## 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 .