Final answer:
To determine the probability of finding a sample mean of 58,500 miles or less, we calculate the standard error using the formula, then use the z-score formula to find the probability. The calculated probability is less than the significance level of 0.05, indicating that the data is inconsistent with the manufacturer's claim.
Step-by-step explanation:
To determine the probability of finding a sample mean of 58,500 miles or less, we need to use the concept of sampling distributions. The mean of the sampling distribution of the sample mean is equal to the population mean. In this case, the population mean is 60,000 miles and the standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size.
Using the formula: standard error = standard deviation / sqrt(sample size), we can calculate the standard error as follows: standard error = 4000 / sqrt(16) = 4000 / 4 = 1000 miles.
Next, we can use the z-score formula: z = (sample mean - population mean) / standard error. Plugging in the values for our problem, we have: z = (58500 - 60000) / 1000 = -1.5.
We can then find the probability using a z-table or calculator. The probability of finding a sample mean of 58,500 miles or less is approximately 0.0668, which is less than 0.05. Since the probability is less than the significance level of 0.05, we can reject the manufacturer's claim and conclude that the data is inconsistent with the claim.