Calculations for the t-Test: One Sample

The following is a discussion on explicit expressions that can be used to carry out various $t$-tests. In each case, the formula for a test statistic that either exactly follows or closely approximates a $t$-distribution under the null hypothesis is given. Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a one-tailed test or a two-tailed test.

Once a $t$-value is determined, a $p$-value can be found using a table of values from Student's $t$-distribution. If the calculated $p$-value is below the threshold chosen for statistical significance (usually the $0.10$, the $0.05$, or $0.01$ level), then the null hypothesis is rejected in favor of the alternative hypothesis.

One-Sample T-Test

In testing the null hypothesis that the population mean is equal to a specified value $\mu_0$, one uses the statistic:

$t=\dfrac { \bar { x } -{ \mu }_{ 0 } }{ s/\sqrt { n } }$

where $\bar { x }$ is the sample mean, $s$ is the sample standard deviation of the sample and $n$ is the sample size. The degrees of freedom used in this test is $n-1$.

Slope of a Regression

Suppose one is fitting the model:

${ Y }_{ i }=\alpha +\beta { x }_{ i }+{ \varepsilon }_{ i }$

where $x_i, i=1, \cdots, n$ are known, $\alpha$ and $\beta$ are unknown, and $\varepsilon_i$ are independent identically normally distributed random errors with expected value $0$ and unknown variance $\sigma^2$, and $Y_i,i=1,\cdots,n$ are observed. It is desired to test the null hypothesis that the slope $\beta$ is equal to some specified value $\beta_0$ (often taken to be $0$, in which case the hypothesis is that $x$ and $y$ are unrelated). Let $\hat{\alpha}$ and $\hat{\beta}$ be least-squares estimators, and let $SE_\hat{\alpha}$ and $SE_\hat{\beta}$, respectively, be the standard errors of those least-squares estimators. Then,

$t\text{-score} = \dfrac{\hat{\beta}-\beta_0}{SE_\hat{\beta}} \sim \tau_{n-2}$

has a $t$-distribution with $n − 2$ degrees of freedom if the null hypothesis is true. The standard error of the slope coefficient is:

$\displaystyle{SE_{\hat{\beta}}=\frac{\sqrt{\frac{1}{n-2} \sum_{i=1}^n \left(Y_i - \hat{y}_i \right) ^2}}{\sqrt{\sum_{i=1}^n \left( x_i - \bar{x} \right) ^2}}}$

can be written in terms of the residuals $\hat{\varepsilon}_i$:

$\hat{\varepsilon}_i = Y_i - \hat{y}_i - (\hat{\alpha} + \hat{\beta}x_i )$

Therefore, the sum of the squares of residuals, or $SSR$, is given by:

$\displaystyle{SSR = \sum_{i=1}^n \hat{\varepsilon}_i^2}$

Then, the $t$-score is given by:

$\displaystyle{t = \frac{ \left( \hat{\beta} - \beta_0\right) \sqrt{n-2}}{\sqrt{ \frac{\text{SSR}}{\sum_{i=1}^n\left( x_i - \bar{x}\right)^2}}}}$