STAT 361 – Fall 2014
Homework 4: Linear models
This Assignment is intended for students’ independent homework and need not
be submitted for marking
1. (Linear model with 2 regression parameters) a) Consider a linear model with
two regression parameters:
yi = aui + bvi + ei ,
i = 1, ..., n,
where, as usual, yi are the responses, and u and v are two stimuli variables. Find the
least squares estimates (LSE’s) of the parameters a and b. Under what condition are
these estimates uniquely defined?
b) Use your results from Part a) to find the LSE’s in the quadratic response model
yi = axi + bx2i + ei ,
i = 1, ..., n.
2. (Determining cargo weight) Upon entering a cargo terminal, a loaded truck was
weighed, and the result was Y1 . Before shipping, the truck was unloaded, and both the
truck and the cargo were weighed separately. The respective results were Y2 and Y3 .
Assume that the readings of weights were prone to independent Gaussian errors with
mean zero and variance σ 2 .
a) Set up a linear model for estimating the weights of the truck (T ) and the cargo (C).
ˆ
b) Find the corresponding least squares estimates Tˆ and C.
ˆ
c) Find the variances of the least squares estimates Tˆ and C.
d) Give an unbiased estimator of σ 2 .
e) Exhibit 95% confidence intervals for the true weights, T and C.
3. (Linear model with 3 regression parameters) a) Consider a linear model with
three regression parameters:
yi = c + aui + bvi + ei ,
i = 1, ..., n,
where, as usual, yi are the responses, and u and v are two stimuli variables. Find the
least squares estimates (LSE’s) of the parameters a and b. Under what condition are
these estimates uniquely defined? Hint: textbook, p. 27.
b) Use your results from Part a) to find the LSE’s in the quadratic response model
yi = c + axi + bx2i + ei ,
1
i = 1, ..., n.
4. (Simple non-linear models) Convert the following non-linear relationships into linear ones by making transformations and defining new stimulus and response variables.
a) y = a/(1 + cx)
b) y = ae−bx
c) y = abx
d) y = x/(a + bx)
e) y = 1/(1 + ebx )
5. (Correlation coefficient) Suppose the grades on a midterm xi and the grades on
a final exam yi have a sample correlation coefficient of 0.5 and both exams have an
average score of 75 and a standard errors of 10.
a) If a students’s score on the midterm is 95, what would you predict his score on the
final to be?
b) If a student scored 85 on the final, what would you guess that her score on the
midterm was? Hint: Use the following representation of the linear model:
y − y¯
x − x¯
=r
,
sy
sx
where r is the correlation coefficient.
6. (Fitting simple linear model to the data and the p-values) Return to the
Problem 7 from the homework Assignment 2. For both submodels i) and ii),
a) Find the p-values for testing significance of both the intercept and the slope.
b) Calculate the coefficients if determination R2 . Which of the two models exhibits a
better fit?
c) Use the F -test in verifying the significance of the intercept. Does the p-value of
this test coincide with that of the corresponding significance test used in Part a)?
Explain, why.
2

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