Lecture 23: Tests Based on the Central Limit Theorem

Suppose that we wish to test a hypothesis about the population mean, which will denote by $\mu$, and we are interested in whether or not $\mu = \mu_0$. For the moment, let us ignore the structure of the alternative hypothesis. Suppose that when the null hypothesis, $H_0 = \{\mu = \mu_0\}$ is true, not only do we know that the population mean is $\mu_0$, but we also know that the population variance is $\sigma_0^2$. Then a test of hypothesis based on a random sample $(Y_1,\dots, Y_N)$ can be constructed using the statistic

$$T(Y_1,\dots,Y_N) = \frac{Y_1 + \dots + Y_N-N\mu_0}{\sqrt{N\sigma_0^2}}.$$

The test is of the general form:

• Accept the null hypothesis if $T(Y_1,\dots,Y_N)\in (a,b)$;
• Reject the null hypothesis if $T(Y_1,\dots,Y_N)\in (a,b)^c$;

where $-\infty \leq a < 0 < b \leq \infty$. As usual, we choose a=-b if we want a two-sided test, and either $a=-\infty$ or $b = \infty$ if we want a one-sided test. The correct values are chosen by specifying a Type I error probability and then using the Central Limit Theorem to approximate the distribution of $T(Y_1,\dots,Y_N)$ as standard normal if the null hypothesis is true. It will be impossible to find Type II error probabilities if the alternative hypothesis does not shed any light on the population variance if the alternative hypothesis is true.

In the case of testing for binomial and Poisson populations, there is only one parameter, and specifying it specifies all moments. The general case is not so simple. To be quite precise, the null hypothesis should be of the form ``the mean is $\mu_0$ and the variance is $\sigma_0^2$, and the alternative hypothesis is some subset of the complement of the null hypothesis. Most commonly the alternative hypothesis is of the form: `` the mean is ... and the variance is $\sigma^2_0$. This is usually refered to as testing about the mean when the variance is known. In such cases the probability of Type II error is a function of a single variable.