## t-Distribution

The t distribution is symmetric but comparatively flatter (see the solid line in the graph below) than the standard normal distribution (the dashed line below). The shape of particular t-distribution is determined by the degrees of freedom (df) = . When the sample mean and the sample standard deviation are obtained from the data of observations, it is often assumed that the test statistic has the t-distribution with degrees of freedom with true population mean . If the true standard deviation is known, use df = +Inf (the infinity ).

We can calculate the critical region corresponding to the level .

Level (p-value) | |

Right-tailed region | |

Two-sided region | |

Left-tailed region |

The appropriateness of this calculation can be ensured if (a) the sample distribution is approximately normal (the use of QQ plot is recommended), or (b) the sample size is adequately large (as a rule of thumb it is desirable to have ).

Conversely when the statistic is given, we can find the corresponding so that the value belongs to the critical region, and call it p-value.