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?

  1. -0.7398998
  2. -0.6726362 X
  3. -0.5246562
  4. -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, SSy for the estimated regression equation.

  1. 3777.24
  2. 2905.57 X
  3. 5520.58
  4. 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, SSreg.

  1. 1314.594 X
  2. 1472.346
  3. 1077.967
  4. 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, r2?

  1. 0.5655492
  2. 0.3890979
  3. 0.5003980
  4. 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 b1?

  1. -0.10826150
  2. -0.05413075
  3. -0.07217434 X
  4. -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 b0?

  1. 32.7462 X
  2. 47.4821
  3. 20.6302
  4. 15.4726

Solution:

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

> b0

[1] 32.7462

7.  Which is the estimated regression equation?

  1. ŷ = 3.27462 - 0.07217x
  2. ŷ = 32.7462 - 0.07217x  X
  3. ŷ = 0.07217 - 32.7462x
  4. ŷ = 32.7462 - 0.72174x

Solution:

Substitute b0 = 32.7462 and b1 = -0.07217 into regression equation to obtain

ŷ = 32.7462 - 0.07217x

8.  What is the standard error of the regression coefficient b1?

  1. 0.008323 X
  2. 0.166466
  3. 0.058263
  4. 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 b1?

  1. -4.7692
  2. -11.532
  3. -8.6713 X
  4. -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 b1?

  1. 0.0000000000001536838 X
  2. 0.0000000000192104814
  3. 0.0000000000000737682
  4. 0.0000000000115262967

Solution:

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

> pvalue

[1] 0.0000000000001536838