Chapter 4: Q. 4.96 (page 181)

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, $S S T = S S R + S S E$ .
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.

Short Answer

Expert verified

Part (a): The three sums of squares, SST, SSR and SSE are 5, 1and 4respectively.

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 0.2.

Part (d): The percentage of variation in the observed values of the response variable that is explained by the regression is 20 %.

Part (e): The regression is not useful for making predictions.

Step by step solution

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,

$S S T = \sum_{i} {(y_{i} - y)}^{2}, w h e r e y = \frac{\sum_{i} y_{i}}{n} S S R = \sum_{i} {({\overset{⏞}{y}}_{i} - y)}^{2}, S S E = S S T - S S R$

Then, $y = \frac{1 + 3 + 2 + 4}{4} = 2.5$

Part (a) Step 3. Construct the table.

On constructing the table,

For $i = 1, 2, 3$ by putting the values of xin equation $\hat{y} = 1.75 + 0.25 x$

Using the table,

$S S T = \sum_{i} {(y_{i} - y)}^{2} = 5 S S R = \sum_{i} {({\overset{⏞}{y}}_{i} - y)}^{2} = 1 S S E = S S T - S S R = 5 - 1 = 4$

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

In regression, the equation,

$S S T = S S R + S S E = 1 + 4 = 5$

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

Consider the coefficient of determination,

$r^{2} = \frac{S S R}{S S T} = \frac{1}{5} = 0.2$

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

As the coefficient of determination is 0.2.

Then we can say that 20%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 not useful for making predictions.

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Short Answer

Step by step solution

Part (a) Step 1. Given information.

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

Part (a) Step 3. Construct the table.

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

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

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

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

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