Q. 4.94 - 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 38,24 and 14 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 0.632.

Part (d): The percentage of variation in the observed values of the response variable that is explained by the regression is 63.2%.

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

Part (a) Step 1. Given information.

Consider the given question,

Part (a) Step 2. Construct the table,

On constructing the table,

From the table,

$$\begin{gathered} SST=38, \\ SSR=24, \\ SSE=14 \end{gathered}$$

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

In regression, the equation,

$$\begin{gathered} SST=SSR+SSE \\ 38=24+14 \\ 38=38 \end{gathered}$$

Thus, the regression identity is verified.

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

Consider the coefficient of determination,

$$\begin{gathered} r^2=1-\frac{SSR}{SST} \\ =1-\frac{14}{38} \\ =1-0.368 \\ =0.632 \end{gathered}$$

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

As the coefficient of determination is 0.632.

Then we can say that 63.2% 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 useful for making predictions as the coefficient of determination is 63.2%. That is, the coefficient of determination is more than 50%.