# Statistics with R

## Student Resources

# Chapter 12: Simple Linear Regression

The following exercises are intended to (1) provide practice analyzing data using simple linear regression and (2) review and reinforce our ability to subset data. The reason we emphasize these two skills together is that, in many instances, we want to analyze data that include only certain observations (and variables) while excluding the others. To this end, we make use of the Cars93 data from the R package named MASS. If necessary, make sure you consult Chapter 2 for speci c instructions on how to access Cars93, with particular attention to the brief discussion immediately preceeding Exercise 1 at the chapter's end. And because regression is the nal analytic methodology of the book, we revisit several of the most useful R functions covered earlier in the book, just for practice.

1. Import the Cars93 data into the object named E12_1. What are the variable names? How many observations are included? Find (1) the minimum and maximum values,(2) the median and mean, (3) the first and third quartiles, and (4) the standard deviation of the two variables, MPG.city and EngineSize. Comment on your initial findings.

library(MASS)

##*## Attaching package: 'MASS'**## The following object is masked from 'package:introstats':*

##*## housing**#Comment1. Import Cars93 into the object named E12_1.*

E12_1 <- Cars93*#Comment2. Use the nrow() function to find the number**#of observations*.

nrow(E12_1)

## [1] 93*#Comment3. Use the names() function to list variable names.*

names(E12_1)

## [1] "Manufacturer" "Model" "Type"

## [4] "Min.Price" "Price" "Max.Price"

## [7] "MPG.city" "MPG.highway" "AirBags"

## [10] "DriveTrain" "Cylinders" "EngineSize"

## [13] "Horsepower" "RPM" "Rev.per.mile"

## [16] "Man.trans.avail" "Fuel.tank.capacity" "Passengers"

## [19] "Length" "Wheelbase" "Width"

## [22] "Turn.circle" "Rear.seat.room" "Luggage.room"

## [25] "Weight" "Origin" "Make"*#Comment4. Use the summary() function to find the basic*

#descriptive statistics for MPG.city and EngineSize.

summary(E12_1$MPG.city)

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 15.00 18.00 21.00 22.37 25.00 46.00

summary(E12_1$EngineSize)

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.000 1.800 2.400 2.668 3.300 5.700*#Comment5. Use the sd() function to find the standard deviation**#of each variable.*

sd(E12_1$MPG.city)

## [1] 5.619812

sd(E12_1$EngineSize)

## [1] 1.037363

**Answer:**The Cars93 data include 93 observations across 27 variables. The descrip-

tive statistics for MPG.city and EngineSize are provided in Comments 4 and 5.

2. Do MPG.city and EngineSize appear related in any systematic way? Comment.

plot(E12_1$EngineSize, E12_1$MPG.city,

pch = 19,

xlab = 'Engine Size (liters)',

ylab = 'City Miles per Gallon ',

main = 'Relationship Between City MPG and Engine Size (liters)')

**Answer: **The pattern of points revealed by the scatterplot suggests that the relationship is negatively related. The important question is whether the relationship is linear; the scatterplot suggests that it is probably more curvilinear than linear. To sort out that issue, we next move on to the residual plot.

3. Make and inspect a residual plot. Does the pattern of points reveal anything that might cause us to question the assumptions underlying the appropriate usage of regression analysis to explore the relationship between MPG.city and EngineSize?

*#Comment1. Use the lm() function to create the model object*

*#named slr1 (the first simple linear regression).*

slr1 <- lm(MPG.city ~ EngineSize, data = E12_1)

*#Comment2. Use the plot() function to create a residual plot.*

*#Note that resid(slr1) must be included as an argument.*

plot(E12_1$EngineSize,

resid(slr1),

abline(h = 0),

pch = 19,

xlab = 'Engine Size (liters)',

ylab = 'Residuals',

main = 'Residuals Against the Independent Variable')

**Answer:** When the engine size is between (roughly) 1.5 and 4.75 liters, the residuals reveal a reasonably linear relationship between MPG.city and EngineSize. However, this pattern tends to break down for both the upper and lower values of engine size: for vehicles having the smallest engine size (below 1.5 liters) and the largest engine size (above 4.75) the pattern of the residuals reveals that the assumptions underlying the correct application of regression are not very well met.

4. There are several possible methods for managing the problem of nonlinear relation- ships among variables, such as what we have encountered in this case. One of theapproaches involves transforming the variables—by way of logarithms, exponents, etc.—in such a way that they are *forced *to be more linearly related. (This class of methods, sometimes referred to as GLM or *general linear model*, is not covered in this book.) Another procedure requires including additional variables into the mul- tiple regression model (the focus of Chapter 13). Instead, the approach we employ here involves subsetting the data according to some specification—such as, subset- ting the data in a way that includes, for example, only vehicles manufactured in the US or all vehicles that have smaller engines (i.e., fewer liters of displacement). The expectation (or hope) is that, by subsetting, the resulting data may meet the assumptions behind the appropriate application of regression analysis. As a first step, subset the data stored in object E12_1 in a way that excludes all vehicles with EngineSize greater than the median. (See Comment 4 under the first exercise above.) Name this new object E12_2. Check to make sure that E12_2 conforms to this requirement. How many observations remain in the new object? List the first 3 observations; list the last 3 observations. If necessary, review the third point of the Chapter 2 Appendix, “Can We Extract A Data Subset From A Larger Data Set?"

*#Comment1. Use indexing [ ]; set median Engine Size value of 2.4.*

E12_2 <- E12_1[E12_1$EngineSize <= 2.40, c(MPG.city, EngineSize)]

*#Comment2. Use the max() and min() functions to find the maximum*

*#and minimum values of EngineSize in E12_2.*

max(E12_2$EngineSize)

## [1] 2.4

min(E12_2$EngineSize)

## [1] 1

*#Comment3. List the first 3 and last 3 observations in E12_2.*

head(E12_2, 3)

## MPG.city EngineSize

## 1 25 1.8

## 6 22 2.2

## 12 25 2.2

tail(E12_2, 3)

## MPG.city EngineSize

## 90 21 2.0

## 92 21 2.3

## 93 20 2.4

*#Comment4. Use the nrow() function to find the number of**#observations stored in E12_2.*

nrow(E12_2)

## [1] 49

**Answer: **The variable EngineSize now runs from a low of 1 to a high of 2.4 liters; *n *= 49 observations are stored in the E12_2 object.

5. For E12_2, do MPG.city and EngineSize appear related in a systematic way?

plot(E12_2$EngineSize, E12_2$MPG.city,

pch = 19,

xlab = 'Engine Size (liters)',

ylab = 'City Miles per Gallon',

main = 'Relationship Between City MPG and Engine Size (liters)')

**Answer: **Yes, the pattern of points revealed in the scatterplot of the E12_2 data

suggests that the two variables, MPG.city and EngineSize, may be negatively and

linearly related. This is not a surprising nding, of course, since it confirms what

we believe about this relationship in the first place.

6. Make and inspect a residual plot. Does the pattern of points appear more linearly arranged than they did in the earlier exercise when the data included vehicles with large engines as well as those with a smaller one?

*#Comment1. Use the lm() function to create the model object named*

*#slr2 (the second simple linear regression).*

slr2 <- lm(MPG.city ~ EngineSize, data = E12_2)

*#Comment2. Use the plot() function to create a residual plot.*

*#Note that resid(slr2) must be included as an argument.*

plot(E12_2$EngineSize,

resid(slr2),

abline(h = 0),

pch = 19,

xlab = 'Engine Size (liters)',

ylab = 'Residuals',

main = 'Residuals Against the Independent Variable')

**Answer:** Yes, apart from 3 or 4 outliers for values of engine size below 1.5 liters, the residual plot does not reveal serious violations of the assumptions.

7. As part of making the residual plot in the preceding exercise, we used the lm() function to create the model object slr2. This is an important step in residual analysis because the model object (slr2) includes all the important information associated with the particular regression problem at hand, including the estimated regression equation itself. What is the estimated regression equation?

slr2

##

## Call:

## lm(formula = MPG.city ~ EngineSize, data = E12_2)

##

## Coefficients:

## (Intercept) EngineSize

## 46.15 -10.87

**Answer: **The estimated regression equation is *ŷ *= *b*_{0} + *b*_{1}*x* = 46.15 – 10.87*x*, where *ŷ* is the predicted dependent variable, MPG.city, and *x* is the independent variable, EngineSize.

8. Find the 95 and 99 percent confidence interval estimates of the regression coefficient *b*_{1}. Describe what these con dence intervals mean.

*#Comment. Use the confint(, level = ) function to find the*

*##confidence interval estimates of the regression coefficient.*

confint(slr2, level = 0.95)

## 2.5 % 97.5 %

## (Intercept) 40.09671 52.207573

## EngineSize -14.03045 -7.714777

confint(slr2, level = 0.99)

## 0.5 % 99.5 %

## (Intercept) 38.07151 54.232777

## EngineSize -15.08657 -6.658656

**Answer:** There is a 95% probability that the regression coefficient falls in the interval from -14.03045 to -7.714777; there is a 99% probability that it falls in the interval from -15.08657 to -6.658656.

9. What does the estimated regression equation tell us?

**Answer:** We can interpret the estimated regression equation *ŷ* = 46.15–10.87x this way: for the class of vehicles with engine sizes of 2.4 liters or less, a change of 1 liter in engine size is associated with a change of 10.87 miles per gallon when the vehicle is driving in a city. Moreover, the negative sign tells us that MPG.city and EngineSize are inversely related: as EngineSize increases (decreases), MPG.city decreases (increases). We know this because the regression coefficient *b*_{1} = –10.87. The intercept term *b*_{0} = 46.15 means less to us, except when we make predictions, because it implies that a vehicle with 0 liters of displacement should get 46.15 miles per gallon in city driving.

10. What is the strength of association between the two variables, MPG.city and Engine Size? Find the coefficient of determination *r*^{2} using the following expression for *r*^{2} (do not use the summary() function to unpack the regression statistics; we will use it later). This exercise provides another opportunity to hone your coding skills.

*#Comment1. Find the total sum of squares, ss_y.*

ss_y <- sum((E12_2$MPG.city - mean(E12_2$MPG.city)) ^ 2)

*#Comment2. Find the residual sum of squares, ss_res.*

ss_res <- sum((resid(slr2)) ^ 2)

*#Comment3. Find the coefficient of determination.*

(ss_y - ss_res) / ss_y

## [1] 0.505143

**Answer:** The coefficient of determination, *r*^{2} = 0.505143.

11. What does the coefficient of determination *r*^{2} reveal about the regression model?

**Answer:** The *r*^{2} indicates the proportion of variation in the dependent variable MPG.city that is explained (or accounted for) by variation in the independent vari- able EngineSize. In this case, that proportion is 0.505143, or roughly 51%. More- over, because *r*^{2} = 0*.*505143, we also know that almost 49% of the variation in MPG.city remains unaccounted for, even once the association with EngineSize has been considered.

12. What is the *t *value of the coefficient *b*_{1} on the independent variable EngineSize? Do not use the summary() function but rather write out the R code (great practice).

**Answer:** *t *= *−*6*.*926538

Because finding the answer to this question requires a slightly more complicated bit of code, we break up the solution into several pieces.

(a) The expression for the *t *value is found by taking the ratio of the coefficient itself to the standard error.

(b) Finding the denominator (i.e., the standard error *s _{b1}*) of the above expression

requires calculating another ratio

where the numerator of this ratio *s _{y/x}* is

s_xy <- sqrt(sum((resid(slr2) ^ 2)) / (nrow(E12_2) - 2))

s_xy

## [1] 4.061098

and where the denominator of this ratio is

ssx <- sqrt(sum((E12_2$EngineSize - mean(E12_2$EngineSize)) ^ 2))

ssx

## [1] 2.587174

The ratio can now be found by dividing the rst value (above) by the second.

This value is *s _{b1}* :

sb1 <- s_xy / ssx

sb1

## [1] 1.569704

(c) The numerator of the t statistic requires the regression coefficient *b*_{1}

b1 <- sum((E12_2$EngineSize - mean(E12_2$EngineSize)) *

(E12_2$MPG.city - mean(E12_2$MPG.city))) /

sum((E12_2$EngineSize - mean(E12_2$EngineSize)) ^ 2)

b1

## [1] -10.87261

(d) Finally, the *t *statistic is found by dividing the regression coefficient *b*_{1 }by the standard error *s _{b}*

_{1}.

t = b1 / sb1

t

## [1] -6.926538

13. What is the p-value of *t *= *−*6*.*926538?

**Answer: **the p-value=(2)(*p*(*t **≤ **t *= *−*6*.*926538*, df *= 47)) = 0*.*00000001056443.

Note: For convenience and accuracy, we use *t *from the preceding exercise as the first argument of the pt() function

*#Comment1. Since the p-value statistic has a very small value, we#can override the default of reporting it in scientific notation.*

options(scipen = 999)

*#Comment2. Use the pt() function with (n-2)=47 degrees of freedom.#Remember that since this is a two-tail test, we need to multiply*

#by 2.

2 * pt(t, 47)

## [1] 0.00000001056443

14. Use the summary() extractor function to check our work. Remember to use the

model object slr2 as the argument.

**Answer:**

*#Comment Use summary() function to extract the desired statistics.*

*summary(slr2)*

##

## Call:

12

## lm(formula = MPG.city ~ EngineSize, data = E12_2)

##

## Residuals:

## Min 1Q Median 3Q Max

## -15.0177 -1.5814 -0.1451 1.7676 12.1568

##

## Coefficients:

## Estimate Std. Error t value Pr(>|t|)

## (Intercept) 46.15 3.01 15.333 < 0.0000000000000002 ***

## EngineSize -10.87 1.57 -6.927 0.0000000106 ***

## ---

## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 '.' 0.1 ' '1

##

## Residual standard error: 4.061 on 47 degrees of freedom

## Multiple R-squared: 0.5051,Adjusted R-squared: 0.4946

## F-statistic: 47.98 on 1 and 47 DF, p-value: 0.00000001056

All the findings arrived at using the summary() function confirm what has been found in the preceding exercises. That is, the estimated regression equation is ŷ = 46*.*15 *−* 10*.*87*x*; the coefficient of determination is *r*^{2} = 0*.*505143; the *t* statistic is *t *= *−*6*.*926538; and the p-value=0.00000001056443.

15. Use the regression equation to find the predicted values of MPG.city for the follow- ing values of EngineSize (liters of displacement): 1.25, 1.50, 1.75, 2.00, 2.25.

**Answer:** The predicted values of MPG.city for EngineSize of 1.25, 1.50, 1.75, 2.00,and 2.25 liters are (in order) 32.56138, 29.84322, 27.12507, 24.40692, and 21.68876 miles per gallon.

*#Comment1. Use data.frame() to create a new object containing 1.25,#1.50, 1.75, 2.00, and 2.25. Name the new object size_new.*

size_new <- data.frame(EngineSize <- c(1.25, 1.50, 1.75, 2.00, 2.25))

*#Comment2. Use the predict() function to provide predicted values#of miles per gallon for vehicles having 1.25, 1.50, 1.75, 2.00, and#2.25 liters EngineSize.*

predict(slr2, size_new)

## 1 2 3 4 5

## 32.56138 29.84322 27.12507 24.40692 21.68876

16. What are the predicted values of MPG.city that were used to calibrate the estimated

regression equation *ŷ* = 46.15–10.87*x*? Import those predicted values into an object

named mileage predicted and list the rst and last three elements.

**Answer:**

*#Comment1. Use fitted(slr2) function to create the predicted#values of the dependent variable. Import those values into#the object named mileage_predicted.*

mileage_predicted <- fitted(slr2)

*#Comment2. Use the head(,3) and tail(,3) functions to list the#first and final three values of the predicted value.*

head(mileage_predicted, 3)

## 1 6 12

## 26.58144 22.23239 22.23239

tail(mileage_predicted, 3)

## 90 92 93

## 24.40692 21.14513 20.05787

17. Add the mileage predicted object (created in the preceding exercise) to E12_2, and name the resulting object E12_3. List the first and last four elements. Find the correlation of the actual and predicted variables; that is, the correlation of MPG.city and mileage predicted. Once you have calculated the correlation, square it (i.e., raise it to the second power). Does the squared correlation coefficient look familiar?

**Answer: **

*#Comment1. Use the cbind() function to bind the column#mileage_predicted #to E12_2. Name the new object E12_3.*

E12_3 <- cbind(E12_2, mileage_predicted)

*#Comment2. List the first and last four elements of E12_3.*

head(E12_3, 4)

## MPG.city EngineSize mileage_predicted

## 1 25 1.8 26.58144

## 6 22 2.2 22.23239

## 12 25 2.2 22.23239

## 13 25 2.2 22.23239

tail(E12_3, 4)

## MPG.city EngineSize mileage_predicted

## 88 25 1.8 26.58144

## 90 21 2.0 24.40692

## 92 21 2.3 21.14513

## 93 20 2.4 20.05787

*#Comment3. Find the correlation of the actual and predicted#dependent variables. Store the value in an object named r.*

r <- cor(E12_3$MPG.city, E12_3$mileage_predicted)

r

## [1] 0.7107341

*#Comment4. Square the value of r.*

r^2

## [1] 0.505143

The square of the correlation of the *actual* dependent variable and *predicted dependent* variable equals the coefficient of determination, *r*^{2}.

18. Create a scatterplot with MPG.city on the vertical axis, Engine Size on the hor- izontal axis. Add labels to both axes as well as a main title. Finally, using the abline() function, add a regression line to the scatterplot.

**Answer: **

plot(E12_2$EngineSize, E12_2$MPG.city,

xlab = 'Engine 'Size (liters)',

ylab = 'City Miles per Gallon',

main = 'The Best Line Through the Scatterplot',

pch = 19,

col = 'blue')

abline(slr2)

19. For additional practice structuring our data before analyzing it, we now subset E12_1 by Origin. For this exercise: (1) create a new object from the original data set, E12 _1, that includes only vehicles of non-USA origin (thus excluding all vehicles of USA origin) and name it E12_4; (2) find the median EngineSize of non-USA vehicles, and (3) create a new object, named E12_5, that includes only (a) those vehicles having MPG.city less than or equal to the median and only (b) the two variables, MPG.city and EngineSize. In other words, subset the original data set, E12_1, to include only the two variables, MPG.city and EngineSize, and only those vehicles that are of non-USA origin and that feature engines with displacement (in liters) at or below the median for the relevant category. Just to make sure E12_5 “looks” as it should, run a few of the same functions that were used in Exercise 1.

*#Comment1. Subset data so that it includes only vehicles of non-USA*

*#origin. Name new object E12_4.*

E12_4 <- E12_1[ which(E12_1$Origin == 'non-USA' ), ]

*#Comment2. Find the median EngineSize for the sample including only#non-USA vehicles*.

median(E12_4$EngineSize)

## [1] 2.2

*#Comment3. Since the median is 2.2 liters, subset the data once#again to include only vehicles with 2.2 liters (or less) engine#displacement. Name the new object E12_5.*

E12_5 <- E12_4[E12_4$EngineSize <= 2.20, c( 'MPG.city', 'EngineSize' )]

*#Comment4. Use the summary() function to find the basic descriptive#statistics for MPG.city and EngineSize*

summary(E12_5$MPG.city)

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 17.00 22.75 25.50 27.54 29.25 46.00

summary(E12_5$EngineSize)

## Min. 1st Qu. Median Mean 3rd Qu. Max.

## 1.000 1.500 1.700 1.708 2.000 2.200

*#Comment5. Use nrow() function to find the number of observations.*

nrow(E12_5)

## [1] 24

*#Comment6. Use the names() function to confirm that E12_5 includes#only two variables, MPG.city and EngineSize.*

names(E12_5)

## [1] "MPG.city" "EngineSize"

**Answer:** The new data set, E12_5, includes 24 observations across the two vari- ables, MPG.city and EngineSize. The descriptive statistics for both variables are reported under Comment 4. Note that the maximum value of EngineSize is 2.2, thus confirming that the data include only what we want, in terms of observations, variables, and restrictions. This practice of “looking under the hood (or bonnet)” is a sound one. It allows us to con rm that the data really do look like they should.

20. For the category of vehicles of non-USA origin, do the two variables, MPG.city and

EngineSize, seem to be related in a systematic way? If so, how?

plot(E12_5$EngineSize, E12_5$MPG.city,

pch = 19,

xlab = 'Engine Size (liters)',

ylab = 'City Miles per Gallon',

main = 'Relationship Between City MPG and Engine Size (liters) ')

**Answer: **Yes, the pattern of points appears to run (approximately) from the upper- lefthand to lower-righthand corners of the scatterplot, implying a negative associa- tion: larger engine sizes are associated with lower miles per gallon (in city driving). But because of a few outlier cases, we do not expect the *r*^{2} to be very high, perhaps not even *r*^{2} = 0*.*50.

21. Make and inspect a residual plot. Does the pattern of points reveal any reason why we should not use regression to analyze these data? Are there any radical departures from the assumptions underlying the appropriate usage of this methodology?

*#Comment1. Use the lm() function to create the model object named#slr3 (slr3 stands for the third simple linear regression).*

slr3 <- lm(MPG.city ~ EngineSize, data = E12_5)

*#Comment2. Use the plot() function to create a residual plot.#Note that resid(slr3) must be included as an argument.*

plot(E12_5$EngineSize,

resid(slr3),

abline(h = 0),

pch = 19,

xlab = 'Engine Size (liters) ',

ylab = 'Residuals',

main = 'Residuals Against the Independent Variable')

**Answer: **The only possible area of trouble revealed by the residual plot might be in the range of the smaller engines, particulary at and below 1.5 liters. For vehicles with larger engines, however, the assumption of constant variation seems satisfied. Therefore, although the data are far from what we might characterize as “well behaved,” the violations do not seem serious enough to cause us to drop regression as a potentially promising analytical methodology.

22. What is the estimated regression equation?

slr3

##

## Call:

## lm(formula = MPG.city ~ EngineSize, data = E12_5)

##

## Coefficients:

## (Intercept) EngineSize

## 51.27 -13.89

**Answer:**The estimated regression equation is

*ŷ*= 51.27-13.89x. The predicted dependent variable is

*ŷ,*MPG.city; the independent variable is x, or EngineSize. Note that to recover this minimal information, we need only enter the model object, slr3.

23. Find the 75 and 90 percent con dence interval estimates of the regression coefficient *b*_{1}. How should we interpret the meaning of these con dence interval estimates?

confint(slr3, level = 0.75)

## 1 2.5 % 87.5 %

## (Intercept) 44.60803 57.94082

## EngineSize -17.72209 -10.06260

confint(slr3, level = 0.90)

## 5 % 95 %

## (Intercept) 41.58615 60.962698

## EngineSize -19.45811 -8.326578

**Answer: **There is a 75% probability that the regression coefficient falls in the interval from -17.72209 to -10.06260; there is a 90% probability that it falls in the interval from -19.45811 to -8.326578.

24. What does the estimated regression equation tell us?

**Answer: **The estimated regression equation *ŷ* = 51.27 – 13.89*x* can be interpreted in this manner: for the category of vehicles of non-USA origin|and with engine sizes of 2.2 or fewer liters|a change of 1 liter in engine size is associated with a change of 13.89 miles per gallon when the vehicle is driving in a city. The negative sign, *b*_{1} = –13.89, also tells us that as EngineSize increases (decreases), MPG.city decreases (increases). Although the intercept term, *b*_{0} = 51.27, is not meaningfully interpretable, we will find it necessary to include it in the regression equation when ever we want to forecast or make predictions.

25. What is the strength of association between the two variables, MPG.city and Engine Size? Find the coefficient of determination *r*^{2} using the following expression for r^{2} (do not use the summary() function to unpack the regression statistics; we will use it later). This exercise provides another opportunity to hone your coding skills.

*#Comment1. Find the total sum of squares, ss_y.*

ss_y <- sum((E12_5$MPG.city - mean(E12_5$MPG.city)) ^ 2)

*#Comment2. Find the residual sum of squares, ss_res.*

ss_res <- sum((resid(slr3)) ^ 2)

*#Comment3. Find the coefficient of determination.*

(ss_y - ss_res) / ss_y

## [1] 0.455044

**Answer: **The coefficient of determination, *r*^{2} = 0.455044.

26. What does the coefficient of determination *r*^{2} tell us about the regression model?

**Answer:** The *r*^{2} is the proportion of variation in the dependent variable MPG.city that is accounted for (or explained) by variation in EngineSize, the independent variable. When *r*^{2} = 0*.*455044, we understand that that proportion is about 45%. We also know that approximately 55% of variation in MPG.city remains unex- plained, even after taking EngineSize into account.

27. What is the *t *value of the coefficient *b*_{1} on the independent variable EngineSize? Do not use the summary() function but rather write out the code (more practice).

**Answer:** *t* = –4.286

Because nding the answer to this question requires a slightly more complicated bit of code, we break up the solution into several pieces.

(a) The expression for the t value is found by taking the ratio of the coefficient

itself to the standard error.

(b) Finding the denominator (i.e., the standard error s_{b1}) of the above expression

requires calculating another ratio

where the numerator of this ratio *s*_{y|x} is

s_xy <- sqrt(sum((resid(slr3) ^ 2)) / (nrow(E12_5) - 2))

s_xy

## [1] 5.304575

and where the denominator of this ratio is

ssx <- sqrt(sum((E12_5$EngineSize - mean(E12_5$EngineSize)) ^ 2))

ssx

## [1] 1.636561

The ratio can now be found by dividing the rst value (above) by the second.

This is the value for *s*_{b1}

sb1 <- s_xy / ssx

sb1

## [1] 3.241293

(c) The numerator of the t statistic requires the regression coefficient *b*_{1}

b1 <- sum((E12_5$EngineSize - mean(E12_5$EngineSize)) *

(E12_5$MPG.city - mean(E12_5$MPG.city))) /

sum((E12_5$EngineSize - mean(E12_5$EngineSize)) ^ 2)

b1

## [1] -13.89235

(d) Finally, the *t* statistic is found by dividing the regression coefficient *b _{1}* by the

standard error

*s*.

_{b1}t = b1 / sb1

t

## [1] -4.286051

28. What is the p-value of *t* = –4.286051?

**Answer:** p-value= (2)(p(t ≤ t = –4.286051; *df* = 22)) = 0.0003000032.

Note: For convenience and accuracy, we use the *t* from the preceding exercise as the

rst argument of the pt() function.

*#Comment. Use the pt() function with (n-2)=22 degrees of freedom.**#Remember that since this is a two-tail, we need to multiply by 2.*

2 * pt(t, 22)

## [1] 0.0003000032

29. Use the summary() extractor function to check our work. Remember to use the

model object slr3 as the argument.

**Answer: **summary(slr3)

##

## Call:

## lm(formula = MPG.city ~ EngineSize, data = E12_5)

##

## Residuals:

## Min 1Q Median 3Q Max

## -16.2144 -1.7278 -0.6574 1.8710 11.5641

##

## Coefficients:

## Estimate Std. Error t value Pr(>|t|)

## (Intercept) 51.274 5.642 9.088 0.00000000668 ***

## EngineSize -13.892 3.241 -4.286 0.0003 ***

## ---

## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '. 0.1 ' '1

##

## Residual standard error: 5.305 on 22 degrees of freedom

## Multiple R-squared: 0.455,Adjusted R-squared: 0.4303

## F-statistic: 18.37 on 1 and 22 DF, p-value: 0.0003

All findings arrived at using the summary() function con rm what has been found in the preceding exercises. The estimated regression equation is *ŷ* = 51.274-13.892*x*; the coefficient of variation is *r*^{2} = 0.455; the t statistic is t = -4.286051; and the p-value=0.0003000032.

30. Use the estimated regression equation to nd the predicted values of MPG.city for the following values of EngineSize (liters of displacement): 1.25, 1.50, 1.75, 2.00, 2.25.

**Answer: **The predicted values of MPG.city for EngineSize of 1.25, 1.50, 1.75, 2.00, and 2.25 liters are (in order) 33.90899, 30.43591, 26.96282, 23.48973, and 20.01665 miles per gallon.

*#Comment1. Use data.frame() to create a new object containing 1.25,#1.50, 1.75, 2.00, and 2.25. Name the new object size_new.*

size_new <- data.frame(EngineSize <- c(1.25, 1.50, 1.75, 2.00, 2.25))

*#Comment2. Use predict() function to provide the predicted values#of miles per gallon for vehicles having 1.25, 1.50, 1.75, 2.00, and*

*#2.25 liters EngineSize.*

predict(slr3, size_new)

## 1 2 3 4 5

## 33.90899 30.43591 26.96282 23.48973 20.01665

31. What are the predicted values of MPG.city that were used to calibrate the estimated regression equation *ŷ*= 51.274–13.892x? Import those predicted values into an object named mileage predicted and list the first and last three elements.

**Answer: **

*#Comment1. Use fitted(slr3) function to create the predicted#values of the dependent variable. Import those values into#the object named mileage_predicted.*

mileage_predicted <- fitted(slr3)

*#Comment2. Use the head(,3) and tail(,3) functions to list the#first and final three values of the predicted value.*

head(mileage_predicted, 3)

## 1 9 40

## 26.26820 37.38208 29.04667

tail(mileage_predicted, 3)

## 86 88 90

## 20.71126 26.26820 23.48973

32. Add the mileage predicted object (created in the preceding exercise) to E12_5, and name the resulting object E12_6. List the rst and last four elements. Find the correlation of the actual and predicted variables; that is, the correlation of MPG.city and mileage predicted. Once you have the correlation, square it (i.e., raise it to the second power). Comment on the square of the correlation. What is it?

**Answer:**

** **#Comment1. Use the cbind() function to bind the column

#mileage_predicted #to E12_5. Name the new object E12_6.

E12_6 <- cbind(E12_5, mileage_predicted)

*#Comment2. List the first and last four elements of E12_6.*

head(E12_6, 4)

## MPG.city EngineSize mileage_predicted

## 1 25 1.8 26.26820

## 39 46 1.0 37.38208

## 40 30 1.6 29.04667

## 42 42 1.5 30.43591

tail(E12_6, 4)

## MPG.city EngineSize mileage_predicted

## 85 25 2.2 20.71126

## 86 22 2.2 20.71126

## 88 25 1.8 26.26820

## 90 21 2.0 23.48973

*#Comment3. Find the correlation of the actual and predicted#dependent variables. Store the value in an object named r.*

r <- cor(E12_6$MPG.city, E12_6$mileage_predicted)

r

## [1] 0.6745695

*#Comment4. Square the value of r.*

r^2

## [1] 0.455044

The square of the correlation of the *actual* dependent variable and *predicted* *dependent* variable equals the coefficient of determination, *r ^{2}*.

33. Create a scatterplot with MPG.city on the vertical axis, Engine Size on the horizontal axis. Add labels to both axes as well as a main title; set blue as the color of the points. Finally, using the abline() function, add a regression line to the scatterplot.

**Answer: **

plot(E12_6$EngineSize, E12_6$MPG.city,

xlab = 'Engine Size (liters)',

ylab = 'City Miles per Gallon,

main = 'The Best Line Through the Scatterplot',

pch = 19,

col = 'blue' )

abline(slr3)

34. This exercise provides further opportunity to (1) find a set of data from an online source (pick a source, any source), (2) create a data frame (see the Chapter 1 Appendix, if necessary), and (3) analyze it using some of the methods associated with simple linear regression. Look up and record the high and low intraday temperatures (in degrees Fahrenheit) for the following 14 cities from around the world: Auckland, Beijing, Cairo, Lagos, London, Mexico City, Mumbai, Paris, Rio de Janeiro, Sydney, Tokyo, Toronto, Vancouver, and Zurich. This information (in degrees Fahrenheit) is easily found after brief search.

(a) Use c() to create 3 objects, one for each city name, one for the high temperature, one for the low temperature. Data are recorded for December 19, 2016.

**Answer:**

city <- c( 'Auckland ',' Beijing ', 'Cairo ', 'Lagos' ,' London','Mexico City', 'Mumbai','Paris','Rio de Janeiro' 'Sydney' , ' Tokyo' , ' Toronto' , ' Vancouver' , 'Zurich' )

high <- c(71, 45, 65, 91, 46, 67, 88, 44, 92, 88, 57, 20, 42, 40)

low <- c(56, 23, 48, 76, 37, 45, 71, 35, 73, 65, 39, 15, 39, 29)

(b) Use data.frame() to create a data frame consisting of each city name and high and low temperatures. Display the result to check your work.

**Answer:**

WorldTemps <- data.frame(City = city, High = high, Low = low)

WorldTemps

## City High Low

## 1 Auckland 71 56

## 2 Beijing 45 23

## 3 Cairo 65 48

## 4 Lagos 91 76

## 5 London 46 37

## 6 Mexico City 67 45

## 7 Mumbai 88 71

## 8 Paris 44 35

## 9 Rio de Janeiro 92 73

## 10 Sydney 88 65

## 11 Tokyo 57 39

## 12 Toronto 20 15

## 13 Vancouver 42 39

## 14 Zurich 40 29

(c) Make a scatterplot of high against low temperatures. Create a main title, label each axis appropriately, and use pch= to specify how the points should appear. Does the pattern of points appear to con rm that the relationship between high and low temperatures is linear?

plot(WorldTemps$Low, WorldTemps$High,

pch = 19,

xlab = "Low",

ylab = "High",

main = "High and Low Intraday Temperatures")

**Amswer: **The scatterplot makes clear that the relationship between high and low intraday temperatures is both positive and linear.

(d) Estimate and write out the regression equation: *ŷ* = *b*_{0} + *b*_{1}*x*. Let the high temp be the dependent variable; the low temp, the independent variable.

**Answer:**

reg_eq_temps <- lm(High ~ Low, data = WorldTemps)

reg_eq_temps

##

## Call:

## lm(formula = High ~ Low, data = WorldTemps)

##

## Coefficients:

## (Intercept) Low

## 7.763 1.148

The estimated regression equation is: *ŷ* = 7.763 + 1.148*x*

(e) What is the *r*^{2}?

**Answer:**

summary(reg_eq_temps)

##

## Call:

## lm(formula = High ~ Low, data = WorldTemps)

##

## Residuals:

## Min 1Q Median 3Q Max

## -10.533 -3.991 -1.051 3.884 10.834

##

## Coefficients:

## Estimate Std. Error t value Pr(>|t|)

## (Intercept) 7.76313 4.27689 1.815 0.0946 .

## Low 1.14795 0.08546 13.433 0.0000000136 ***

## ---

## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '. ' 0.1 ' '1

##

## Residual standard error: 5.918 on 12 degrees of freedom

## Multiple R-squared: 0.9376,Adjusted R-squared: 0.9325

## F-statistic: 180.5 on 1 and 12 DF, p-value: 0.00000001363

The *r*^{2} is 0.9376, indicating that approximately 93.76% of variation in the dependent variable is explained by variation in the independent variable.

(f) What is the p-value? Is the estimate regession equation signi cant? Why or why not

**Answer: **

Since p-value=0.0000000136 is far less than the usual values we set for α—e.g., 0.05, 0.01, etc.—we say that the estimated regression equation is significant.

35. A dependent variable *y* is regressed on an independent variable* x*; the sample size is *n* = 32.

(a) If *SS _{reg}* = 808.89 and

*SS*= 317.16, what is the

_{res}*r*

^{2}?

**Answer:** 0.7183.

*SS*_{y} = *SS*_{reg} + *SS*_{res} = 808.89 + 317.16 = 1126.05

(b) If *b1 *= –0.041215 and *sb1* = 0.004712, what is the value of the test statistic *t*?

**Answer: **

(c) What is the p-value?

**Answer: **= 2(*p(t* < –8.75; df = n–k–1)) = 2(*p(t* < –8.75; *df* = 30)) = 0.00000000093

2 * pt(-8.75, 30)

## [1] 0.0000000009313949

(d) Is the estimated regression equation significant at the *α* = 0.01 level?

**Answer: **Yes, since p-value = 0.00000000093 <α = 0.01, we conclude that the estimated

regression equation is significant.

(e) If *b*_{0} = 29.599855, write out the regression equation, *ŷ* = *b _{0} + b_{1x}*.

**Answer: ***ŷ* = 29.599855 – 0.041215*x*

36. This exercise uses the mtcars data set that is installed in R. (Remember that to see all the installed data sets, simply enter data() at the R prompt in the Console; to view the mtcars data set itself, enter mtcars at the R prompt; to learn more about the data set, including the variables and observations, enter ?mtcars at the prompt and wait for the R help page to open.) In this case, we are interested in the relationship between an automobile's quarter mile time and gross horsepower.

(a) Create a scatterplot of the 2 variables. What does the pattern of points suggest about the relationship (if any) between the variables?

**Answer: **The scatterplot makes clear that the relationship between gross horse power and quarter mile time (seconds) is both negative and (approximately) linear.

plot(mtcars$hp, mtcars$qsec,

pch = 19,

xlab = "Gross Horse Power",

ylab = "Quarter Mile Time (seconds)")

(b) Letting the quarter mile time be the dependent variable, estimate the regression equation. Write out the regression equation, *ŷ* = *b*_{0}* + b** _{1x}*.

**Answer: **

reg_eq_mtcars <- lm(qsec ~ hp, data = mtcars)

reg_eq_mtcars

##

## Call:

## lm(formula = qsec ~ hp, data = mtcars)

##

## Coefficients:

## (Intercept) hp

## 20.55635 -0.01846

The estimated regression equation is: *ŷ* = 20.55635 – 0.01846*x*.

(c) What is the *r ^{2}*?

**Answer: **

summary(reg_eq_mtcars)

##

33

## Call:

## lm(formula = qsec ~ hp, data = mtcars)

##

## Residuals:

## Min 1Q Median 3Q Max

## -2.1766 -0.6975 0.0348 0.6520 4.0972

##

## Coefficients:

## Estimate Std. Error t value Pr(>|t|)

## (Intercept) 20.556354 0.542424 37.897 < 0.0000000000000002 ***

## hp -0.018458 0.003359 -5.495 0.00000577 ***

## ---

## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##

## Residual standard error: 1.282 on 30 degrees of freedom

## Multiple R-squared: 0.5016,Adjusted R-squared: 0.485

## F-statistic: 30.19 on 1 and 30 DF, p-value: 0.000005766

The *r2* is 0.5016, indicating that approximately 50.16% of variation in the dependent variable is explained by variation in the independent variable.

(d) What is the p-value?

**Answer:** p-value=0.000005766

(e) Is the estimated regression equation signi cant at the *α *= 0.05 level?

**Answer: **Since p-value=0.000005766 is far less than the usual values for *α*, we say that the estimated regression equation is significant.

37. Please use the mtcars data set to answer the following questions.

(a) Find the predicted quarter mile time for the values of gross horsepower from the data set used in the original analysis. Report the predicted values for the last four observations.

**Answer: **

tail(fitted(reg_eq_mtcars) , 4)

## Ford Pantera L Ferrari Dino Maserati Bora Volvo 142E

## 15.68336 17.32615 14.37282 18.54440

(b) Find the predicted values of quarter mile time for the following values of gross horsepower: 100, 125, 160, 225, and 250.

**Answer:**

new_values <- data.frame(hp <- c(100, 125, 160, 225, 250))

predict(reg_eq_mtcars, new_values)

## 1 2 3 4 5

## 18.71052 18.24906 17.60302 16.40323 15.94178

(c) Can we use the estimated regression equation to make predictions of quarter mile time when gross horsepower is 40 or 350? Why or why not?

**Answer:** When we learn that the minimum and maximum values of the variable hp are 52 and 335, respectively, we should not use the estimated regression equation to make predictions based on values that fall above or below that range. Some analysts will do so anyway, but one should proceed very carefully when making any predictive claims based on this estimated regression equation.

min(mtcars$hp)

## [1] 52

max(mtcars$hp)

## [1] 335

38. This exercise explores the relationship (if any) between 2 of the 5 variables included in the polling data (found on the companion website): *x*_{1} =Age, measured in years, and *x*_{3} =Same Sex, which is measured on a 1-to-7 Likert scale as a response to the statement," approve of the right of same-sex couples to marry." A respondent registers strong disapproval with a 1, strong approval with a 7, and relative indi erence with a response in the middle of the range from 1-to-7.

(a) Make a scatterplot of x_{3} against x_{1}. Do you see any possible violations of the assumptions underlying the correct application of simple linear regression to these data? What does the nature of the pattern tell you? Do you think regression can be used to explore the relationship between the 2 variables?

**Answer: **The scatterplot reveals the negative and (relatively) linear relationship between a person's age and the extent to which he approves of the right of same-sex couples to marry. That is, in general, resistance to the idea that same-sex couples should have the right to marry seems to increase with one's age. However, as with most plots, the relationship is not a perfect one. Even so, regression analysis would seem to be a promising means by which to explore the relationship between these 2 variables.

plot(polling$x1, polling$x3,

xlab = "Age",

ylab = "Views of Same-Sex Marriage",

pch = 19)

(b) Write out the regression equation. In this case, does it make more sense to specify *x _{1}* or

*x*as the dependent variable? That is, should you de ne the model as

_{3}*x*? Or as

_{3}= b_{0}+ b_{1}x_{1}*x*=

_{1}*b*

_{0}+

*b*

_{3}

*x*

_{3}? Why?

**Answer: **We would most likely specify *x _{3}*, Same-Sex Marriage, as the dependent variable and

*x*, Age, as the independent variable. Although we do not use regression analysis to demonstrate causality, it makes more sense to say that approval of the right of same-sex couples to marry falls with age than the reverse.

_{1}The estimated regression equation is *ŷ* = 11.757 – 0.168x.

reg_eq_polling <- lm(x3 ~ x1, data = polling)

reg_eq_polling

##

## Call:

## lm(formula = x3 ~ x1, data = polling)

##

## Coefficients:

## (Intercept) x1

## 11.757 -0.168

(c) What is* r ^{2}*?

summary(reg_eq_polling)

##

## Call:

## lm(formula = x3 ~ x1, data = polling)

##

## Residuals:

## Min 1Q Median 3Q Max

## -1.8535 -1.0235 -0.2935 0.6705 2.8025

##

## Coefficients:

## Estimate Std. Error t value Pr(>|t|)

## (Intercept) 11.7573 1.2276 9.577 0.000000159 ***

## x1 -0.1680 0.0265 -6.340 0.000018289 ***

## ---

## Signif. codes: 0 '***' 0.001 '*'* 0.01 '*' 0.05 '.' 0.1 '' 1

##

## Residual standard error: 1.296 on 14 degrees of freedom

## Multiple R-squared: 0.7417,Adjusted R-squared: 0.7232

## F-statistic: 40.19 on 1 and 14 DF, p-value: 0.00001829

**Answer:** *r*^{2} = 0.7417

(d) What is the p-value?

**Answer:** p-value=0.00001829

(e) Is the regression equation significant at the *α* = 0.05 level?

**Answer:** Since p-value=0.00001829 is less than the usual values for *α*, we say that the

estimated regression equation is significant.

(f) Find the 95% confidence interval estimate of the regression coefficient?

**Answer: **

confint(reg_eq_polling, level = 0.95)

## 2.5 % 97.5 %

## (Intercept) 9.1242484 14.3903218

## x1 -0.2248278 -0.1111626

(g) State in words the meaning of the con dence interval estimate.

**Answer:** There is a 95% probability that the regression coefficient falls in the interval

from -0.2248278 to -0.1111626.

(h) What are your conclusions about the regression analysis.

**Answer:** The estimated regression equation *ŷ* = 11.757-0.168*x* allows us to conclude that a change of 1 year is associated with a change of -0.168 in approval. (We know this because the regression coefficient is *b*_{1} =-0.168.) In this case, the meaning of the intercept term, *b*_{0} = 11.757, is less clear because it represents the predicted value of approval for a person whose age is 0. Even so, it is important to retain the intercept term in the equation because it must be included when we want to make predictions.