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# DATA 303/473 Assignment 2 solution

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## Q1.(20 marks)

We’ll continue to use the CarDekho data from Assignment 1. As a reminder the variables
in the cardekho2.csv dataset are:
• price: Selling price in thousand Indian Rupees (INR)
• make: Car make grouped into eight categories: Ford, Honda, Hyundai, Mahindra, Maruti, Tata,
Toyota, 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 Manual
• 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)

The residual diagnostic plot showed evidence of non-linear relationships between price and some predictors,
non-normality and non-constant variance. To address non-constant variance, use log(price) as the response
variable for this assignment.
a. (3 marks) Fit a model with log(price) as the response variable and include all predictors without
transformations or interactions. Use the plot function to carry out residual diagnostics for your fitted
model. Based on these plots, are there any observations you might consider excluding from further

Some data cleaning is done and a new dataset, cardekho3.csv, (available on Canvas) is created. Use this
new dataset to answer the rest of Question 1.
b. (3 marks) Read in dataset cardekho3.csv and fit the same model as in part (a). Plot the residuals
from your fitted model against each of the numerical predictors kms, mileage, esize and power. Is
there an indication of a non-linear relationships with log(price) for any of these predictors? If so,
which ones?

c. (3 marks) Based on the model fitted in part (b), calculate and give an interpretation for the difference
in price for a petrol car compared to a diesel car when all other predictors are held constant?
d. (4 marks) Based on the dataset and model in part(b), provide two plots that give graphical evidence
that a log transformation is the most appropriate transformation for kms in a model for log(price).

e. (3 marks) Apply stepwise regression based on the AIC criterion for the model in part (b). Are there
any predictors you would exclude from the model? Explain your answer briefly.
1
f. (4 marks) Fit a model you would use to investigate whether the effect of mileage on log(price)
depends on the value of tx. Based on your model, give the change in E(log(price)) associated with a
unit increase in mileage for a car with:

(i) Automatic transmission
(ii) Manual transmission.

## Q2.(20 marks)

Data were collected on 158 cruise ships in operation around the world in 2013. Complaints
had been raised by customers about overcrowding on cruises and there was interest in investigating whether
there was a trend of overcrowding on certain types of ships. As part of the investigation, a regression analysis
was carried out to explore the connection between passenger density (no. of passengers per unit area) and
ship characteristics. The variables in the dataset were:
• name: Ship Name

• line: Cruise Line
• line_grp: Cruise Line grouped
• age.2013: Age (as of 2013)
• tonnage: Weight of ship (1000s of tonnes)
• passengers.100: Maximum no. of passengers (100s)
• length: Length of ship (100s of feet)
• cabins: No. of passenger cabins (100s)

• pass.density: Passenger density (no. of passengers per square foot)
• crew.100: No. of crew member (100s)
The data are available in the file cruise_ship.csv. The dataset was imported into R and the scatterplot
matrix below was obtained.
age.2013
0 150 4 10 0 15
10 30
0 100
−0.69

tonnage
−0.57
0.94
passengers.100
0 20 40
4 8 12
−0.63
0.94
0.87

length
−0.57
0.93
0.98
0.86
cabins
0 10 20
0 10 20
−0.59
0.93
0.91
0.88
0.94

crew.100
10 40
−0.23
0.04
0
−0.23
30
0.07

0
−0.18
15
−0.06
20 40 60
20 60
pass.density
2

The scatterplot matrix indicates severe multicollinearity among the predictors tonnage, passengers.100,
length, and crews.100. These four predictors are all related to the size of a ship, so only a subset will be
used.
a. [8 marks] Fit a model for pass.density using the predictors line_grp, age.2013, passengers.100
and length. Using residual diagnostic checks, determine whether any transformations of the predictors
or response variable are necessary.

you may need to transform. Provide output of any graphical checks or hypothesis tests you perform.
For the rest of the question use log(pass.density) as the response variable.
b. [3 marks] Fit a model with log(pass.density) as the response variable including all the predictors
in part (a) without any transformations. Apply stepwise regression based on the BIC criterion. Are
there any predictors you would exclude from the model? Explain your answer briefly.

c. [3 marks] Fit a GAM for log(pass.density) and smooth terms for each of the predictors age.2013,
passengers.100 and length. Comment on the non-linearity and significance of smooth terms.
d. [2 marks] Is there evidence that more basis functions are required for any of the smooth terms?