Q. 4.98 - Elementary Statistics

Question

Tax Efficiency. Following are the data on the percentage of investments in energy securities and tax efficiency from Exercise 4.58.

a. Compute $SST,SSR,SSE$

b. Compute the coefficient of determination 

c. Determine the percentage of variation in the observed values of the response variable explained by the regression, and interpret your answer.

d. State how useful the regression equation appears to be for making predictions.

Step-by-Step Solution

Answer

(a) $$\begin{gathered} SST=1532.865 \\ SSR=1456.6898 \\ SSE=76.1752 \end{gathered}$$

(b) $0.9503$

(c) $95.03\%$

(d)  Utilising the regression equation to create predictions is quite effective, and the regression can explain roughly $95\%$of the variation.

Part (a) Step 1: Given information

The given data is 

Part (a) Step 2: Explanation

The table given lists the ten mutual fund portfolios that were utilised to investigate the relationship between mutual fund investments and tax efficiency. The tax efficiency is denoted by $y$, while the percentage of investments in energy securities is denoted by $x$.

The formulas to calculate the sum of squares is  

$SST=\sum y_i^2-{\left(\sum y_i\right)}^2/n$

$SSR=\frac{{\left[\sum x_i{y}_i-\left(\sum x_i\right){\left(\sum y_i\right)}^{\text{'}n}\right]}^2}{\sum x_i^2-{\left(\sum x_i\right)}^2/n}$

$SSE=SST-SSR$

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

$$\begin{gathered} SST=70838.49-\frac{832.5^2}{10} \\ SST=1532.865 \end{gathered}$$

$$\begin{gathered} SSR=\frac{[4376.95-55.9\times 832.5÷10{]}^2}{365.05-55.9^2÷10} \\ SSR=1456.6898 \end{gathered}$$

$$\begin{gathered} SSE=1532.865-1456.6898 \\ SSE=76.1752 \end{gathered}$$

Part (b) Step 1: Given information

The given data is

Part (b) Step 2: Explanation

The coefficient of determination is  

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

    $$\begin{gathered} =\frac{1456.6898}{1532.865} \\ =0.9503 \end{gathered}$$

Part (c) Step 1: Given information

The given data is 

Part (c) Step 2: Explanation

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

$0.9503=95.03\%$

Part (d) Step 1: Given information

The given data is 

Part (d) Step 2: Explanation

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

The computed $r^2=0.9503$, which is extremely near to $1$.

As a result, utilising the regression equation to create predictions is quite effective, and the regression can explain roughly $95\%$ of the variation.