Description
Problem 1: (20pts)
This problem uses the dataset cars from the faraway library. The cars data
records the speed of cars and the distances taken to stop. Note that the data
were recorded in the 1920s. Using this dataset, first perform a simple linear regression with dist as the response variable and speed as the predictor variable.
Then using this model, answer the following questions
1. Check the constant variance assumption for the errors.
2. Check the normality assumption for the errors.
3. Check for outliers and nominate potential outliers, if they exist.
4. Without using the anova command, check the goodness-of-fit of this simple linear regression model.
Problem 2: (10pts)
Table 1 displays the male death (per millions) due to lung cancer vs per capita
cigarette consumption for several countries.
(a) Using the above data, fit a simple linear regression. Do any of the points
look particularly influential ? Delete the United States, refit a simple
linear regression and note how the regression line changes. Now, put the
U.S.A back and delete Great Britain and note how the regression line
changes.1
.
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If you have better things to do than to manually enter the above data into your R session,
you can load the dataset as dataset E8.12 in the package SenSrivastava
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Country Y X
Ireland 58 220
Sweden 115 310
Denmark 165 380
United States 190 1280
Switzerland 250 530
Great Britain 465 1145
Norway 90 250
Canada 150 510
Australia 170 455
Holland 245 460
Finland 350 1115
Table 1: Male death per millions (Y) vs per capita ciragette consumption (X)
in 1930 for several selected countries.
(b) What is a plausible explanation for the outliers or influential points from
(a) ?
Problem 3: (10pts)
Install the package aprean3 to get access to data from the book “Applied Regression Analysis” by Draper and Smith. The following problem uses the dataset
dse13e which records the growth rate and densities of ice crystals. Ice crystals
are introduced into a chamber where the interior is maintained at a fixed temperature (−5 degree centigrade) and humidity. The growth of the crystals with
time is observed. The variables here are t and m denoting the times in seconds
from the introduction of the crystals and the mass of the crystals in nanograms,
respectively.
Using the above dataset, answer the following question
1. Perform a simple linear regression with m as the response variable and t as
a predictor variable. Discuss whether or not this simple linear regression
model is appropriate.
2. Try to see if you can “improve” the simple linear regression model in part
(1) by transforming either (or both) the predictor or the response variable.
Problem 4: (20pts)
Install the package aprean3 to get access to data from the book “Applied Regression Analysis” by Draper and Smith. The following problem uses the dataset
dse07c which records the number of motor vehicle deaths and the number of
drivers for 50 states in the US in 1966. We augment the data for the 50 state
with the statistics for Washington DC and clean up the variable names as follows.
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library(“aprean3”)
data(dse07c)
data(state)
y <- c(dse07c$y, 115)
x1 <- c(dse07c$x1, 35)
x2 <- c(dse07c$x2, 12524)
x3 <- c(dse07c$x3, NA)
x4 <- c(dse07c$x4, “No”)
x5 <- c(dse07c$x5, 44)
x6 <- c(dse07c$x6, 23)
df <- data.frame(deaths = y, drivers = x1, density = x2, rural.mileage = x3, more.male = x4,
january.temp = x5, fuel.consumption = x6, state.name = c(state.name, “DC”))
Here the variable drivers is in units of 104
, the variable density is number
of persons per square mile, rural.mileage is the total length of rural road in
the state (the unit is in thousand of miles), january.temp is the average high
temperature in January in the state, and fuel.consumption is the total amount
of gallons consumed per year (the unit is in 107 gallons).
Given this augmented data, answer the following question.
1. Plot the number of deaths against the number of drivers. Argue why a
simple linear regression with log(deaths) as the response variable against
log(drivers) as the predictor variable is to be preferred over a simple
linear regression with deaths as the response variable and drivers as the
predictor variable.
2. Perform a simple linear regression with log(deaths) as the response variable against log(drivers) as the predictor variable. Plot the residuals
against the fitted value for this regression model. Which of the residuals
are potential outliers ?
3. It is claimed that by adding another predictor variable, the new model
will no longer exhibit any potential outliers, thereby yielding a model
with better fit. Looking at the list of remaining variables, which would be
the most logical candidate to be added to the current model ? You are
not required to fit a new regression model.
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