Reference no: EM132400918

Michelin Tire Company -

Founded in 1889, the French Michelin Tire Company is one of the three largest tire manufacturers in the world with 113,000 employees worldwide and $20+ billion in annual revenue. The largest tire the company makes is the "Earthmover", a 10,000 pound behemoth used, for example, by heavy-duty mining equipment.

It is challenging to build such a large tire and for years the company has relied on a special proprietary blend of several different types of rubber. This blend has worked fine for many years, but the company's research and development team has concocted a new synthetic rubber that they claim will last longer than current tires.

You have calculated that the switch to synthetic rubber would be profitable for the company if the new tires have a mean lifespan of 7,500 miles or more. The plan is to first do a "pilot" study, manufacturing a sample of 40 new tires under carefully- controlled conditions. Then, based on the results of the pilot, you will decide whether or not to switch to synthetic rubber.

a) In this problem what is a Type I error? What is a Type II error? Use non-technical language.

b) In terms of the pilot sample's mean lifespan in miles, what decision rule ("action threshold") should you use? Use a 10% significance level and assume that you know that the population standard deviation (σ) across individual tires is 1,000 miles. Provide a clear statement of your decision-making rule using both numerical values and words.

c) What is the power of this test when the true population mean lifespan of the new tires is 8,000 miles? Use the decision rule you determined in part (b), and continue to assume that the standard deviation across individual tires in the population (σ) is 1,000 miles at the mean value of 8,000.

d) Draw a sketch of the normal distribution curve that illustrates the power of the test when the true population mean is 8,000 miles with a standard deviation of 1,000 miles. Your figure should correspond as closely as possible to part (c) above, including continuing to use the decision rule (action threshold) from part (b). Your figure should indicate the numerical value of the population mean assumed to be true, and the numerical value of the decision rule (action threshold), and you should clearly label the area corresponding to the power of the test.

e) Your manager does not like your decision rule. She tells you that she wants the company to use a 5% significance level, and she also wants there to be only a 5% probability of a Type II error when the true mean lifespan of the new tires is 8,000 miles. Assume that the standard deviation across individual tires in the population (σ) is 1,000 miles at the mean value of 7,500 and also at a mean value of 8,000. What is the minimum sample size you would need for a test that would meet these specifications? Show your work.