Understanding Coefficient of Determination ($r^2$)

The coefficient of determination, denoted as $r^2$, indicates the proportion of the variance in the dependent variable ($y$) that is predictable from the independent variable ($x$) using the linear regression model. It is calculated using the formula:

$ r^2 = 1 - \frac{SSE}{SST} $

Where:

• SSE is the Residual Sum of Squares (sum of the squared differences between actual $y$ and predicted $\hat{y}$).
• SST is the Total Sum of Squares (sum of the squared differences between actual $y$ and the mean of $y$, $\bar{y}$).

The question provides SSE = 25.8.

Calculating Total Sum of Squares (SST)

First, find the mean of the given $y$ values: $y = [8, 9, 7, 16]$.

$ \bar{y} = \frac{8 + 9 + 7 + 16}{4} = \frac{40}{4} = 10 $

Next, calculate the SST using the formula:

$ SST = \sum_{i=1}^{n} (y_i - \bar{y})^2 $

Calculate the squared differences:

• $(8 - 10)^2 = (-2)^2 = 4$
• $(9 - 10)^2 = (-1)^2 = 1$
• $(7 - 10)^2 = (-3)^2 = 9$
• $(16 - 10)^2 = (6)^2 = 36$

Sum the squared differences:

$ SST = 4 + 1 + 9 + 36 = 50 $

Determining the Coefficient of Determination ($r^2$)

Now, substitute the values of SSE and SST into the $r^2$ formula:

$ r^2 = 1 - \frac{SSE}{SST} = 1 - \frac{25.8}{50} $

$ r^2 = 1 - 0.516 $

$ r^2 = 0.484 $

Rounding to two decimal places, $r^2 \approx 0.48$. This value falls within the range of 0.45 to 0.5.

Final Answer

The coefficient of determination ($r^2$) for the given dataset is approximately 0.48.