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MGSC 372
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Brian Smith
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Lecture 3

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Management Science

MGSC 372

Brian Smith

Fall

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Simple Linear Regression Review
Simple Linear Regression
Simple linear regression analysis is used to analyze the nature of the relationship between two variables.
The dependent (response) variable is designated by π and the independent (predictor) variable is
designated by π. For a given independent variable, there may be many values of the dependent
variable.
The decision regarding which variable to designate π and which variable to designate π must be based
upon theory, knowledge of the subject matter, and the objectives of the analysis. The relationship
between the two variables is estimated and then used to make predictions for π.
Scatter Diagram
A scatter diagram is a graph showing the shape and direction of the underlying relationship between the
independent variable π and the dependent variable π.
Observations are plotted in pairs (π₯,π¦) with one variable plotted on each axis.
Linear Relationships Between Two Variables Intercept and Slope
The relationship between the two variables is described by a straight line mode in general form:
True regression line:
π = π½ 0 π½ π 1 π or πΈ π = π½ + π½ π0 1
Estimatedregression line:
Μ Μ Μ
π = π½ +0π½ π 1
Parameter Estimation
π½0is a point estimation of π½0
Μ
π½1is a point estimation of π½1
We note that π½ and π½ are variables while π½ and π½ are constants.
0 1 0 1
Simple Linear Regression Model Definitions of Terms
π = the independent (predictor) variable
π = the dependent (response) variable
π½0= the true π-intercept
π½ = the true slope
1
π = the error term
Μ
π½0= the estimated π-intercept
π½ = the estimated slope
1
Simple Linear Regression Model Assumptions
1. In Simple Linear Regression, we have the assumption of linearity.
2. For each value of π the π values are normally distributed.
3. For each value of π the variance of the πvalues is the same (homoscedasticity).
4. Independence (independent sample of π are chosen for different values oπ i.e. error terms are
not correlated).
Intercept and Slope
The π-intercept π½0is the point on the π-axis where the true regression line crosses and is the average
value of π when π = 0.
The slope of the true regression line1π½ represents the average change in π when π is increased by one
unit.
π½0and π½ 1re called the parameters of the regression line.
Residual Error
The difference between an observed value π and an estimated π is called a residual.
SSE - Sum of Squares Error
2 2
πππΈ = βπ = π π¦ β π¦ π Μπ)
SSE represents the sum of the squared deviations between the observed π-values in the data set and
the π-values predicted by the estimated regression line. This is the amountof variation in π that is not
explained by the regression line.
Μ Μ
Note: The Least Squares regression line is the line that has an intercep0 π½ and slope1π½ that will
minimize SSE.
Minimizing SSE To find the line that best fits the points in the scatter diagram, we minimize the quantity:
πππΈ = β(π β π) = β(π β π½ β π½ π) Μ Μ 2
π π π 0 1
Μ Μ
We note that πππΈ = π(π½ ,π½ 0. W1 obtain the two partial derivatives:
πΏ(πππΈ) πΏ(πππΈ)
πΏπ½Μ and πΏπ½
0 1
And solve the equations:
πΏ(πππΈ) = 0 and πΏ(πππΈ) = 0
πΏπ½Μ0 πΏπ½Μ1
Least Squares Estimates of Regression Parameters
βπ₯π¦ β ππ₯Μ
π¦ Μ
π½1=
βπ₯ β ππ₯Μ
2
π½0= π¦ Μ
β π½ 1Μ
Interpretation of Regression Coefficients
The regression equation is:
π = 7.905 + 0.715π
π½1= 0.715 implies that, on average, a one unit increase in π will result in an increase of 0.715 in π, i.e.
if advertising increases by $1000 then profit will increase, on average, by $715.
π½ = 7.905 implies that the average profit in stores with no advertising budget will be $7905.
0
Extrapolation
The interpretation of π½ 0 7.905 is not reliable.
The problem is that we are making a claim about a value of π for which we have no experimental
evidence.
All of our experimental data is for values of π in the range $3000 to $7000. Therefore we cannot make a
reliable claim about the relationship between π and π when π = 0.
This is called extrapolation.
Point Estimates The regression equation can be used to predict a value of π based on a given value of π by substituting
the π-value into the regression line.
Example
Estimate the profit of a store which spends $3500 on advertising.
Let π = 3.5
Then π = 7.905 + 0.715 3.5 = 10.4 (i.e. $10400)
Caution: Do not use the regression line to predict π with values of the independent variable significantly
beyond the range of those represented in the sample. The nature of the relationship outside the range
of π-values represented in the sample may not be linear and extrapolation may lead to false
conclusions.
Partitioning Total Deviation
Μ
Total deviation = π π π
Unexplained deviation = π β π Μπ
Explained deviation = π βππ Μ
Μ
Μ Μ Μ
ππβ π = (π β π) + ππ β π)π
Total deviation = Unexplained deviation + Explained deviation
Sum of Squares
We can compute sums of squares in regression analysis and construct an analysis of variance (ANOVA)
table for the regression.
Partitioning Sums of Squares
It can be shown that:

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