Final answer:
To find the probability that a sample mean of tire mileage differs from the population mean by less than 166 miles, calculate the standard error of the mean, determine Z-scores for the mileage limits, and look up the Z-scores in a standard normal distribution table to find the cumulative probabilities. Subtract the smaller probability from the larger to get the final probability.
Step-by-step explanation:
The student is asking about the probability that a sample mean would differ from the population mean of tire mileage by less than 166 miles, given a sample of 33 tires. To address this, we will use the standard normal distribution, as it can be applied to sample means using the Central Limit Theorem when the sample size is sufficiently large (typically n >= 30).
We calculate the standard error of the mean (SEM) which is the standard deviation of the sample mean distribution, using the formula SEM = σ/√n, where σ represents the population standard deviation and n is the sample size. For this problem, σ = 2536 miles and n = 33 tires, so SEM = 2536 / sqrt(33) = 441.6 miles (approx).
Next, we use the Z-score formula to convert the mileage difference into a standardized score: Z = (X - μ) / SEM, where X is the value for which we want to find the probability, μ the population mean, and SEM the standard error of the mean. We are looking at a difference of less than 166 miles, so we calculate the Z-score for both 37,558 (37724+166) and 37,558 (37724-166). Then, we would look up these Z-scores in the standard normal distribution table to find the probabilities.
The final probability is the difference in the cumulative probabilities of these two Z-scores. This value represents the likelihood that the sample mean falls within 166 miles of the population mean. However, to provide an exact numerical probability, one would need to use software or a standard normal distribution table. Assuming we have calculated these probabilities, we would then present a calculated value rounded to four decimal places.