Solution
How Much Variance Does the Line Explain
Show the problem again
Five observations $(x, y)$ are: (0, 1), (1, 3), (2, 2), (3, 6), (4, 3). Fit the least-squares line of $y$ on $x$. What is the coefficient of determination $R^2$? Give it as a reduced fraction.
Worked solution
With $\bar x = 2$ and $\bar y = 3$: $S_{xx} = 10$, $S_{xy} = 7$, $S_{yy} = 14$. For simple regression $R^2 = r^2 = S_{xy}^2/(S_{xx}S_{yy}) = 49/140 = 7/20$. Equivalently, the fitted slope is $7/10$ and the explained sum of squares $S_{xy}^2/S_{xx} = 4.9$ is $7/20$ of the total $S_{yy} = 14$.
Source: Original exercise written for AxiomIQ.