# Statistics with R

## Student Resources

# Chapter 12: Simple Linear Regression

1. Using the Cars93 data (see the exercises at the end of Chapter 2 for more information about Cars93, if necessary), suppose we want to investigate whether two variables---MPG.city and Horsepower---are related. As a first step, what is the correlation of these two variables?

- -0.7398998
- -0.6726362 X
- -0.5246562
- -0.3699499

**Solution:**

> cor(Cars93$MPG.city, Cars93$Horsepower)

[1] -0.6726362

2. Letting MPG.city be the dependent variable and Horsepower the independent variable, find the total sum of squares, SS_{y} for the estimated regression equation.

- 3777.24
- 2905.57 X
- 5520.58
- 2121.07

**Solution:**

> sum((Cars93$MPG.city - mean(Cars93$MPG.city))^2)

[1] 2905.57

3. Referring to preceding exercise, find the regression sum of squares, SS_{reg}.

- 1314.594 X
- 1472.346
- 1077.967
- 1262.011

**Solution:**

>slr <- lm(MPG.city ~ Horsepower, data = Cars93)

> sum((predict(slr)-mean(Cars93$MPG.city))^2)

[1] 1314.594

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

- 0.5655492
- 0.3890979
- 0.5003980
- 0.4524394 X

**Solution:**

> sum((predict(slr) - mean(Cars93$MPG.city)) ^ 2) / sum((Cars93$MPG.city - mean(Cars93$MPG.city)) ^ 2)

[1] 0.4524394

5. What is the estimated regression coefficient b_{1}?

- -0.10826150
- -0.05413075
- -0.07217434 X
- -0.13532691

**Solution:**

> b1 <- sum((Cars93$Horsepower - mean(Cars93$Horsepower)) * (Cars93$MPG.city - mean(Cars93$MPG.city))) / sum((Cars93$Horsepower - mean(Cars93$Horsepower)) ^ 2)

> b1

[1] -0.07217434

6. What is the estimated intercept term b_{0}?

- 32.7462 X
- 47.4821
- 20.6302
- 15.4726

**Solution:**

> b0 <- mean(Cars93$MPG.city) - b1 * mean(Cars93$Horsepower)

> b0

[1] 32.7462

7. Which is the estimated regression equation?

- ŷ = 3.27462 - 0.07217x
- ŷ = 32.7462 - 0.07217x X
- ŷ = 0.07217 - 32.7462x
- ŷ = 32.7462 - 0.72174x

**Solution:**

Substitute b_{0} = 32.7462 and b_{1} = -0.07217 into regression equation to obtain

ŷ = 32.7462 - 0.07217x

8. What is the standard error of the regression coefficient b_{1}?

- 0.008323 X
- 0.166466
- 0.058263
- 0.004161

**Solution:**

> syx <- sqrt(sum((Cars93$MPG.city - predict(slr))^2 / (nrow(Cars93) - 2)))

> ssx <- sqrt(sum((Cars93$Horsepower - mean(Cars93$Horsepower))^2))

> std_error_b1 <- syx / ssx

> std_error_b1

[1] 0.008323347

9. What is the t statistic associated with the regression coefficient b_{1}?

- -4.7692
- -11.532
- -8.6713 X
- -6.9370

**Solution:**

Dividing b1 (see exercise 5) by std_error_b1 (see exercise 8), we have t

> t <- b1 / std_error_b1

> t

[1] -8.671312

10. What is the p-value associated with the regression coefficient b_{1}?

- 0.0000000000001536838 X
- 0.0000000000192104814
- 0.0000000000000737682
- 0.0000000000115262967

**Solution:**

> pvalue <- 2 * pt(-8.671312, 91)

> pvalue

[1] 0.0000000000001536838