e-Statistics > 4480-5480 Probability and Statistics II

Hypothesis Test of Mean

The test procedure, known as t-test, is based upon the sample mean $ \bar{X} =$ and the sample standard deviation $ S =$ from a data set of size $ n =$ . Then the discrepancy between the sample mean $ \bar{X}$ and the ``assumed'' null value $ \mu_0$ of population mean is measured by the test statistic

$ T = \displaystyle\frac{\bar{X} - \mu_0}{S/\sqrt{n}} =$

Here we are interested in the plausibility of the hypothesis

$ H_A:\hspace{0.05in}\mu$ $ \mu_0 =$

regarding the ``true'' population mean $ \mu$. $ H_A$ is called an alternative hypothesis, and together with null hypothesis $ H_0$ it forms the basis of hypothesis testing. $ H_0$ becomes the opposite of $ H_A$, and is used in the context of determining whether we can reject ``$ H_0$ in favor of $ H_A$.''

The significance level $ \alpha =$ has to be chosen from $ \alpha = 0.01$ or $ 0.05$ ( $ \alpha = 0.1$ is not common in this particular test). Under the null hypothesis $ H_0$, it is ``unlikely'' that the t-statistic $ T$ lies in the critical region specified in the table below. If so, it suggests significant evidence against the null hypothesis $ H_0$.

Hypotheses Critical region to reject $ H_0$
$ H_0: \mu = \mu_0$ versus $ H_A: \mu \neq \mu_0$. $ \vert T\vert > t_{\alpha/2,n-1} =$
$ H_0: \mu \ge \mu_0$ versus $ H_A: \mu < \mu_0$. $ T < -t_{\alpha,n-1} =$
$ H_0: \mu \le \mu_0$ versus $ H_A: \mu > \mu_0$. $ T > t_{\alpha,n-1} =$

Alternatively, the p-value can be calculated so that ``p-value $ < \alpha$'' is equivalent to the t-statistic $ T$ being observed in the critical region. When the null hypothesis $ H_0$ is rejected (i.e., p-value $ < \alpha$), it is reasonable to calculate the confidence interval estimating the population mean $ \mu$.