Sorry for my bad english, i'll try to explain my difficulty.
1° Question
In the last exam an exercise asked to derivate OLS estimators and calculate their value with the data provided (ex.: $\sum x_iy_i=...$, $\sum Y_i=...$). Now, I know how to derivate them ($\hat{\beta _{0}}=\bar{Y} -\hat{\beta _{1}}\bar{Y}$ and $\hat{\beta _{1}}=\frac{\sum_{i=1}^{n}(X_i-\bar{X})(Y_i-\bar{Y})}{\sum_{i=1}^{n}(X_i-\bar{X})^2}$) but i have difficulties calculating them because data can be "big" ($X$) or "little" ($x$). My prof. tried to explain me that $x=X-\bar{X}$ but my difficulties remain. For example: assuming that $\sum x_iy_i=...$, $\sum Y_i=...$, $\sum X_{i}^{2}=...$ and $\sum X_i=...$: all data are uncertain except the last. I know that $\hat{\beta _{1}}$ can be written like $=\frac{\sum_{i=1}^{n}X_iY_i- n\bar{X}\bar{Y}}{\sum_{i=1}^{n}X_{i}^{2}-n\bar{X}^2}$: so, how can i transform $x_iy_i$ in $X_iY_i$ to use it in the formula?
2° Question
In light of the problems that i have for the 1° question, i can't write a good form so i ask you if formulas below are corrected:
- mean for observed variables: $\bar{x}=\frac{1}{n}\sum x_i$
- variance for observed variables: ${s_{X}}^{2}=\frac{1}{n}\sum (x_i-\bar{x})^2=\frac{1}{n}\sum ({x_{i}}^{2}-2x_i\bar{x}+\bar{x}^2)=\bar{x^2}-\bar{x}^2=\frac{1}{n}\sum {x_{i}}^{2}-\bar{x}^2$
- standard deviation for observed variables: ${s_{X}}^{2}=\sqrt{\frac{1}{n}\sum (x_i-\bar{x})^2}$
- covariance for observed variables: ${s_{XY}}=\frac{1}{n}\sum (x_i-\bar{x})(y_i-\bar{y})=\bar{xy}-\bar{x}\bar{y}=\frac{1}{n}\sum x_iy_i-(\frac{1}{n}\sum x_i)(\frac{1}{n}\sum y_i)$
- correlation for observed variables: ${r_{XY}}=\frac{s_{XY}}{s_{X}s_{Y}}=\frac{\frac{1}{n}\sum x_iy_i-(\frac{1}{n}\sum x_i)(\frac{1}{n}\sum y_i)}{\sqrt{\frac{1}{n}\sum (x_i-\bar{x})^2\cdot\frac{1}{n}\sum (y_i-\bar{y})^2}}=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2\cdot\sum (y_i-\bar{y})^2}}=\frac{\sum x_iy_i-n\bar{x}\bar{y}}{({\sum x_{i}}^{2}-n\bar{x}^2)({\sum y_{i}}^{2}-n\bar{y}^2)}$
- sample mean: $\bar{X}=\frac{1}{n}\sum X_i$
- sample variance: ${S_{e}}^{2}=\frac{1}{n}\sum (X_i-\bar{X})^2$
- standard error: $SE=\sqrt{\frac{1}{n}\sum (X_i-\bar{X})^2}$
- corrected estimator for sample variance: $S^2=\frac{1}{n-1}\sum (X_i-\bar{X})^2$ with $S^2=\frac{n}{n-1}{S_{e}}^{2}$
- $R^2$: $R^2=(r_{XY})^2$ or $R^2=\frac{ESS}{TSS}=\frac{\sum_{i=1}^{n}(\hat{y_i}-\bar{y})^2}{\sum_{i=1}^{n}(y_i-\bar{y})^2}\Rightarrow ESS=R^2\cdot TSS=(r_{XY})^2\cdot (\sum {y_{i}}^{2}-n\bar{y}^2)\Rightarrow RSS=TSS-ESS$
- estimate of the variance of residuals: $SER=\sqrt{\frac{1}{n-2}\sum (e_i-\bar{e})^2}=\frac{RSS}{n-2}$