Refer to Exercise 7.137.
The EPA sets an airborne limit of 5 parts per million (ppm) on vinyl chloride, a colorless gas used to make plastics, adhesives, and other chemicals. It is both a carcinogen and a mutagen (New Jersey Department of Health, Hazardous Substance Fact Sheet, 2010). A major plastics manufacturer, attempting to control the amount of vinyl chloride its workers are exposed to, has given instructions to halt production if the mean amount of vinyl chloride in the air exceeds 3.0 ppm. A random sample of 50 air specimens produced the following statistics: x̅ = 3.1 ppm, s = .5 ppm.
a. In the context of the problem, define a Type II error.
b. Calculate β for the test described in part a of Exercise 7.137, assuming that the true mean is μ = 3.1 ppm.
c. What is the power of the test to detect a departure from the manufacturer's 3.0 ppm limit when the mean is 3.1 ppm?
d. Repeat parts b and c, assuming that the true mean is 3.2 ppm. What happens to the power of the test as the plant's mean vinyl chloride level departs further from the limit?