Question

The operation manager at a tire manufacturing company believes that the mean mileage of a tire is 25,835 miles, with a variance of 18,207,290.

What is the probability that the sample mean would differ from the population mean by less than 104 miles in a sample of 89 tires if the manager is correct? Round your answer to four decimal places.

Answer 1

Answer: 0.9641

Step-by-step explanation:

We can use the central limit theorem to approximate the distribution of the sample mean. The mean of the sample mean is equal to the population mean, and the variance of the sample mean is equal to the population variance divided by the sample size:

mean of sample mean = 25,835
variance of sample mean = 18,207,290 / 89 = 204,618.2

To find the probability that the sample mean differs from the population mean by less than 104 miles, we can standardize the difference using the central limit theorem:

Z = (X - μ) / (σ / sqrt(n))
Z = (104) / (sqrt(204,618.2) / sqrt(89))
Z = 1.807

We can use a standard normal distribution table or calculator to find the probability that Z is less than 1.807. This probability is 0.9641 when rounded to four decimal places. Therefore, the probability that the sample mean would differ from the population mean by less than 104 miles in a sample of 89 tires if the manager is correct is 0.9641.

Answer 2

To calculate the probability that the sample mean would differ from the population mean by less than 104 miles, we can use the Central Limit Theorem and the standard error formula.

The standard error (SE) is calculated by dividing the population standard deviation by the square root of the sample size:

SE = √(population variance / sample size)
= √(18,207,290 / 89)
≈ 425.1928

Next, we calculate the z-score, which measures the number of standard errors the sample mean differs from the population mean:

z = (sample mean - population mean) / SE
= (25,835 - 25,835) / 425.1928
= 0

Since the difference between the sample mean and population mean is 0, the z-score is 0.

To find the probability that the sample mean would differ by less than 104 miles, we calculate the area under the standard normal distribution curve to the left of the z-score of 0. This can be done using a standard normal distribution table or a statistical calculator.

The probability is equal to 0.5000, which means there is a 50% chance that the sample mean would differ from the population mean by less than 104 miles in a sample of 89 tires.

Please note that this calculation assumes a normal distribution and the use of the Central Limit Theorem.