1. To help the manufacturer get a clear picture of type I and type II error probabilities, draw a β versus α chart for sample sizes of 30, 40, 60, and 80. If β is to be at most 1% with α = 5%, which sample size among these four values is suitable?

2. Calculate the exact sample size required for α = 5% and β = 1%. Construct a sensitivity analysis table for the required sample size for μ ranging from 2,788 to 2,794 psi and β ranging from 1% to 5%.

3. For the current practice of n = 40 and α = 5% plot the power curve of the test. Can this chart be used to convince the manufacturer about the high probability of passing batches that have a strength of less than 2,800 psi?

4. To present the manufacturer with a comparison of a sample size of 80 versus 40, plot the OC curve for those two sample sizes. Keep an α of 5%.

5. The manufacturer is hesitant to increase the sample size beyond 40 due to the concomitant increase in testing costs and, more important, due to the increased time required for the tests. The production process needs to wait until the tests are completed, and that means loss of production time. A suggestion is made by the production manager to increase α to 10% as a means of reducing β. Give an account of the benefits and the drawbacks of that move. Provide supporting numerical results wherever possible.

When a tire is constructed of more than one ply, the interply shear strength is an important property to check. The specification for a particular type of tire calls for a strength of 2,800 pounds per square inch (psi). The tire manufacturer tests the tires using the null hypothesis where μ is the mean strength of a large batch of tires.

H0: μ ≥ 2,800 psi

From past experience, it is known that the population standard deviation is 20 psi.

Testing the shear strength requires a costly destructive test and therefore the sample size needs to be kept at a minimum. A type I error will result in the rejection of a large number of good tires and is therefore costly. A type II error of passing a faulty batch of tires can result in fatal accidents on the roads, and therefore is extremely costly. (For purposes of this case, the probability of type II error, β, is always calculated at μ = 2,790 psi.) It is believed that β should be at most 1%. Currently, the company conducts the test with a sample size of 40 and an α of 5%.