Question

The operation manager at a tire manufacturing company believes that the mean mileage of a tire is 37,724 miles, with a standard deviation of 2536 miles. What is the probability that the sample mean would differ from the population mean by less than 166 miles in a sample of 33 tires if the manager is correct?

Answer 1

To find the probability that the sample mean would differ from the population mean by less than 166 miles in a sample of 33 tires, use the standard deviation and sample size to standardize the sample mean and find the probability using the standard normal distribution.

Step-by-step explanation:

To find the probability that the sample mean would differ from the population mean by less than 166 miles in a sample of 33 tires, we can use the standard deviation of the population and the sample size.

Since the data is normally distributed and the sample size is large (n = 33), we can use the central limit theorem to approximate the sampling distribution of the sample mean.

We can then standardize the sample mean 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.

Finally, we can use the standard normal distribution to find the probability by finding the area under the curve between -166 and 166. We can do this by subtracting the cumulative probability of Z = -166 from the cumulative probability of Z = 166.

Using a standard normal distribution table or a calculator, the probability is approximately 0.9693.