Question

A company produces aluminum sheets for use in aerospace applications and wants to ensure that the thickness of the sheets is within tolerance. The target thickness for the sheets is 2.5 millimeters with an upper tolerance limit of 2.6 millimeters and a lower folerance limit of 2.4 millimeters. The company has been taking samples every hour and recording the thickness of each sheet. The average thickness of the sheets dher the last 30 samples is 2.51 millimeters, and the population standard deviation is known to be 0.02 millimeters.

A) Construct an appropriate hypothesis test to demonstrate that the mean thickness of the aluminum sheets is different from 2.5 millimeters.
B) Find a 95 percent confidence interval on the mean thickness of the aluminum sheets.
C) Test the hypothesis for a significance level a=0.05. What are your conclusions?
D) Assume that the company requires the thickness of the aluminum sheets to be berween 2.4 and 2.6 millimeters. What is the probability of mistakenly failing to reject the null bypothesis if the true population mean thickness is 2.55 millimeters?

Answer 1

The question is about constructing a hypothesis test for the mean thickness of aluminum sheets, calculating a confidence interval, testing the hypothesis at a 5% significance level, and finding the probability of a Type II error if the true mean thickness is different from the hypothesized mean.

The correct answer is none of all.

Step-by-step explanation:

The question involves using statistical methods to verify the quality of aluminum sheets produced by a company. It requires setting up a hypothesis test, calculating a confidence interval, testing the hypothesis at a given level of significance, and determining the probability of a Type II error in the context of quality control for manufacturing processes. Here are the steps to answer each part of the question:

• Construct an appropriate hypothesis test: The null hypothesis H0 would be μ = 2.5 mm, and the alternative hypothesis H1 would be μ ≠ 2.5 mm. This is a two-tailed test because the company wants to ensure the thickness is not different from the target, either higher or lower.
• Find a 95 percent confidence interval: The confidence interval can be calculated using the sample mean, the population standard deviation, and the z-score corresponding to the 95% confidence level.
• Test the hypothesis for a significance level of α=0.05: Calculate the test statistic (z-score) and compare it with the critical values for a two-tailed test with significance level α=0.05. Based on this comparison, we will either reject or fail to reject the null hypothesis.
• Probability of a Type II error: To calculate the probability of a Type II error (failing to reject the null hypothesis when the true mean is 2.55 mm), we need to determine where a mean of 2.55 would fall in terms of standard deviations from the hypothesized mean of 2.5 and then use the standard normal distribution to find the probability.