Editing Student's t-test (section)

===Slope of a regression line===
Suppose one is fitting the model

: <math> Y = \alpha + \beta x + \varepsilon, </math>

where {{math|''x''}} is known, {{math|''α''}} and {{math|''β''}} are unknown, {{math|''ε''}} is a normally distributed random variable with mean 0 and unknown variance {{math|''σ''<sup>2</sup>}}, and {{math|''Y''}} is the outcome of interest. We want to test the null hypothesis that the slope {{math|''β''}} is equal to some specified value {{math|''β''<sub>0</sub>}} (often taken to be 0, in which case the null hypothesis is that {{math|''x''}} and {{math|''y''}} are uncorrelated).

Let

: <math>
\begin{align}
 \hat\alpha, \hat\beta &= \text{least-squares estimators}, \\
 SE_{\hat\alpha}, SE_{\hat\beta} &= \text{the standard errors of least-squares estimators}.
\end{align}
</math>

Then

:<math>
 t_\text{score} = \frac{\hat\beta - \beta_0}{ SE_{\hat\beta} } \sim \mathcal{T}_{n-2}
</math>

has a ''t''-distribution with {{math|''n'' − 2}} degrees of freedom if the null hypothesis is true. The [[Simple linear regression#Normality assumption|standard error of the slope coefficient]]:

: <math>
 SE_{\hat\beta} = \frac{\sqrt{\displaystyle \frac{1}{n - 2}\sum_{i=1}^n (y_i - \hat y_i)^2}}{\sqrt{\displaystyle \sum_{i=1}^n (x_i - \bar{x})^2}}
</math>

can be written in terms of the residuals. Let

: <math>
\begin{align}
 \hat\varepsilon_i &= y_i - \hat y_i = y_i - (\hat\alpha + \hat\beta x_i) = \text{residuals} = \text{estimated errors}, \\
 \text{SSR} &= \sum_{i=1}^n {\hat\varepsilon_i}^2 = \text{sum of squares of residuals}.
\end{align}
</math>

Then {{math|''t''}}<sub>score</sub> is given by

: <math> t_\text{score} = \frac{(\hat\beta - \beta_0) \sqrt{n-2}}{\sqrt{\frac{SSR}{\sum_{i=1}^n (x_i - \bar{x})^2}}}. </math>

Another way to determine the {{math|''t''}}<sub>score</sub> is

: <math> t_\text{score} = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}, </math>

where ''r'' is the [[Pearson correlation coefficient]].

The {{math|''t''}}<sub>score, intercept</sub> can be determined from the {{math|''t''}}<sub>score, slope</sub>:

: <math> t_\text{score,intercept} = \frac{\alpha}{\beta} \frac{t_\text{score,slope}}{\sqrt{s_\text{x}^2 + \bar{x}^2}}, </math>

where {{math|''s''<sub>x</sub><sup>2</sup>}} is the sample variance.