McGill
MGCR 271
Winter 2024, Chapter 17 Notes
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
Welcome to Wizeprep
These not es were creat ed on Apr 3rd, 2024
We're always updating our content. Check back for more.
👋 Welcome t o Your Course Not es
I'm Lawron, your Wizeprep statistics tutor. I put these notes and the
corresponding online course together especially for MGCR 271 at McGill. It's
formulated to tell you everything you need to know, in a quick and easy
format so you can get better grades, spend less time studying, and more time
living.
Lawron 4.8/5
BCom
Find Your Course Online
These course notes correspond to an online course full of video lectures, practice
problems, instructor Q&A and more. Access it with this QR code or at
wizeprep.com/in-course-experience/Mgcr271-McGill
98%
Of Wizeprep
Students Get
Better Grades
After discovering Wizeprep at the
beginning of my second semester, my
grades have gone up significantly. I feel
so much more confident when taking
my exams.
Emily, Undergraduat e
St udent
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
Your Wizeprep Resources
Get Bet t er Grades
Really Underst and Concept s
Cut Your St udy T ime in Half
98% of students who study
with Wizeprep reported
higher grades
Our instructors know how to
make complex topics
feel simple
Quick, curated lessons allow
you to focus your study time
where it matters
Find in These Course Not es
Relevant T heory
All the theory and expert knowledge you need to
fully understand your course.
Pract ice Quest ions
Tons of practice problems, similar to those
expected on your exam.
Exam T ips
Unique exam writing tips proven to help you
score higher.
Find Online
Bit e-Sized Video Lessons
Each section corresponds to a minutes-long video
explanation by your expert instructors.
Solut ions t o Problems
See the solutions to the practice problems as well
as a step-by-step breakdown of the answers.
24/7 Inst ruct or Q&A
Need help clarifying a concept? You have direct
access to your instructor.
Not subscribed yet ?
Get started for free on Wizeprep.com
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Table of Contents
Chapt er 17. Mult iple Regression
17.1. Mult iple Regression Model
17.1.1. Assessing a Multiple Regression Model
17.1.2. F-test for Multiple Regression
17.1.3. Solving for the F-score
17.1.4. Example
17.1.5. Practice
17.2. Adjust ed R Square
17.2.1. Adjusted R Square
17.2.2. Adjusted R Square
17.3. Hypot hesis Test ing for Mult iple Regression
17.3.1. Hypothesis Testing for Multiple Regression (F-test)
17.3.2. Example
17.3.3. Example
17.3.4. Practice
17.3.5. Practice
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
17. Multiple Regression
17.1
Multiple Regression Model
1 7 .1 .1
Multiple Regression
So far, for simple linear regression, we only used one explanatory variable x to explain or predict
one response variable y. In reality, it may take more than one explanatory variable to explain y.
Suppose we use the number of hours spent studying (x) to predict one’s grade (y) and we get this
simple linear regression equation:
y^ = 48.56 + 3.599x
​
Given this linear regression model, r 2 = 0.671. This means only 67% of grade is explained by the
number of hours spent studying.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Where is t he ot her 33%?
It is explained by other explanatory variables other than hours spent studying, such as IQ, GPA, and
hours spent playing video games.
When more explanatory variables are added to a simple regression model to strengthen the ability to
explain y, the simple regression model is converted into a mult iple regression model.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
When we have mult iple explanat ory variables to explain y, we are using a mult iple regression
model:
Y = βo + β1 X1 + β2 X2 + ...... + βk Xk + ε
​
​
​
​
​
​
​
​
k = # of explanatory variables,xi ; i = 1, 2, 3, ..., k
​
ε = random error term. It represents everything that the model does not explain for y.
The estimated regression equation is:
y^ = bo + b1 x1 + ..... + bk xk
​
​
​
​
​
​
​
To predict grade using a multiple regression line, it may look like this:
y^ = bo + b1 (Study) + b2 (IQ) + b3 (GP A) + b4 (Games)
​
​
​
​
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Simple Linear Regression vs. Mult iple Regression
Simple Linear Regression
● k = 1: There is one explanatory variable X to explain Y
● there is one t-score
● correlation r can be used
● R2 may be used
Mult iple Regression
● k > 1: There are multiple explanatory variables X1 , X2 , ...Xn to explain Y
● each explanatory variable has their own t-score
● correlation r cannot be used
● R2 may be used
​
​
​
● F-score
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Wat ch t he video t ut orial for t his lesson (05:25)
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=52443&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .1 .2
F-test for Multiple Regression
What the "F"?
You have probably noticed the F-score in the ANOVA table. What is it? In regression, it is important
to know the difference between a t -t est and an F-t est .
● The F-st at only tells us if the overall model is sufficient. However, it does not tell us which
individual explanatory variables are significant.
● Each explanatory variable will have its own t -score so you will be able to assess the
significance of each one by running t-tests.
● There is only one F-score in a regression model.
Solution available online
Wat ch t he video t ut orial for t his lesson (04:53)
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=52433&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .1 .3
Solving for the F-score
We use the F-score to test to see if the regression model improves our ability to predict y. In other
words, we test the overall significance of the regression model.
Ho : β1 = β2 = β3 = ... = βk = 0 ("The overall model is not significant.”)
Ha : at least one βi 
=0
("The overall model is significant.”)
​
​
​
​
​
Solution available online
F=
SSR
k
SSE
(n−k−1)
​
​
​
=
MSR
MSE
​
​
where,
● k = # of explanatory variables, xi ′ s
​
The F-t est has two degrees of freedom:
● df numerator= k
● df denominator= n − k − 1
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Summary:
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Example
Determine the F-statistic:
F=
SSR
k
SSE
(n−k−1)
​
​
​
=
MSR
MSE
​
Solution available online
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
The follow rules are only true in simple linear regression where is there is only one explanatory
variable (k = 1):
● p-value(F) = p-value(t)
Solution available online
● t2 = F
Solution available online
WAT C H O U T !
The above rules are only true if k = 1. This is not the case in multiple regression where k > 1.
Wat ch t he video t ut orial for t his lesson (08:46)
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=71846&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .1 .4
Example: Solving for the F-score
Solve for F by completing the table.
Solution available online
Wat ch t he video t ut orial for t his lesson (02:25)
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=71847&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .1 .5
Solve for F using the clues provided.
R = 0.8
Se = 12.362
k=3
n = 60
Try this on your own before looking at the video solution! Click on "Hint" for useful formulas.
Ente r F-stat with at le ast 2 de cimal place s (e .g. 8.85)
View Solut ions on Wizeprep.com
Solutions to these questions, as well as step-by-step breakdowns of the answers at:
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=78362&activity_type=QuizQuestion
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
17.2
Adjusted R Square
1 7 .2 .1
Adjusted R Square
The final component in the multiple regression output is the Adjusted R Square.
Recall that R2 is the coefficient of determination, which tells us what % of y us explained by x. The
more explanatory variables you add to the model, the bigger R2 gets. This makes sense, since the
model becomes better at explaining y when you keep adding more explanatory variables.
However, we established in the previous section that we do just fine with a “leaner-and meaner”
reduced model where only the significant explanatory variables are included and the insignificant
variables are removed. The Adjusted R Square penalizes us for adding too many “filler” explanatory
variables to the model. While t he R Square goes up when you add more and more insignificant
explanat ory variables t o t he model, t he Adjust ed R Square goes down.
Adjusted R2 = 1 − (1 − R2 ) (
n−1
)
n−k −1
​
WI Z E T I P
Pretend R Square and Adjusted R Square are hockey coaches.
R Square wants quantity: “The more the merrier ”
Adjusted R Square wants quality: “I only want the best players in the team.”
View t his lesson online
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=71929&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .2 .2
T here are 7 explanat ory variables and 20 observat ions. R2 = 0.753. What is t he Adjust ed R
Square?
Round your answe r to 3 de cimal place s
View Solut ions on Wizeprep.com
Solutions to these questions, as well as step-by-step breakdowns of the answers at:
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=105952&activity_type=QuizQuestion
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
17.3
Hypothesis Testing for Multiple
Regression
1 7 .3 .1
Hypothesis Testing for Multiple Regression
In a mult iple regression model, there are more than one explanatory variables used to explain or
predict one response variable y.
● We conduct an F-t est to see if the overall model is significant in predicting y.
● Specifically, we are assessing how good all the explanat ory variables are, collectively, at
predicting y.
Hypotheses for F-test:
Ho : β1 = β2 = β3 = ... = βk = 0 ("The overall model is not significant.”)
Ha : at least one βi 
=0
("The overall model is significant.”)
​
​
​
​
​
Solution available online
F=
SSR
k
SSE
(n−k−1)
​
​
​
=
MSR
MSE
​
​
where,
● k = # of explanatory variables, xi ′ s
● df numerator = k
​
● df denominator = n − k − 1
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Import ant
● The F-stat only tells us if the overall model is sufficient.
○ It does not tell us which individual explanatory variables are significant!
● Each explanatory variable will have its own t-score so you will be able to assess the
significance of each one by running t-tests.
● There is only one F-score in a regression model.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
F-Dist ribut ion
The F-distribution is one-sided and skewed to the right. It start at 0 and goes to infinity. T he larger
t he F score, t he bet t er t he model is overall.
Wat ch t he video t ut orial for t his lesson (06:38)
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=52520&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .3 .2
Example: Hypothesis Testing for Multiple Regression
We want to predict the earnings of an Instagram "Influencer" y based on three explanatory variables:
x1 = number of followers (in thousands)
x2 = hours of volunteer work
x3 = grade
Ho : β1 = β2 = β3 = 0
Ha : at least one βi 
=0
​
​
​
​
("The overall model is not significant.”)
("The overall model is significant.”)
PARTIAL OUTPUT
(a) Determine the F-statistic:
F=
SSR
k
SSE
(n−k−1)
​
​
​
=
MSR
MSE
​
Solution available online
(b) What are the degrees of freedoms? (For the F-stat, there are two df's.)
Solution available online
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
(c) At the 5% significance level, what is the critical value for F? [Use F-table]
Solution available online
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
(d) At the 1% significance level, what is the critical value for F? [Use F-table]
Solution available online
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
(e) What is the p-value for the F-statistic? [Use F-table]
Solution available online
Wow! T he F-score is huge! Does t hat mean all t he explanat ory variables are significant ?
NO!
Recall:
● The F-stat only tells us if the overall model is sufficient. It does not tell us which individual
explanatory variables are significant!
● Each explanatory variable will have its own t-score so you will be able to assess the
significance of each one by running t-tests.
● There is only one F-score in a regression model.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Let's see the full ANOVA table:
● You see that only Followers is a significant explanatory variable because its p-value is low. How
much time a Instagram "Influencer" volunteers and their grade are not good predictors of their
earnings, based on their large p-value.
● Also notice that only the confidence interval for the Followers coefficient β1 does not contain 0.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Ho : β1 = β2 = β3 = 0
Ha : at least one βi 
=0
​
​
​
​
("The overall model is not significant.”)
("The overall model is significant.”)
It is true: at least one explanatory variable is significant. In this example, it's just Followers. That is
enough to reject the null hypothesis and conclude that the overall model is significant.
Finally, notice how high R2 is. This suggests that the one significant explanatory variable, Followers,
is doing almost all the work in explaining earnings.
Wat ch t he video t ut orial for t his lesson (12:34)
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=52962&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .3 .3
Example: Hypothesis Testing for Multiple Regression
We wish to predict grade using 4 predictor variables:
x1 = hours of studying
x2 = student′ s IQ
x3 = student′ s cumulative GP A
x4 = hours spent playing video games
y = grade
We randomly sampled 16 students. Results:
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
We test if the overall model is appropriate to predict grade. Hypotheses:
Ho : β1 = β2 = β3 = β4 = 0
Ha : at least one βi 
=0
​
​
​
​
​
We conduct an F-test to test if the overall model is sufficient:
F=
MSR
MSM
=
MSE
MSE
​
​
​
df numerator = k
df denominator = n − k − 1
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
(i) What percent of grade is explained by t he model?
Solution available online
(ii) Based on t he F-st at and it s p-value, how do you conclude?
(a) The overall model is sufficient.
(b) The overall model is not sufficient.
(c) All the explanatory variables in the model are significant.
(d) None of the explanatory variables in the model are significant.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Solution available online
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
(iii) What does t he coefficient b4 (i.e. hours spent playing video games) t ell us?
For each hour spent playing video games, your grade is increase/reduced by 1.45 percent, all else
equal. Better grade arises if a student spends more/less time playing video games.
Solution available online
(iv) Which of t he following must be t rue about x4 ?
(a) It is a significant explanatory variable on its own.
(b) It is a significant explanatory variable in this multiple regression model.
(c) It is a not significant explanatory variable on its own.
(d) It is a not significant explanatory variable in this multiple regression model.
Solution available online
(v) Sammy st udied for 40 hours, has an IQ of 130, has a GPA of 3.50, and played 3 hours of
Mario Kart . Predict his grade.
Solution available online
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
(vi) At t he 5% significance level, t est if GPA should be included in t he model.
Solution available online
Solution available online
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
GPA is/is not a significant variable for explaining grade; it should/should not be removed from the
model.
Solution available online
(vii) Given t he t -st at s for each explanat ory variable, what do you recommend for t he model?
Hours of study:
IQ:
GPA:
Hours of playing video games:
KEEP
KEEP
KEEP
KEEP
REMOVE
REMOVE
REMOVE
REMOVE
Solution available online
Wat ch t he video t ut orial for t his lesson (25:30)
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=52444&activity_type=CourseLesson
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .3 .4
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Michaela is good at statistics but is famous for her cooking website. She believes that the number of
new membership subscriptions per month depends on money spent on advertising, number of new
recipes posted, number of times her page is shared on social media, and number of guest appearances
she makes on TV. She randomly samples 18 months and applies multiple regression:
Part 1
(i) How is the model overall for explaining new membership subscriptions? Select the null hypothesis.
(Check all that applies.)
Ho : The overall model is significant.
​
Ho : The overall model is not significant.
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Ho : β1 = β2 = β3 = β4 = 0
​
​
​
​
​
Ho : All the explanatory variables are not statistically significantly different from zero.
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Michaela is good at statistics but is famous for her cooking website. She believes that the number of
new membership subscriptions per month depends on money spent on advertising, number of new
recipes posted, number of times her page is shared on social media, and number of guest appearances
she makes on TV. She randomly samples 18 months and applies multiple regression:
Part 2
(ii) How is the model overall for explaining new membership subscriptions? Select the alternative
hypothesis. (Check all that applies.)
Ha : The overall model is significant.
​
Ha : The overall model is not significant.
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Ha : β1 
= β2 
= β3 
= β4 
=0
​
​
​
​
​
Ha : at least one βi 
=0
​
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Michaela is good at statistics but is famous for her cooking website. She believes that the number of
new membership subscriptions per month depends on money spent on advertising, number of new
recipes posted, number of times her page is shared on social media, and number of guest appearances
she makes on TV. She randomly samples 18 months and applies multiple regression:
Part 3
(iii) At the 1% significance level, what is the critical value for the F-test?
3.012
3.179
5.205
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
14.374
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Michaela is good at statistics but is famous for her cooking website. She believes that the number of
new membership subscriptions per month depends on money spent on advertising, number of new
recipes posted, number of times her page is shared on social media, and number of guest appearances
she makes on TV. She randomly samples 18 months and applies multiple regression:
Part 4
(iv) How significant is the overall model for explaining new membership subscriptions? Select the
correct conclusion.
Reject Ho : The overall model is significant.
​
Reject Ho : The overall model is not significant.
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Fail to reject Ho : The overall model is significant.
​
Fail to reject Ho : The overall model is not significant.
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Michaela is good at statistics but is famous for her cooking website. She believes that the number of
new membership subscriptions per month depends on money spent on advertising, number of new
recipes posted, number of times her page is shared on social media, and number of guest appearances
she makes on TV. She randomly samples 18 months and applies multiple regression:
Part 5
(v) At the 1% significance level, what is the critical value to test the significance of an individual
explanatory variable?
2.576
2.650
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
3.012
5.2053
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Michaela is good at statistics but is famous for her cooking website. She believes that the number of
new membership subscriptions per month depends on money spent on advertising, number of new
recipes posted, number of times her page is shared on social media, and number of guest appearances
she makes on TV. She randomly samples 18 months and applies multiple regression:
Part 6
(vi) At the 1% significance level, which of the following explanatory variables contribute significantly
to the prediction of new membership subscriptions? [Check all that applies.]
Advertising
Recipes
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Shares
Appearances
View Solut ions on Wizeprep.com
Solutions to these questions, as well as step-by-step breakdowns of the answers at:
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=78360&activity_type=QuizQuestion
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
1 7 .3 .5
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
An account manager's salary (Y ) is estimated using a regression model. There are 3 explanatory
variables: years of experience, number of complaints, and height (cm). Salary is in $'000.
Use the partial Excel output provided below to answer the series of questions.
Part 1
(i) Which is the correct regression equation?
y^ = 90.58x1 + 2.31x2 − 2.45x3 + 0.34x4
​
​
​
​
​
y^ = 90.58 + 2.31x1 − 2.45x2 + 0.34x3
​
​
​
y^ = 90.58 + 2.31x1 + 2.45x2 + 0.34x3
​
​
​
y^ = 90.58 + 2.31b1 − 2.45b2 + 0.34b3
​
​
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
An account manager's salary (Y ) is estimated using a regression model. There are 3 explanatory
variables: years of experience, number of complaints, and height (cm). Salary is in $'000.
Use the partial Excel output provided below to answer the series of questions.
Part 2
(ii) Athena has 7 years of experience, received 4 complaints, and is 168cm tall. Predict her salary.
(Salary is in $'000.)
$154,000
$174,000
$194,000
$214,000
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
An account manager's salary (Y ) is estimated using a regression model. There are 3 explanatory
variables: years of experience, number of complaints, and height (cm). Salary is in $'000.
Use the partial Excel output provided below to answer the series of questions.
Part 3
(iii) Interpret the coefficient for Complaints.
Increasing salary by $2,450 reduces the number of complaints by one, on average.
Each additional $1 spent on complaints increases salary by $2,450.
Each additional complaint received increases salary by $2,450.
Each additional complaint received reduces salary by $2,450.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
An account manager's salary (Y ) is estimated using a regression model. There are 3 explanatory
variables: years of experience, number of complaints, and height (cm). Salary is in $'000.
Use the partial Excel output provided below to answer the series of questions.
Part 4
(iv) Compute the R2 .
Ente r R-square value with at le ast 2 de cimal place s (e .g. 0.52)
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
An account manager's salary (Y ) is estimated using a regression model. There are 3 explanatory
variables: years of experience, number of complaints, and height (cm). Salary is in $'000.
Use the partial Excel output provided below to answer the series of questions.
Part 5
(v) The null hypothesis is Ho = β1 = β2 = β3 = 0
​
​
​
Which is the correct alternative hypothesis? (Check all that applies.)
Ha : β1 
= β2 
= β3 
=0
​
​
​
​
Ha : The overall model is significant.
​
Ha : at least one βi is statistically significantly different from zero.
​
​
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
Ha : all explanatory variables are statistically significantly different from zero.
​
An account manager's salary (Y ) is estimated using a regression model. There are 3 explanatory
variables: years of experience, number of complaints, and height (cm). Salary is in $'000.
Use the partial Excel output provided below to answer the series of questions.
Part 6
(vi) At the 5% significance level level, is the overall model significant?
Yes
No
Not enough information
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
An account manager's salary (Y ) is estimated using a regression model. There are 3 explanatory
variables: years of experience, number of complaints, and height (cm). Salary is in $'000.
Use the partial Excel output provided below to answer the series of questions.
Part 7
(vii) At the 5% significance level level, which explanatory variables are significant?
Experience
Complaints
Height
None are significant
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
wizeprep.com
View Solut ions on Wizeprep.com
Solutions to these questions, as well as step-by-step breakdowns of the answers at:
https://www.wizeprep.com/in-course-experience/Mgcr271-McGill?
activity_id=78441&activity_type=QuizQuestion
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
Thanks for Studying With Us
Good Luck!
Find this, and much, much more on
Wizeprep.com
Bit e-Sized Video Lessons
Solut ions t o Problems
24/7 Inst ruct or Q&A
Each section corresponds to a
minutes-long video
explanation by your expert
instructors.
See the solutions to the
practice problems as well as a
step-by-step breakdown of
the answers.
Need help clarifying a concept?
You have direct access to your
instructor.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
Also on Wizeprep.com
Crash Courses
A live review of all testable concepts, exam-like practice problems, tips & tricks, and Q&A.
Led by an instructor who is an expert on your course.
Live Online Session
Booklet
Solutions
Recording
Weekly Tutorials
A weekly, live review of lecture topics led by an instructor who knows your course inside
and out.
Live Online Session
Booklet
Solutions
Recording
First week free!
Mock Exam Walkthroughs
A realistic practice exam based on past exams from your course. An instructor experienced
with your course will walk through the solutions.
Live Online Session
Booklet
Solutions
Recording
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
Wizeprep MCAT
Biology
Chemistry
Org Chem
Biochem
Physics
Two Plans
Bot h Plans Include...
ELIT E 515 LIVE
144 hrs of expert
instruction
Flexible live schedules, face-time with
our MCAT instructors.
CARS
Psych
All AAMC materials
Personalized study plan
15 full-length practice
exams
6 top-quality textbooks
SELF-PACED
100+ practice passages
Watch 144 hours of expert MCAT
instruction whenever you have time.
Unlimited Q&A with
MCAT experts
405+ passage-based
questions
515+ performance guarant ee
Performance Guarant ee
The Elite 515 program promises you a score of at
least 515 on the MCAT or money back. A 515
puts you within the top 2% of scores!
Find Free MCAT Resources on Wizeprep.com/MCAT
Free Live Event s
Free Diagnost ic Exam
Free Trial
Learn about the med school
application process and
more.
Predict your MCAT score and
assesses strengths and
weaknesses.
Don’t just take our word for it.
Try out the first few lessons
yourself.
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.
Other Courses at McGill
MATH 133
Linear Algebra and Geometry
MATH 141
Calculus 2
MATH 140
Calculus 1
CHEM 110
General Chemistry 1
BIOL 200
Molecular Biology
PHYS 101
Introductory Physics - Mechanics
PHGY 209
Mammalian Physiology 1
MGCR 211
Introduction to Financial Accounting
BIOL 111
Principles: Organismal Biology
ECON 208
Microeconomic Analysis and
Applications
Find more on Wizeprep.com
© Wizedem y Inc. All Rights Reserved. No part of this publication m ay be reproduced or transm itted in any form or by any m eans, or sorted in a data base
or retrieval system , without the prior written perm ission of Wizedem y Inc.