Standard +0.3 This question tests understanding of hypothesis testing concepts and basic calculations with normal distributions. Part (i) is definitional recall, parts (ii) and (iv) require conceptual understanding to sketch graphs, and part (iii) involves straightforward application of normal distribution with known variance—all standard S4 material requiring no novel insight.
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).
3 (i) Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.\\
(ii) 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.\\
(iii) 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.\\
(iv) Without carrying out any further calculations, draw a sketch of the operating characteristic for the test in part (iii).
\hfill \mbox{\textit{OCR MEI S4 2013 Q3 [24]}}