Description
1. Consider the MinnWater data set from R package alr4.
(a) [2 pts] Produce a pairs plot of all variables in the data set.
(b) [2 pts] Three of the variables appear to have especially high sample correlations with
each other. Which are they?
(c) [2 pts] Fit the linear regression model with muniUse as the response, and all other
variables as the regressors. Produce a summary of the model fit.
(d) [2 pts] Compute the variance inflation factors (VIFs) for the variables. Using the
threshold of 10 to determine if a VIF indicates a problem of (approximate) collinearity,
which variables have a VIF indicating a possible problem?
(e) [2 pts] Fit the linear regression model with muniUse as the response, but only allUse,
irrUse, muniPrecip, and statePop as the regressors (the variables that were
individually significant in the original analysis). Produce a summary of the model fit.
Are all of those variables still significant?
(f) [2 pts] Compute the VIFs for the reduced set of variables in the previous part. Relative
to the original fit, have the VIFs for those variables increased? Decreased? Stayed the
same? Using the threshold of 10, which have a VIF indicating a possible problem?
2. Nineteenth century economist W. Stanley Jevons was concerned about the loss of value in
coins due to their loss in weight while in circulation. He collected, cleaned, and weighed 274
gold sovereigns, then grouped them roughly according to age (in decades):
Age Number Average Weight (g) Standard Dev.
1 123 7.9725 0.01409
2 78 7.9503 0.02272
3 32 7.9276 0.03426
4 17 7.8962 0.04057
5 24 7.8730 0.05353
(a) [2 pts] Perform the usual (unweighted) simple linear regression of Average Weight on
Age. Produce a summary of the results.
(b) [2 pts] Form a 95% confidence interval for the regression slope, that is, the average
weight change per decade. (You may use the function confint.)
(c) [2 pts] Perform a weighted regression of Average Weight on Age, using the group sizes
as weights (for weighted least squares). Produce a summary of the results.
(d) [2 pts] What are the weight matrix W and the matrix Σ for the analysis of the
previous part?
1
(e) [2 pts] Under this weighted model, form a 95% confidence interval for the slope.
(Again, you may use the function confint, which also works for weighted models.)
(f) [2 pts] Notice that the standard deviations seem to be different for different groups.
Perform another weighted regression of Average Weight on Age, this time using
weights that also account for the different standard deviations of the different groups.
(Refer to the notes on Weighted Least Squares for a formula that incorporates both
group sizes and standard deviations.) Produce a summary of the results.
(g) [2 pts] What are the weight matrix W and the matrix Σ for the analysis of the
previous part?
(h) [2 pts] Under this new weighted model, form a 95% confidence interval for the slope.
3. The lakemary data from R package alr4 contains ages and lengths of 78 bluegill fish
captured from Lake Mary, Minnesota.
(a) [2 pts] Fit a simple linear regression with Length as the response and Age as the
predictor. Produce a summary of the regression results.
(b) [2 pts] Produce a plot of Length versus Age, and include a fitted regression line (from
your least-squares fit in the previous part) on the same plot.
(c) [2 pts] Briefly explain why it is possible to perform a lack-of-fit test for this data, even
though the error variance is unknown.
(d) [2 pts] Perform a lack-of-fit test for the simple linear regression. (Show both your R
code and the results.) Interpret your results. (What can you say about lack of fit?)
(e) [2 pts] Compute an estimate ˜σ
2 of the pure error variance. (This is the sum of squares
for pure error divided by its degrees of freedom.)
(f) [2 pts] Now fit a quadratic regression: Use the formula Length ~ Age + I(Age^2).
Produce a summary of the regression results.
(g) [2 pts] You can perform a lack-of-fit test for this quadratic regression model in the
same way as for the simple linear regression. Perform that lack-of-fit test and interpret
the results. (Show your code and output.)
4. [ GRADUATE SECTION ONLY ] Consider the linear model Y = Xβ + e where E(e) = 0
and Var(e) = σ
2Σ, for known, invertible Σ.
(a) [2 pts] Find Var
βˆ
for the ordinary least squares estimator βˆ =
XTX
−1XTY .
(b) [2 pts] Find Var
βˆG
for the generalized least squares estimator
βˆG =
XT Σ−1X
−1XT Σ−1Y . (Simplify your answer.)
5. [ GRADUATE SECTION ONLY ] Revisit the weighted model you fit in 2(c).
(a) [2 pts] Compute the unweighted residuals as eˆW = y − yˆW , where yˆW = XβˆW and
βˆW is the weighted least squares estimate.
(b) [2 pts] Compute the weighted residuals as δb = W1/2eˆW , where W1/2
is the diagonal
matrix with non-negative elements such that W1/2W1/2 = W.
(c) [2 pts] What values does the R residuals function compute for your model
from 2(c)? Are they unweighted? Weighted? Something else
