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

Step 1: Recall what the coefficient of determination measures.
\(r^2\) tells us what fraction of the total spread in \(y\) is explained by the regression line, out of the total spread the data actually has around its own mean. It is defined as
\[ r^2 = 1 - \frac{SSE}{SST} \]
where SSE is the residual sum of squares (the error the line cannot explain) and SST is the total sum of squares (the total variation in \(y\) about its mean).

Step 2: Find the mean of \(y\) and then SST.
The four \(y\) values are 8, 9, 7 and 16.
\[ \bar{y} = \frac{8+9+7+16}{4} = \frac{40}{4} = 10 \]
SST is the sum of the squared distance of each \(y\) value from this mean.
\[ SST = (8-10)^2 + (9-10)^2 + (7-10)^2 + (16-10)^2 = 4+1+9+36 = 50 \]

Step 3: Substitute SSE and SST into the \(r^2\) formula.
We are told SSE = 25.8.
\[ r^2 = 1 - \frac{25.8}{50} = 1 - 0.516 = 0.484 \]

Final Answer:
Rounded to two decimal places, \(r^2 = 0.48\). \[ \boxed{0.48} \]