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 and the sample standard deviation from a data set of size . Then the discrepancy between the sample mean and the assumed'' null value of population mean is measured by the test statistic

Here we are interested in the plausibility of the hypothesis

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

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

 Hypotheses Critical region to reject versus . versus . versus .

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