91Ó°ÊÓ

The given data is

$\hat{y}=5-x$

The regression equation is

$\hat{y}=5-x$

The formulas to calculate the sum of squares is

$$\begin{gathered} SST=∑{\left(y_i-\stackrel{¯}{y}\right)}^2 \\ SST=∑{\left({\hat{y}}_i-\stackrel{¯}{y}\right)}^2 \\ SST=∑{\left(y_i-\hat{y}\right)}^2 \end{gathered}$$

As shown in the table below, the relevant sums can be determined.

$$\begin{gathered} SST=10 \\ SSR=4 \\ SSE=6 \end{gathered}$$

The given data is

$\hat{y}=5-x$

$SST=SSR+SSE$

$$\begin{gathered} =4+6 \\ =10 \end{gathered}$$

The given data is

$\hat{y}=5-x$

The coefficient of determination is

$r^2=\frac{SSR}{SST}$

$$\begin{gathered} =\frac{4}{10} \\ =0.4 \end{gathered}$$

The given data is

$\hat{y}=5-x$

The coefficient of determination restated as a percentage is the percentage of variation:

$0.4=40\%$

The given data is

$\hat{y}=5-x$

The regression equation can be used to generate predictions if the estimated $r^2$is near to $1$.

The estimated $r^2$value is $0.4$, which is not equal to $1$.

As a result, utilising the regression equation to generate predictions is useless, as the regression can only explain $40\%$of the variation.