# Tutorial 4 - City University of Hong Kong ```MA3518: Applied Statistics
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Department of Mathematics
Faculty of Science and Engineering
City University of Hong Kong
MA 3518: Applied Statistics
Tutorial 4
Question 1:
Consider the following simple linear regression model without intercept:
Yi = b Xi + ei, i = 1, 2, …, n,
where ei are i.i.d. with common distribution N(0,  2)
(a) Find the least squares estimator Be for the parameter b.
(b) E(Be) = b
Question 2:
Consider the following simple linear regression model:
Yi = a + b Xi + ei, i = 1, 2, …, n,
where the random errors {ei} are i.i.d. with common distribution N(0,  2)
Let Ae and Be denote the least squares estimators of the unknown parameters a and b,
respectively. Prove that
Be = Sxy / Sxx and Ae = Y – Be X
n
where Sxy =

i 1
n
(Xi – X ) (Yi – Y ) and Sxx =

i 1
(Xi – X )2
MA3518: Applied Statistics
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Question 3:
Consider again the simple linear regression model in Question 2. Prove that
(a) E(Ae) = a
(b) E(Be) = b
Question 4:
Suppose an experiment is conducted to examine the impact of the heating degree day on
the daily gas consumption. Let X and Y represent the daily gas consumption and the
heating degree day, respectively. The data collected for X and Y over different months are
displayed as follows:
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
X
12.2
22.6
27.8
30.3
36.6
38.1
38.6
11.2
18.8
22.4
30.1
33.1
Y
3.0
3.2
3.8
4.0
5.1
6.2
6.8
3.2
3.6
3.1
2.6
2.8
(a) Create a scatter plot of the data by SAS and comment on the plot
(b) Fit a simple linear regression model to the data
(c) Let b denote the slope of the regression line. Perform a statistical test on the
hypotheses H0: b = 0 against H0: b  0 at 5% significance level
~ End of Tutorial 4~
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