e-Statistics

Hypothesis Test

Here we are interested in the plausibility of the hypothesis

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

regarding the “true” population mean $\mu$ . is called an alternative hypothesis, and together with the null value $\mu_0$ it forms the basis of hypothesis testing. The null hypothesis $H_0: \mu = \mu_0$ is used in the context of rejecting “ in favor of .”

The test procedure, known as t-test, is based on the sample mean $\bar{X}$ = and the sample standard deviation = from data of sample 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

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

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 , it is “unlikely” that the t-statistic lies in the critical region obtained from t-distribution and summarized in the table below. If so, it suggests significant evidence against the null hypothesis in favor of .

Critical region	Alternative	Reject if
1. Two-sided	$H_A: \mu \neq \mu_0$	$\vert T\vert > t_{\alpha/2,n-1}$ =
2. Left-tailed	$H_A: \mu < \mu_0$	$T < -t_{\alpha,n-1}$ =
3. Right-tailed	$H_A: \mu > \mu_0$	$T > t_{\alpha,n-1}$ =

Alternatively,

p-value =

can be calculated so that “p-value < $\alpha$ ” is equivalent to the t-statistic being observed in the critical region. When the null hypothesis is rejected (i.e., p-value < $\alpha$ ), it is reasonable to calculate the Confidence Interval estimating the population mean $\mu$ .