Question

A bearing used in an automotive application is required to have a nominal inside diameter of 1.5 inches. Arandom sample of 25 bearings is selected and the average inside diameter of these bearings is 1.4975 inches. Bearing diameter is known to be normally distributed with standard deviation σ=0.01 inch. (a) Test the hypotheses H 0:μ=1.5 versus H 1 :μ

=1.5 using α=0.01. The true mean hole diameter significantly different from 1.5 in. at α=0.01. (b) What is the P-value for the test in part (a)? P.value = Round your answer to two decimal places (e.g. 98.76 ). (c) Compute the power of the test if the true mean diameter is 1.495 inches. Power of the test = Round your answer to two decimal places (e.g. 98.76). (d) What sample size would be required to detect a true mean diameter as low as 1.495 inches if we wanted the power of the test to be at least 0.88 ? bearings

Answer 1

The hypothesis testing involves a two-tailed test with the null hypothesis that the true mean diameter is 1.5 inches. The test statistic is calculated using the sample mean, standard deviation, and alpha level, with the power of the test determined based on the true mean diameter of 1.495 inches and a desired power level of 0.88.

Step-by-step explanation:

Hypothesis Testing for Bearing Diameter

The hypothesis testing procedure involves comparing the sample mean to the assumed population mean to see if there is a significant difference. Given the normal distribution of bearing diameters with a standard deviation (σ) of 0.01 inch, we can conduct a two-tailed test to determine if the true mean (μ) is significantly different from the nominal size.

The null hypothesis (H0) claims that the true mean diameter (μ) is 1.5 inches, and the alternative hypothesis (H1) suggests that the true mean (μ) is not equal to 1.5 inches. The sample mean is 1.4975 inches, and the given standard deviation is 0.01 inch for a sample size of 25. Using α=0.01, we calculate the test statistic and compare it to the critical value from the normal distribution table or use a statistical software to find the p-value.

To compute the power of the test when the true mean diameter is 1.495 inches, we need to consider the probability of correctly rejecting the null hypothesis. This involves finding the non-centrality parameter and using software or a power table.

For part (d), determining the sample size required to achieve a specific power level can be done through the calculation of effect size and again using software or power analysis tables to find the appropriate sample size for the given power of 0.88.