# ECON 1310 Havard University Simple Linear Regression Exercises

Description

16 statistics problems

5 attachmentsSlide 1 of 5attachment_1attachment_1attachment_2attachment_2attachment_3attachment_3attachment_4attachment_4attachment_5attachment_5

Unformatted Attachment Preview

Today’s Topics:
 Simple Linear Regression (SLR)
What is it? Why is it used?
Estimation (using Excel and not hand calculation).
Interpretation of Excel analysis results.
ECON1310
Introductory Statistics for Social Sciences
 Statistics
LECTURE 11
Simple Linear Regression– Part 1
1 hour of online YouTube videos on Blackboard
to be viewed to complete Lecture 11.
Sample slope coefficient, b1
Coefficient of Determination, r2
Standard error of the estimate, se
 Confidence interval for the slope coefficient.
1
What is Simple Linear Regression (SLR)?
2
Why use Simple Linear Regression?
 used to predict the value of one variable
(dependent variable) based on a given value of
another variable (independent variable).
 a procedure to find the equation of the line of
best fit between two variables of the form:
Y = mX + c
used to explain the impact of a change in the
independent variable on the dependent variable.
SLR is an inferential statistics technique allowing
parameter based on a sample statistic.
where
m = the slope of the line
c = the y-axis intercept
3
4
Questions.
1. What is the impact on a retail store’s sales revenue
with an increase in advertising expenditure?
2. Do house prices decrease with an increase in
distance from the Central Business District (CBD)?
How can these questions be analysed?
 Firstly, collect observed data on the two
variables of interest.
 Plot the data on a graph with one variable on
the X axis and the other on the Y axis.
3. Do home values increase as land size increases?
4. Is there a relationship between gross box office movie
revenue and the number of videos sold once the
video is released a short time later?
5
 Observe the scatter plot (graph).
Note: a scatter plot can be drawn using Excel by
setting up a table of XY coordinates, then using
insert/scatter.
6
1

2. Scatter plot
2000
1800
1600
1400
1200
1000
800
600
400
200
0
House price (\$’000)
sales (\$000)
1. Scatter plot
0
20
40
60
1000
900
800
700
600
500
400
300
200
100
0
80
0
20
40
60
Distance from CBD (km)
3. Scatter plot
4. Scatter plot
number of videos sold
1000
900
800
appraised value (\$000)
700
600
500
400
300
70
60
50
40
30
20
10
0
0
100
200
300
400
200
Box Office Gross (\$m)
100
0
0
500
1000
1500
2000
2)
Land (m
2500
3000
3500
10
What can scatter plots reveal?
Correlation Coefficient, r

Is there a linear (straight line) relationship?
 It has no units (it is just a number)

Is there a curvilinear (eg: parabolic shape)
Y
relationship? Y
 Can only have a value between –1 and 1
X

-1  r  1
 r close to 1 implies a strong positive linear relationship
X
 r close to –1 implies a strong negative linear relationship
If the relationship looks linear, is the line
sloping upward (positive relationship) or
downward sloping (negative relationship)?
 r closer to 0 implies a weaker linear relationship
 r = 0 implies no linear relationship exists
 To calculate r using Excel, use:
Data/Data Analysis/Correlation or Insert/ fx /correl
Is the linear relationship weak or strong?
11
12
2
Correlation coefficients and the strength
of linear relationship.
Line of best fit in scatter plot
House prices (\$’000)
sales (\$000)
rTV advertising & sales = 0.83
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0
20
40
60
80
rdistance and price = -0.90
1000
900
800
700
600
500
400
300
200
100
0
0
r
number of videos sold
appraised value (\$000)
20
40
60
Distance from CBD (km)
rland and value = 0.61 (weak)
1000
Y
Y
900
800
700
600
500
400
300
200
X
= 0.85
Box office & videos sold
70
60
50
40
30
20
10
0
100
200
300
-10 0
400
X
r = -1
Perfectly negatively
correlated = a very
strong relationship
r = +1
Perfectly positively
correlated = a very
strong relationship
Box Office Gross (\$m)
100
land (m2)
0
0
500
1000
1500
2000
2500
14
3000
3500
Correlation coefficients and strength of
linear relationship.
Y
Correlation coefficients and strength of
linear relationship.
X
r = +0.2
A weak, positive (upward
sloping) linear relationship.
Y
Y
Y
X
r = -0.6
A negative (downward
sloping) linear relationship
that is neither strong nor
weak.
X
X
r=0
r=0
NO linear relationship exists.
The slope of the regression line = 0
15
Correlation summary.
16
Simple Linear Regression
 A scatter diagram can indicate if some kind of
relationship might exist between two variables.
(line of best fit using sample data)
Ŷi  b 0  b1X i
 Correlation analysis is used to measure the
strength of the linear relationship between two
variables.
 the relationship between the two variables X
and Y is described as a linear function.
Note:
Correlation does NOT imply a causal effect
(ie: a change in X does not cause a change in Y)
17
 changes in Y are assumed to be caused by
changes in X.
18
3
Sample Linear Regression Equation
Ŷi  b 0  b1X i
Y
Observed value
Yi
of Yi for Xi
Predicted
value of Y for Xi
(estimated Y
value = Ŷi )
Defining and Interpreting the terms
Ŷi
(Xi ,Yi)
ei  Yi  Ŷi
n = 12 (number
of observed data
points in
scatter plot)
1 unit
(0 , b0)
0
 (Xi ,Yi) is a measured (observed) data point in the
sample (of size n) used to help estimate the
sample linear regression equation.
(Xi ,Ŷi )
b1 units
X
 Ŷi is the predicted value (estimated value) of Y
for a particular chosen value of Xi
X
Xi
 b0 = the value of Ŷ when the value of X is zero.
It is the “estimate sample regression line” y
intercept value.
 b1 = the slope of the estimated regression line.
rise
change in y

run
change in x
 If b1 is positive, for every one unit increase in X
there is b1 units increase in Y.
 If b1 is negative, for every one unit increase in X
there is b1 units decrease in Y.
21
Population Simple Linear Regression Equation
(for the true, but unknown, relationship)
Y
Y  β 0  β1X i  ε i
error
εi= random
for Xi value
Expected value
of Y for Xi
(0 , β0) X
1 unit
0
(Xi , Y)
Population SLR Equation (for the true,
but unknown, relationship)
Population Population slope Independent Random
error
Y intercept coefficient
variable
term
Dependent
variable
Yi  β 0  β1X i  ε i
where β0 and β1 are the parameters respectively
for the y intercept value and slope coefficient of X.
ε i = an error term to allow for a range of values of Y to
occur for any given Xi . In the population, there may be
many different Y values for the same X value.
22
Finding the equation to the Simple Linear
Regression Equation (based on a sample):
 The error at any value of Xi is defined as the
difference between the observed data value of
Yi and the predicted value Ŷi
β1 units
N = very large for
population model
Xi
20
 Note: some data points are above the estimated
regression line (a positive error) and some
below the line (a negative error).
(Xi ,Yi)
Sample SLR
equation
 Yi is called the dependent variable (or response
variable). The value of Y changes when X
changes. (ie: Y depends on the value of X).
(units)
Defining and Interpreting the terms.
Observed value
Yi
of Yi for Xi
 Xi is called the independent variable (or
explanatory variable).
X
(units)
ei  Yi  Ŷi
24
4
Observations of the Simple Linear
Regression Equation (based on a sample).
 the error (ei) at any of Xi is always measured in the
vertical direction.
So far, what do we know?
 Simple Linear Regression aims to find a straight line
relationship between two variables, X and Y, using
sample data.
 The estimated equation is the line of best fit (the
Sample Linear Regression Model) is given by the
equation:
Ŷ  b 0  b1X
 the slope of the SLR equation has the same sign
as r.
 if b1 positive, r > 0, the line slopes upwards

## Reviews, comments, and love from our customers and community:

This page is having a slideshow that uses Javascript. Your browser either doesn't support Javascript or you have it turned off. To see this page as it is meant to appear please use a Javascript enabled browser.

Peter M.
So far so good! It's safe and legit. My paper was finished on time...very excited!
Sean O.N.
Experience was easy, prompt and timely. Awesome first experience with a site like this. Worked out well.Thank you.
Angela M.J.
Good easy. I like the bidding because you can choose the writer and read reviews from other students
Lee Y.
My writer had to change some ideas that she misunderstood. She was really nice and kind.
Kelvin J.
I have used other writing websites and this by far as been way better thus far! =)
Antony B.
I received an, "A". Definitely will reach out to her again and I highly recommend her. Thank you very much.