# Chapter 13: Multiple Regression

1.  A regression equation with 2 independent variables is calibrated on a sample of size 18. The ANOVA table reports that the regression sum of squares is 112,909 and the total sum of squares is 138,162.  Complete the missing (numbered) 8 entries in the ANOVA table, one by one.   That is, for exercise 1, provide the residual sum of squares.  For exercise 2, find the correct value of df for regression.

What is the residual sum of squares? That is, what value should be entered in cell 1?

1. 251071
2. 22728
3. 209074
4. 25253 X

Solution:

> 138162 - 112909

[1] 25253

2.  What is the df value associated with regression? That is, what value should be entered in cell 2?

1. 18
2. 17
3. 2 X
4. 15

Solution:

The degrees of freedom (df) value associated with regression equals the number of independent variables.  Since there are 2 independent variables, the df value is 2.

3.  What is the df value associated with residual? That is, what value should be entered in cell 3?

1. 2
2. 15 X
3. 18
4. 17

Solution:

> 18 - 2 - 1

[1] 15

The degrees of freedom (df) value associated with residual equals the total number of observations (18) minus the number of independent variables (2) minus 1.

4.  What is the total df value? That is, what value should be entered in cell 4?

1. 2
2. 17 X
3. 15
4. 18

Solution:

> 2 + 15

[1] 17

The total df value (17) equals the sum of the regression df value (2) plus the residual df value (15).

5.  What is the mean square value (MS) associated with regression? That is, what value should be entered in cell 5?

1. 56454.5 X
2. 69081
3. 9210.8
4. 1683.533

Solution:

> 112909 / 2

[1] 56454.5

The mean square value (MS) associated with regression is found by dividing regression sum of squares by its own df.

6.  What is the mean square value (MS) associated with residual? That is, what value should be entered in cell 6?

1. 56454.5
2. 69081
3. 9210.8
4. 1683.533 X

Solution:

> 25253 / 15

[1] 1683.533

The mean square value (MS) associated with residual is found by dividing residual  sum of squares by its own df.

7.  What is the value of the F statistic?  That is, what value should be entered in cell 7?

1. 40.24001
2. 23.84593
3. 33.53334 X
4. 28.31704

Solution:

> (112909 / 2) / (25253 / 15)

[1] 33.53334

8.  What is the p-value for the test of significance of the overall regression model?  That is, what value should be entered in cell 8?

1. 0.000005827238
2. 0.000002913619 X
3. 0.000014568090
4. 0.000043704281

Solution:

> pf((112909 / 2) / (25253 / 15), 2, 15, lower.tail = FALSE)

[1] 0.000002913619

9.  What is the coefficient of determination r2?

1. 0.8989444
2. 0.6537774
3. 0.7354996
4. 0.8172218 X

Solution:

> 112909 / 138162

[1] 0.8172218

10.   What is the adjusted coefficient of determination, or the adjusted r2?

1. 0.7928514 X
2. 0.7294233
3. 0.6659952
4. 0.6184241

Solution:

> (112909 / 138162) - (2)*(1 - 112909/138162) / (18 - 2 - 1)

[1] 0.7928514