.