Question

Using linear regression (least squares), the best-fit line for the given dataset is \(y = 2.2x + 2.3\). Given the residual sum of squares is 25.8, the coefficient of determination \((r^2)\) is . (rounded off to two decimal places)

x	2	3	4	5
y	8	9	7	16

Hint:

Find SST from the spread of the y values about their mean, then use \(r^2 = 1 - SSE/SST\).

Answer 1

Correct Answer: 0.48

Another way to reach the same $r^2$ is to work out how much of the variation the regression line actually explains, rather than starting from what it fails to explain.

Step 1: Find the total variation in y (SST). The y values are 8, 9, 7 and 16, with mean $\bar{y} = 40/4 = 10$. Squaring each deviation from this mean and adding them gives $SST = 4 + 1 + 9 + 36 = 50$.

Step 2: Find the explained variation (SSR) instead of starting from the error. The regression splits total variation into two parts: the part the line explains (SSR) and the part it misses (SSE, the residual sum of squares), so $SST = SSR + SSE$. Since $SSE = 25.8$: $SSR = SST - SSE = 50 - 25.8 = 24.2$.

Step 3: The coefficient of determination is the explained share of the total variation. $r^2 = SSR / SST = 24.2 / 50 = 0.484$.

Both routes must agree, because $1 - SSE/SST$ and $SSR/SST$ are just two ways of writing the same split of total variation into explained and unexplained parts.

Let's summarize:

• SST measures the total spread of y around its own mean, here 50.
• SSR = SST - SSE = 24.2 is the part of that spread the fitted line accounts for.
• $r^2 = SSR/SST = 0.48$ rounded to two decimal places.

So the coefficient of determination is 0.48.