Description
Data Analysis 1
We are interested in analyzing the data contained in the attached file (cardata.txt)
The variables in the data sets are as follows: the car’s manufacturer, the suggested retail price (namely, what the manufacturer thinks the vehicle is worth,
including adequate profit for the automaker and the dealer, in U.S. dollars),
the dealer’s cost (what the dealership pays the manufacturer, in U.S. dollars);
the width (in inches), the length (in inches) and weight of the car (in pounds);
the horse power; the engine size (in liters); the number of cylinders; the fuel
consumption in miles per gallons in the city and on the highway; the wheelbase
(the distance between the centers of the front and rear wheels); and whether
the car is a hybrid (1) or not.
Specifically we wish to study the e↵ects of di↵erent aspects of a car on its
retail price.
1) Answer the question above using the following car’s features: engine size,
number of cylinders, horse power, the highway mpg, the weight, the wheelbase and whether it is a hybrid. Specifically, build a linear regression
model with these seven variables as regressors; determine whether or not
it is a valid model and check whether the model assumptions are satisfied (look at the fitted line; consider residual plots; study the leverage
points and outliers; draw the marginal plots). Determine whether the
variables (regressors and response) should be transformed and if so, study
the transformed model. For the transformed model (or the original model,
if no transformation was needed), interpret the model (for example, stating which variables are important and why, interpreting the regression
coecients in the model). Draw added variable plots.
Using an F-test, compare the model with a smaller model without the
wheelbase and the fuel consumption variables. Can we drop these two
variables?
2) If one is interested in estimating the e↵ect of a car’s manufacturer on the
retail price, how can the model be expanded?
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Data Analysis 2
Consider the data set, nyc.cvs, which contains data on Italian restaurant in NY
collected in 2004. The variables are: Price (the price in US dollars of a dinner,
including one drink and the tip); Food (the customer rating of the food, out of
30); Decor (the customer rating of the d´ecor, out of 30); Service (the customer
rating of the service, out of 30) and a variable (East) indicating whether the
restaurant is east (1) or west of Fifth Avenue.
We want to study the e↵ect on the price of all other variables.
1) Build a regression model using the ratings on food, decor, service and the
location as regressors. Check the model validity (following the same steps
as above) and comment on the results
2) Consider adding to the model interaction terms of all regressors with
the variable East. Compare this model with the previous one and state
whether there is evidence to support the need for di↵erent models for the
East and West
Transforming the regressors
Consider a multivariable linear model. Do the regression coecient estimates
change if the regressors are centered, and if so how? By centering we mean we
shift the regressor to have mean zero
x˜T
j = (x1j ,…,xnj ) 7! (x1j x¯j ,…,xnj x¯j )
with ¯xj = Pn
i=1 xij . What if we standardize each variable (that is, if we center
it and then divide it by its standard deviation: sj = pPn
i=1(xij x¯j )2/(n 1)
How do the regression estimate changes if the response is centered?
After you obtain the results mathematically, you may want to check it using
the function scale() (notice that by default it scales and centers the data, that
is, the default values of the two options are scale=TRUE, center=TRUE).
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