Question

This represents the proportion of the variability on the distribution of Y that can be explained for by the mode (15 points) Show that R2 2 where is the sample correlation coefficient that is defined T- , (Y -Y Hint Use the fact that and that

Answer 1

We have:

$\\ \hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1X_i \\ \text{Summing both sides over i and dividing by n, we get:} \\ \Rightarrow \bar{Y} = \hat{\beta}_0 + \hat{\beta}_1\bar{X}$

Now, consider R²:

$\begin{align*} R^2 &= \frac{\sum_i (\hat{Y}_i - \bar{Y})^2}{\sum_i (Y_i - \bar{Y})^2} \\ &= \frac{\sum_i (\hat{\beta}_0 + \hat{\beta}_1\hat{X}_i - \hat{\beta}_0 - \hat{\beta}_1\bar{X})^2}{\sum_i (Y_i - \bar{Y})^2} \\ &= \hat{\beta}_1^2 * \frac{\sum_i ( \hat{X}_i - \bar{X})^2}{\sum_i (Y_i - \bar{Y})^2} \\ \text{Sub} &\text{stituting the value of } \hat{\beta}_1 \text{, we get:} \\ &= \frac{(\sum_i (X_i - \bar{X})(Y_i - \bar{Y}))^2}{(\sum_i (X_i - \bar{X})^2)^2} * \frac{\sum_i ( \hat{X}_i - \bar{X})^2}{\sum_i (Y_i - \bar{Y})^2} \\ &= \frac{(\sum_i (X_i - \bar{X})(Y_i - \bar{Y}))^2}{\sum_i (X_i - \bar{X})^2 \sum_i (Y_i - \bar{Y})^2} \\ &= \begin{bmatrix} \frac{\sum_i (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum_i (X_i - \bar{X})^2 \sum_i (Y_i - \bar{Y})^2}}\end{bmatrix}^2 \\ &= r^2 \end{align*}$

Hence Proved.

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