Q. 4.89 - Elementary Statistics

Question

We repeat the data and provide the regression equations.

Part (a): Compute the three sums of squares, SST, SSR and SSE using the defining formulas.

Part (b): Verify the regression identity,$SST=SSR+SSE$.

Part (c): Compute the coefficient of determination.

Part (d): Determine the percentage of variation in the observed values of the response variable that is explained by the regression.

Part (e): State how useful the regression equation appears to be for making predictions.

Step-by-Step Solution

Answer

Part (a): The three sums of squares, SST, SSR and SSE are 14,8 and 6 respectively.

Part (b): Substitute the values of SSR and SSE in the formula of SST, to verify the given regression identity.

Part (c): The coefficient of determination is 0.571.

Part (d): The percentage of variation in the observed values of the response variable that is explained by the regression is 57.1%.

Part (e): The regression is moderately useful for making predictions.

Part (a) Step 1. Given information.

Consider the given question,

Part (a) Step 2. Write the formulas of three sums of squares, SST, SSR and SSE.

We know,

$$\begin{gathered} SST=\sum _i{\left(y_i-\overline{y}\right)}^2, where \overline{y}=\frac{{\displaystyle \sum _i{y}_i}}{n} \\ SSR=\sum _i{\left({\overbrace{y}}_i-\overline{y}\right)}^2, \\ SSE=SST-SSR \end{gathered}$$

Then, $$\begin{gathered} \overline{y}=\frac{-4+0+5}{3} \\ =-3 \end{gathered}$$

Part (a) Step 3. Construct the table.

On constructing the table,

For $i=1,2,3$ by putting the values of x in equation $\hat{y}=1-2x$.

Using the table,

$$\begin{gathered} SST=\sum _i{\left(y_i-\overline{y}\right)}^2 \\ =14 \\ SSR=\sum _i{\left({\overbrace{y}}_i-\overline{y}\right)}^2 \\ =8 \\ SSE=SST-SSR \\ =14-8 \\ =6 \end{gathered}$$

Part (b) Step 1. Verify the equation of SST.

In regression, the equation,

$$\begin{gathered} SST=SSR+SSE \\ =8+6 \\ =14 \end{gathered}$$

Part (c) Step 1. Compute the coefficient of determination.

Consider the coefficient of determination,

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ =\frac{8}{14} \\ =0.571 \end{gathered}$$

Part (d) Step 1. Determine the percentage of variation.

As the coefficient of determination is 0.571.

Then we can say that 57.1% of the variation in the observed value is explained by the regression.

Part (e) Step 1. State usefulness of the regression equation.

On stating the usefulness of the regression equation,

We can say that here the regression is moderately useful for making predictions.