e-Statistics

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 $ \bar{X}$ and the sample standard deviation $ S$ are obtained from data of the sample size

n = ,

it is assumed that the test statistic

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

has the t-distribution with df = n-1 degrees of freedom with true population mean $ \mu$. If the true standard deviation $ \sigma$ is known, use df = +Inf (the infinity $ +\infty$).

We can calculate the critical region corresponding to the level $ \alpha$.

Level (p-value) $ \alpha =$
Right-tailed region $ T > t_{\alpha,df} =$
Two-sided region $ \vert T\vert > t_{\alpha/2,df} =$
Left-tailed region $ T < -t_{\alpha,df} =$

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 $ n$ is adequately large (as a rule of thumb it is desirable to have $ n \ge 30$).

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.


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