Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
A test is to be carried out concerning a parameter \(\theta\). The null hypothesis is that \(\theta\) has the particular value \(\theta _ { 0 }\). The alternative hypothesis is \(\theta \neq \theta _ { 0 }\). Draw a sketch of the operating characteristic for a perfect test that never makes an error.
The random variable \(X\) is distributed as \(\mathrm { N } ( \mu , 9 )\). A random sample of size 25 is available. The null hypothesis \(\mu = 0\) is to be tested against the alternative hypothesis \(\mu \neq 0\). The null hypothesis will be accepted if \(- 1 < \bar { x } < 1\) where \(\bar { x }\) is the value of the sample mean, otherwise it will be rejected. Calculate the probability of a Type I error. Calculate the probability of a Type II error if in fact \(\mu = 0.5\); comment on the value of this probability.
Without carrying out any further calculations, draw a sketch of the operating characteristic for the test in part (iii).