Question

Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 125125 millimeters, and a variance of 4949. If a random sample of 4141 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 2.22.2 millimeters? Round your answer to four decimal places.

Answer 1

a. The mean diameter for the sample is 125 millimeters and the standard deviation is approximately 1.195 millimeters. b. The probability that 50 randomly selected screws will be within the stated tolerance levels is approximately 0.0338. The company's diameter claim is plausible.

Step-by-step explanation:

a. To find the mean diameter for the sample, we can use the fact that the sample mean is equal to the population mean. So the mean diameter for the sample will also be 125 millimeters.

To find the standard deviation for the sample, we can use the fact that the standard deviation for the sample is equal to the square root of the population variance divided by the sample size. The standard deviation for the sample will be sqrt(49/41) ≈ 1.195 millimeters.

b. To find the probability that 50 randomly selected screws will be within the stated tolerance levels (differ from the mean by less than or equal to 2.2 millimeters), we can use the standard normal distribution.

1. Calculate the z-score for the upper tolerance level: (2.2 - 0) / 1.195 ≈ 1.8408
2. Look up the z-score in the standard normal distribution table. The area to the left of z = 1.8408 is approximately 0.9662.
3. To find the probability of being within the tolerance level, subtract the area to the left of z = 1.8408 from 1: 1 - 0.9662 = 0.0338.

The probability that 50 randomly selected screws will be within the stated tolerance levels is approximately 0.0338. Since this probability is quite low, it suggests that the company's diameter claim is plausible.