Q. 4.98 - Elementary Statistics

Question

Following are the data percentage of investments in energy securities and tax efficiency from Exercise 4.85.

Part (a): Compute the SST,SSR, and SSE using Formula 4.2 on page 179.

Part (b): Compute the coefficient of determination, $r^2$.

Part (c): Determine the percentage of variable in the observed values of the response variable explained by the regression and interpret your result.

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

Step-by-Step Solution

Answer

Part (a): The SST, SSR, and SSE is 1532.87, 1456.69 and 76.18.

Part (b): The coefficient of determination is 0.95.

Part (c): 95% of the variation in the observed value is explained by the regression.

Part (d): The regression is very useful for making predictions.

Part (a) Step 1. Given information.

Consider the given question,

Part (a) Step 2. Construct the table for computing required results.

On constructing the table,

Therefore, total sum of squares,

$$\begin{gathered} \mathrm{SST}={{\mathrm{\Sigma y}}_{\mathrm{i}}}^2-\frac{{\left({\mathrm{\Sigma y}}_{\mathrm{i}}\right)}^2}{n} \\ =70838.49-\frac{832.5}{10} \\ =1532.87 \end{gathered}$$

Part (a) Step 3. Find the regression sum of squares.

We know,

$$\begin{gathered} SSR={\frac{\left[\Sigma x_i-{\displaystyle \frac{\left(\Sigma x_i\right)\left(\Sigma y_i\right)}{n}}\right]}{\Sigma {x_i}^2{\displaystyle \frac{{\left(\Sigma xi\right)}^2}{n}}}}^2 \\ =\frac{\left[4376.95-{\displaystyle \frac{\left(55.9\right)\left(832.5\right)}{}}\right]}{} \\ =365.05-\frac{{\left(55.9\right)}^2}{10} \\ =1456.69 \end{gathered}$$

Error sum of squares is given below,

$$\begin{gathered} SSE=SST-SSR \\ =1532.87-1456.69 \\ =76.18 \end{gathered}$$

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

Consider the coefficient of determination,

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ =\frac{1456.69}{1532.87} \\ =0.95 \end{gathered}$$

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

As the coefficient of determination is 0.95.

Then we can say that 95% of the variation in the observed value is explained by the regression.

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

On stating the usefulness of the regression equation,

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