e-Statistics

t-Distribution

Let $\mu$ be the true population mean of interest. When the sample mean $\bar{X}$ and the standard error $S/\sqrt{n}$ are obtained from data of size n = , it is assumed that the standardized score (test statistic)

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

has the t-distribution with degrees of freedom (df). 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 df.

Provided df = we can calculate the critical region (right-tailed, two-sided, or left-tailed) corresponding to the significance level $\alpha =$ . The numerical value $t_{\alpha,df}$ of right-tailed critical region is called critical value.

Critical region	T is extreme when
Right-tailed	$T > t_{\alpha,df} =$
Two-sided	$\vert T\vert > t_{\alpha/2,df} =$
Left-tailed	$T < -t_{\alpha,df} =$

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

Critical values can be obtained by t-distribution table. The appropriateness of t-distribution 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 $n \ge 30$ ). If the true standard deviation $\sigma$ is known, use df = +Inf (the infinity $+\infty$ ).