# 10-9. Formula List

### 10.9 Formula List

  $$SS\_{xx}=Σx^2− \frac{1}{n}(Σx)^2$$ ,    $$SS\_{xy}=Σxy− \frac{1}{n}(Σx)(Σy)$$ ,   $$SS\_{yy}=Σy^2− \frac{1}{n}(Σy)^2$$ 

**Correlation coefficient:**\
                                              $$r= \frac{SS\_{xy}} {\sqrt{SS\_{xx}⋅SS\_{yy}}}$$

**Least squares regression equation** (equation of the least squares regression line):\
                                              $$\hat{y}=\hat{β\_1}x+\hat{β\_0}$$ \
                      where $$\hat{β\_1}= \frac{SS\_{xy}} {SS\_{xx}}$$  and $$\hat{β\_0} =\bar{y}−\hat{β\_1} \bar{x}$$ 

**Sum of the squared errors for the least squares regression line**: \
                                            $$SSE=SS\_{yy}−\hat{β\_1}SS\_{xy}$$.

**Sample standard deviation of errors**: \
                                                    $$s\_ε=\sqrt{ \frac {SSE}{n−2}}$$ 

$$100(1−α)%$$ **confidence interval for** $$β\_1$$ :\
                                                  $$\hat{β\_1}±t\_{α∕2}  \frac  {s\_ε} {\sqrt{SS\_{xx}}}$$

**Standardized test statistic** for hypothesis tests concerning $$β\_1$$:\
                                         $$T=(\hat{β\_1}−B\_0) / \frac {s\_ε}  {\sqrt{SS\_{xx}}}$$ ( $$df=n−2$$ )

**Coefficient of determination**: \
                                         $$r^2 = \frac{SS\_{yy}−SSE} {SS\_{yy}} = \frac{SS^2\_{xy}}{ SS\_{xx}SS\_{yy}} = \hat{β\_1} \frac{SS\_{xy}}{SS\_{yy}}$$

$$100(1−α)%$$ **confidence interval for the mean value of** $$y$$ at $$x=x\_p$$ : \
                               $$\hat{y\_p} ±t\_{α∕2} s\_ε \sqrt{\frac{1}{n} + \frac{(x\_p−\bar{x})^2} {SS\_{xx}}}$$              ( $$df=n-2$$ )

$$100(1−α)%$$ **prediction interval for an individual new value of** $$y$$ at $$x=x\_p$$:\
&#x20;                             $$\hat{y\_p} ±t\_{α∕2} s\_ε \sqrt{1 + \frac{1}{n} + \frac{(x\_p−\bar{x})^2} {SS\_{xx}}}$$              ( $$df=n-2$$ )<br>