# Statistics with R

## Student Resources

# 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?

- 251071
- 22728
- 209074
- 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?

- 18
- 17
- 2 X
- 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?

- 2
- 15 X
- 18
- 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?

- 2
- 17 X
- 15
- 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?

- 56454.5 X
- 69081
- 9210.8
- 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?

- 56454.5
- 69081
- 9210.8
- 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?

- 40.24001
- 23.84593
- 33.53334 X
- 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?

- 0.000005827238
- 0.000002913619 X
- 0.000014568090
- 0.000043704281

**Solution:**

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

[1] 0.000002913619

9. What is the coefficient of determination r^{2}?

- 0.8989444
- 0.6537774
- 0.7354996
- 0.8172218 X

**Solution:**

> 112909 / 138162

[1] 0.8172218

10. What is the adjusted coefficient of determination, or the adjusted r^{2}?

- 0.7928514 X
- 0.7294233
- 0.6659952
- 0.6184241

**Solution:**

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

[1] 0.7928514