## Description

## Assignment 1 Questions

Q1. (33 marks) Data on selected used car sales in India were obtained from the CarDekho website

(https://www.cardekho.com/). The car sales occurred between 1983 and 2020. The data are available in

the dataset cardekho.csv and include the variables:

• price: Selling price in thousand Rupees (Rs)

• make: Car make grouped into eight categories: Ford, Honda, Hyundai, Mahindra, Maruti, Tata,

Tpyota, Other

• kms: Kilometres driven (x 1000)

• fuel: Fuel type: Diesel or Petrol

• seller: Seller type: Dealer, Individual or Trustmark Dealer

• tx: Transmission type: Automatic or Diesel

• owner: Current owner is: First, Second or Third or above owner

• mileage: Fuel economy in kilometres per litre (kmpl)

• esize: Engine size in cubic centimetres (CC)

• power: Maximum engine power in brake horse power (bhp)

We will analyse the data with price as the response variable and the rest of the variables listed above as

predictors.

a. (4 marks) Use the summary() command to obtain a summary of the variables in the cardekho.csv

dataset. Based on the results:

i) Identify any numerical variables (if any) that may have obviously incorrect values.

ii) Identify any variables (if any) that have missing observations.

Data cleaning is done to filter out some observations and a new dataset, cardekho2.csv, (available on

Canvas) is created. This new dataset should be used to answer the rest of Question 1.

b. (4 marks) Read in the cardekho2.csv dataset and create a scatterplot matrix of all numerical

variables in the dataset.

i) Identify any predictors (if any) that have a non-linear relationship with the response variable

price.

ii) Will there be a need to apply a transformation to the response variable price when fitting a

linear regression model? Explain your answer briefly.

c. (3 marks) Fit a linear model for price including all predictors with no transformations or interactions.

Present a summary of the model in a table. Give an estimate of σ

, the error variance.

d. (2 marks) List the predictor values for a car whose predicted price, E\[Y |X], equals the intercept

βˆ

0 = −745843.41 INR.

e. (4 marks) Based on the model fitted in part (c), give an interpretation of the coefficient for:

i) txManual

ii) mileage

f. (3 marks) Obtain 95% confidence and prediction intervals for the last three observations in the

dataset. Explain briefly why the prediction intervals are wider than the confidence intervals.

g. (4 marks) Use the plot function to carry out residual diagnostics for the model you fitted in part

(c). Comment on what the residual plots indicate about regression assumptions or the existence of

influential observations.

h. (4 marks) Perform hypothesis tests in R to test the assumptions of normality and constant variance

in the errors. Do the results confirm the conclusions you reached in part (g) about these assumptions?

In your response, include the hypotheses being tested in each test.

i. (2 marks) Use the VIF statistic to check whether or not there is evidence of severe multicollinearity

among the predictors. Comment on the results.

j. (4 marks) Based on a global usefulness test, is it worth going on to further analyse and interpret a

model of price against each of the predictors? Carry out the test, give the conclusion and justify your

answer.

Q2. (7 marks) Data were collected on the fatty acid composition of 572 olive oil samples from Italy. There

was interest in investigating the relationship between palmitic acid, and some of the other constituents:

linoleic acid, stearic acid and oleic acid. The data are available in the file olive.csv and were read into R

and analysed as follows:

olive<-read.csv(“olive.csv”, header=T)

str(olive)

## ’data.frame’: 572 obs. of 10 variables:

## $ region : chr “Southern Italy” “Southern Italy” “Southern Italy” “Southern Italy” …

## $ area : chr “North-Apulia” “North-Apulia” “North-Apulia” “North-Apulia” …

## $ palmitic : num 10.75 10.88 9.11 9.66 10.51 …

## $ palmitoleic: num 0.75 0.73 0.54 0.57 0.67 0.49 0.66 0.61 0.6 0.55 …

## $ stearic : num 2.26 2.24 2.46 2.4 2.59 2.68 2.64 2.35 2.39 2.13 …

## $ oleic : num 78.2 77.1 81.1 79.5 77.7 …

## $ linoleic : num 6.72 7.81 5.49 6.19 6.72 6.78 6.18 7.34 7.09 6.33 …

## $ linolenic : num 0.36 0.31 0.31 0.5 0.5 0.51 0.49 0.39 0.46 0.26 …

## $ arachidic : num 0.6 0.61 0.63 0.78 0.8 0.7 0.56 0.64 0.83 0.52 …

## $ eicosenoic : num 0.29 0.29 0.29 0.35 0.46 0.44 0.29 0.35 0.33 0.3 …

palmitic

0.5 65 0.0

6 12

0.0 0.6

0.5 2.0

0.80

palmitoleic

−0.14

−0.17

stearic

1.5 3.0

65 75

−0.84

−0.86

0.11

oleic

0.46

0.64

−0.16

−0.84

linoleic

6 10

0.0 0.4

0.31

0.09

0.06

−0.17

−0.12

linolenic

0.10

−0.01

0.14

−0.19

0.17

0.40

arachidic

0.0 0.6

0.0 0.3 0.6

6

0.50

14

0.40

1.5 3.5

0.11

−0.42

6 14

0.09

0.58

0.0 1.0

0.23

eicosenoic

fit2<-lm(palmitic ~ linoleic + stearic, data=olive)

summary(fit2)

##

## Call:

## lm(formula = palmitic ~ linoleic + stearic, data = olive)

##

## Residuals:

## Min 1Q Median 3Q Max

## -5.193 -1.238 0.069 1.078 4.519

##

## Coefficients:

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

## (Intercept) 10.15814 0.51779 19.618 <2e-16 ***

## linoleic 0.30855 0.02625 11.755 <2e-16 ***

## stearic -0.37847 0.17344 -2.182 0.0295 *

## —

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

##

## Residual standard error: 1.493 on 569 degrees of freedom

## Multiple R-squared: 0.2188, Adjusted R-squared: 0.216

## F-statistic: 79.67 on 2 and 569 DF, p-value: < 2.2e-16

fit3<-lm(palmitic ~ linoleic + stearic + oleic, data=olive)

summary(fit3)

##

## Call:

## lm(formula = palmitic ~ linoleic + stearic + oleic, data = olive)

##

## Residuals:

## Min 1Q Median 3Q Max

## -1.25377 -0.20139 -0.00748 0.19258 1.53568

##

## Coefficients:

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

## (Intercept) 70.70728 0.61290 115.36 <2e-16 ***

## linoleic -0.67472 0.01149 -58.75 <2e-16 ***

## stearic -0.80706 0.04020 -20.08 <2e-16 ***

## oleic -0.68283 0.00678 -100.72 <2e-16 ***

## —

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

##

## Residual standard error: 0.344 on 568 degrees of freedom

## Multiple R-squared: 0.9586, Adjusted R-squared: 0.9584

## F-statistic: 4381 on 3 and 568 DF, p-value: < 2.2e-16

a. (2 marks) In the fit2 model, the coefficient for linoleic is positive, while it is negative in the fit3

model. Making reference to the scaterplot matrix, give a possible explanation for the change in sign

for the linoleic acid coefficient.

b. (2 marks) Use the model in fit3 to obtain 95% confidence and prediction intervals of palmitic for

an olive oil sample with:

• linoleic= 0.3

• stearic = 2.2

• oleic= 73.0

c. (1 mark) If all regression assumptions hold, what other condition would have to be met for the

prediction in part (b) to be valid.

d. (2 marks) Some of the olive oil samples originated from the same region of Italy. What regression

assumption is violated in this case? Explain your answer briefly.

Assignment total: 40 marks