Final answer:
To find the probability of a sample mean being less than ten minutes, calculate the standard error, find the z-score, and then refer to the standard normal distribution table to find the associated probability.
Step-by-step explanation:
The question asks for the probability that a random sample of N = 40 oil changes results in a sample mean time less than ten minutes, given that the population mean time is 11.1 minutes with a standard deviation of 4.7 minutes. To solve this, we can use the Central Limit Theorem which tells us that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, even if the population distribution is not normal.
We first calculate the standard error of the mean by dividing the population standard deviation by the square root of the sample size (N).
Standard Error = 4.7 / √40
Then we calculate the z-score for the sample mean of ten minutes using the following formula:
Z = (Sample mean - Population mean) / Standard Error
Z = (10 - 11.1) / Standard Error
After finding the z-score, we refer to the standard normal distribution table to find the probability that Z is less than this z-score, which gives us the probability that the sample mean is less than ten minutes.