Question

Suppose a batch of metal shafts produced in a manufacturing company have a population standard deviation of 1.3 and a mean diameter of 208 inches. If 60 shafts are sampled at random from the batch, what is the probability that the mean diameter of the sample shafts would differ from the population mean by more than 0.1 inches

Answer 1

To find the mean diameter and standard deviation for the sample, use the formulas for the mean and standard deviation of a sample. To find the probability that the mean diameter of the sample shafts would differ from the population mean by more than 0.1 inches, use the z-score formula and the standard normal distribution table.

Step-by-step explanation:

To find the mean diameter and standard deviation for the sample, we can use the formulas for the mean and standard deviation of a sample. The mean diameter of the sample is the same as the population mean, which is 208 inches. The standard deviation of the sample can be found using the formula:

Standard Deviation of Sample = Population Standard Deviation / Square Root of Sample Size

So, the standard deviation of the sample is 1.3 / √60 ≈ 0.168 inches.

To find the probability that the mean diameter of the sample shafts would differ from the population mean by more than 0.1 inches, we can use the z-score formula and the standard normal distribution table. The z-score formula is:

Z = (Sample Mean - Population Mean) / (Population Standard Deviation / Square Root of Sample Size)

Using this formula, we can calculate the z-score and find the probability using the standard normal distribution table.