# 10-9. Formula List

### 10.9 Formula List

&#x20; $$SS_{xx}=Σx^2− \frac{1}{n}(Σx)^2$$ ,    $$SS_{xy}=Σxy− \frac{1}{n}(Σx)(Σy)$$ ,   $$SS_{yy}=Σy^2− \frac{1}{n}(Σy)^2$$&#x20;

**Correlation coefficient:**\
&#x20;                                             $$r= \frac{SS_{xy}} {\sqrt{SS_{xx}⋅SS_{yy}}}$$

**Least squares regression equation** (equation of the least squares regression line):\
&#x20;                                             $$\hat{y}=\hat{β_1}x+\hat{β_0}$$ \
&#x20;                     where $$\hat{β_1}= \frac{SS_{xy}} {SS_{xx}}$$  and $$\hat{β_0} =\bar{y}−\hat{β_1} \bar{x}$$&#x20;

**Sum of the squared errors for the least squares regression line**: \
&#x20;                                           $$SSE=SS_{yy}−\hat{β_1}SS_{xy}$$.

**Sample standard deviation of errors**: \
&#x20;                                                   $$s_ε=\sqrt{ \frac {SSE}{n−2}}$$&#x20;

$$100(1−α)\%$$ **confidence interval for** $$β_1$$ :\
&#x20;                                                 $$\hat{β_1}±t_{α∕2}  \frac  {s_ε} {\sqrt{SS_{xx}}}$$

**Standardized test statistic** for hypothesis tests concerning $$β_1$$:\
&#x20;                                        $$T=(\hat{β_1}−B_0) / \frac {s_ε}  {\sqrt{SS_{xx}}}$$ ( $$df=n−2$$ )

**Coefficient of determination**: \
&#x20;                                        $$r^2 = \frac{SS_{yy}−SSE} {SS_{yy}} = \frac{SS^2_{xy}}{ SS_{xx}SS_{yy}} = \hat{β_1} \frac{SS_{xy}}{SS_{yy}}$$

$$100(1−α)\%$$ **confidence interval for the mean value of** $$y$$ at $$x=x_p$$ : \
&#x20;                              $$\hat{y_p} ±t_{α∕2} s_ε \sqrt{\frac{1}{n} + \frac{(x_p−\bar{x})^2} {SS_{xx}}}$$              ( $$df=n-2$$ )

$$100(1−α)\%$$ **prediction interval for an individual new value of** $$y$$ at $$x=x_p$$:\
&#x20;                             $$\hat{y_p} ±t_{α∕2} s_ε \sqrt{1 + \frac{1}{n} + \frac{(x_p−\bar{x})^2} {SS_{xx}}}$$              ( $$df=n-2$$ )\
