# MGEB12H3 Lecture Notes - Lecture 5: Confidence Interval, Multicollinearity, Standard Deviation

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MGEB12: Quantitative Methods in Economics-II

TUTORIAL-5

Question-1: A company sells chirping crickets (a type of Bug), found only in remote bush

locations. The company claims that the number of seconds between chirps of the bug (Y,

measured in seconds) is determined by the temperature (X, measured in C0). An experiment was

conducted on a bug at various temperatures and the time between chirps measured. The data

obtained and partially summarized in the following table:

Y X Y*Y X*X X*Y

21 -10 441 100 -210

28 -4 784 16 -112

. . . . .

. . . . .

47 29 2209 841 1363

Total 364 130 13862 3296 5603

[Note the sample size is 10]

a) Find the estimated simple regression equation and interpret the coefficients. (One sentence for

each coefficient)

Solution:

( )

( )( )

( )

5423.0

10/1303296

10/3641305603

/

/

22

2

1=

−

−

=

−

−

=∑ ∑

∑ ∑ ∑

nxx

nyxyx

b

ii

iiii

350.29

10

130

5423.0

10

36411

10

=

−=−= ∑∑

ii

x

n

by

n

b

The estimated regression equation is

)(5423.0350.29 XY +=

.

Interpretation: One C0 increase in the temperature is associated with an increase of

0.5423 seconds between chirps of the bug.

Intercept: When the temperature is zero the average number of seconds between

chirps of the bug is on average 29.350.

Page 2 of 8

b) Add the fitted regression line to the plot. Which assumptions of the linear regression model

appear to be violated? Explain in one sentence why?

The assumption of Homocedasticity appears to be violated: variance increases with

temperature.

Time

Temperature

-10

0

10

20

30

20

30

40

50

*

*

*

*

*

*

*

*

*

*

Page 3 of 8

c) Using your regression can you infer that the number of seconds between chirps of the

bug is determined by the temperature? To answer this question you need to do a test and

explain your answer in two sentences.

Solution: Using the regression we can only infer whether there is an association or not.

The causality requested is by-product of the sample design. Since the sample

appears experimental if we establish a significant association. (Note if the sample

was observational this conclusion would not have been possible) It will be

sufficient to infer that the number of seconds between chirps of the bug is

determined by the temperature

0:

10

=

β

H

(i.e., Model is not significant)

0: 11 ≠

β

H

(i.e., Model is significant)

( )

195.5

1044.0

5423.0

1044.0

1606

5.17

)1(

,5.17

8

02.140

02.1404.612)7714.01(

7714.08783.0

4.6121606

871

9

4.612

110

4.612

4.612

10

364

13862

1

3

2

2

2

2

2

2

2

2

2

1

==

≅=

−

===

=×−=−=

==

×

=

=

=

−

=

=−=−= ∑ ∑

t

Sn

S

SS

SSRSSTSSE

SS

S

R

S

y

n

ySST

x

b

yx

xy

y

ii

ε

ε

p-value is very small (<0.01) therefore the regression is significant. You can also use t

from the table = 2.306 => significant at 5%.

Reject the null => the regression is significant.