The Sledge Tire and Rubber Company plans to warranty its new mountain bike tire for 12 months. However, before it does this, the company wants to be sure that the mean lifetime of the tires is at least 18 months under normal operations. It will put the warranty in place unless the sample data strongly suggest that the mean lifetime of the tires is less than 18 months. The company plans to test this statistically using a random sample of tires. The test will be conducted using an alpha level of 0.03.
a. If the population mean is actually 16.5 months, determine the probability the hypothesis test will lead to incorrectly failing to reject the null hypothesis. Assume that the population standard deviation is known to be 2.4 months and the sample size is 60.
b. If the population mean is actually 17.3, calculate the chance of committing a Type II error. This is a specific example of a generalization relating the probability of committing a Type II error and the parameter being tested. State this generalization.
c. Without calculating the probability, state whether the probability of a Type II error would be larger or smaller than that calculated in part b if you were to calculate it for a hypothesized mean of 15 months. Justify your answer.
d. Suppose the company decides to increase the sample size from 60 to 100 tires. What can you expect to happen to the probabilities calculated in part a?