Question

Let μ denote the true average tread life of a certain type of tire. consider testing h0: μ = 30,000 versus ha: μ > 30,000 based on a sample of size n = 16 from a normal population distribution with σ = 1500. a test with α = 0.01 requires zα = z0.01 = 2.33. the probability of making a type ii error when μ = 31,000 is

Answer 1

The probability of making a type II error is given by one minus the power of the hypothesis test.

In general for the alternative hypothesis , $H_a:\mu\ \textgreater \ \mu_0$

The power of a hypothesis test is given by:

$\beta(\mu')=\phi\left(X \ \textless \ \mu_0+z_{1-\alpha}\frac{\sigma}{\sqrt{n}}|\mu'\right) \\ \\ =\phi\left(z_{1-\alpha}+\frac{\mu_0-\mu'}{\sigma/\sqrt{n}}\right) =\phi\left(z_{1-0.01}+\frac{30000-31000}{1500/\sqrt{16}}\right) \\ \\ =\phi\left(z_{0.99}+\frac{-1000}{1500/4}\right)=\phi\left(2.33-\frac{1000}{375}\right)=\phi(2.33-2.667) \\ \\\phi(-0.33)=0.3682$

The probability of making a type ii error when μ = 31,000 is given by

$1-\beta(\mu')=1-0.3682 \\ \\ =0.6318$