Question: The Power of a Test The probability of a type II error, b, represents the likelihood of accepting the null hypothesis when the alternative is true. The power of a statistical test is π = 1 - β, the probability of (correctly) rejecting the null hypothesis, of detecting a difference in the hypothesized value of the population parameter. Try the Statistical Power applet on the text website. A hot torsion test is used to determine the workability of a metal. Suppose a carbon steel rod is designed to fail (break) with mean axial load of 800 newtons (N) (under certain temperature and speed conditions). A random sample of 25 rods is obtained, and the axial load failure is measured for each. Suppose the underlying distribution of axial load is normal and σ = 50.
a. Consider a hypothesis test of H0: μ = 800 versus H0: μ, 800 with α = 0.025. Find the probability of a type II error and the power of this test for μ = 775; that is, find β(775) and π(775).
b. Use a calculator or computer to compute the power, p(ma), for μa = 730, 735, 740, . . . , 800.
c. Use the values from part (b) to carefully sketch a plot of p(ma) versus ma. The resulting plot is called a power curve.