STATS 413 - Homework 5
Due Friday, February 18, 2022
Online submission on Canvas required by 11:59pm EDT
1. (Chapter 6, Problem 1) Using the sat dataset, fit a model with the to-
tal SAT score as the response and expend, salary, ratio and takers as
predictors. Perform regression diagnostics on this model to answer
the following questions. Display any plots that are relevant. Do not
provide any plots about which you have nothing to say. Suggest
possible improvements or corrections to the model where appro-
priate. If you decide to modify your model, your should perform
diagnostic analysis for the new model as well.
(a) Check the constant variance assumption for the errors.
(b) Check the normality assumption.
(c) Check for large leverage points.
(d) Check for outliers.
(e) Check for influential points.
(f) Check the structure of the relationship between the predictors
and the response.
Hints: Note that for this particular dataset, the partial regression plot
is not successful at discovering the non-linear relationship, while the
partial residual plot is a little better, but still not sensitive enough.
The plot of the residuals vs each prediction variable works the best
for this particular dataset.
2. (Leverages in simple linear regression) Consider the simple linear
regression model, yi = β0 + β1xi+ ϵi, where E(ϵi) = 0 and Var(ϵi) =
σ2, i = 1, . . . , n.
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(a) Show that the leverage hi is equal to
hi = Hii =
1
n
+
(xi − x¯)2
∑ni=1(xi − x¯)2
where x¯ = ∑i xi/n.
(b) In a two-dimensional plot of the response versus the predic-
tor in a simple linear regression problem, explain how high-
leverage points can be identified.
(c) Make up a data set in simple linear regression (in particular,
x1, x2, . . . , xn) so that the value of the leverage for one of the
observations is equal to 1.
Hints: First show that
(XTX)−1 = 1
n∑i x2i − n2x¯2
(
∑i x2i −nx¯−nx¯ n
)
.
Then plug it into the definition of the leverage formula and carry out
the derivation.
Please use the report template to write your solutions for problem 1. Limit
your solutions to at most 7 pages (including code and figures).
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