simple regression-1.pdf

Chap 12-1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 12
Simple Regression
Statistics for
Business and Economics
6th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-2
Correlation Analysis
§ Correlation analysis is used to measure
strength of the association (linear relationship)
between two variables
§ Correlation is only concerned with strength of the
relationship
§ No causal effect is implied with correlation
§ Correlation was first presented in Chapter 3

Correlation Analysis
§ The population correlation coefficient is
denoted ρ (the Greek letter rho)
§ The sample correlation coefficient is
y
x
xy
s
s
s
r =
1
n
)
y
)(y
x
(x
s i
i
xy
-
-
-
=
å
where

Introduction to
Regression Analysis
§ Regression analysis is used to:
§ Predict the value of a dependent variable based on
the value of at least one independent variable
§ Explain the impact of changes in an independent
variable on the dependent variable
Dependent variable: the variable we wish to explain
(also called the endogenous variable)
Independent variable: the variable used to explain
the dependent variable
(also called the exogenous variable)

Linear Regression Model
§ The relationship between X and Y is
described by a linear function
§ Changes in Y are assumed to be caused by
changes in X
§ Linear regression population equation model
§ Where b0 and b1 are the population model
coefficients and e is a random error term.
i
i
1
0
i ε
x
β
β
Y +
+
=

i
i
1
0
i ε
X
β
β
Y +
+
=
Linear component
Simple Linear Regression
Model
The population regression model:
Population
Y intercept
Population
Slope
Coefficient
Random
Error
term
Dependent
Variable
Independent
Variable
Random Error
component

(continued)
Random Error
for this Xi value
Y
X
Observed Value
of Y for Xi
Predicted Value
of Y for Xi
i
i
1
0
i ε
X
β
β
Y +
+
=
Xi
Slope = β1
Intercept = β0
εi
Model

i
1
0
i x
b
b
y +
=
ˆ
The simple linear regression equation provides an
estimate of the population regression line
Equation
Estimate of
the regression
intercept
Estimate of the
regression slope
Estimated
(or predicted)
y value for
observation i
Value of x for
observation i
The individual random error terms ei have a mean of zero
)
)
ˆ
( i
1
0
i
i
i
i x
b
(b
-
y
y
-
y
e +
=
=

Least Squares Estimators
§ b0 and b1 are obtained by finding the values
of b0 and b1 that minimize the sum of the
squared differences between y and :
2
i
1
0
i
2
i
i
2
i
)]
x
b
(b
[y
min
)
y
(y
min
e
min
SSE
min
+
-
=
-
=
=
å
å
å
ˆ
ŷ
Differential calculus is used to obtain the
coefficient estimators b0 and b1 that minimize SSE

§ The slope coefficient estimator is
§ And the constant or y-intercept is
§ The regression line always goes through the mean x, y
X
Y
xy
n
1
i
2
i
n
1
i
i
i
1
s
s
r
)
x
(x
)
y
)(y
x
(x
b =
-
-
-
=
å
å
=
=
x
b
y
b 1
0 -
=
x
x
Least Squares Estimators
(continued)

Finding the Least Squares
Equation
§ The coefficients b0 and b1 , and other
regression results in this chapter, will be
found using a computer
§ Hand calculations are tedious
§ Statistical routines are built into Excel
§ Other statistical analysis software can be used

Linear Regression Model
Assumptions
§ The true relationship form is linear (Y is a linear function
of X, plus random error)
§ The error terms, εi are independent of the x values
§ The error terms are random variables with mean 0 and
constant variance, σ2
(the constant variance property is called homoscedasticity)
§ The random error terms, εi, are not correlated with one
another, so that
n)
,
1,
(i
for
σ
]
E[ε
and
0
]
E[ε 2
2
i
i !
=
=
=
j
i
all
for
0
]
ε
E[ε j
i ¹
=

§ b0 is the estimated average value of y
when the value of x is zero (if x = 0 is
in the range of observed x values)
§ b1 is the estimated change in the
average value of y as a result of a
one-unit change in x
Interpretation of the
Slope and the Intercept

Example
§ A real estate agent wishes to examine the
relationship between the selling price of a home
and its size (measured in square feet)
§ A random sample of 10 houses is selected
§ Dependent variable (Y) = house price in $1000s
§ Independent variable (X) = square feet

Sample Data for House Price
Model
House Price in $1000s
(Y)
Square Feet
(X)
245 1400
312 1600
279 1700
308 1875
199 1100
219 1550
405 2350
324 2450
319 1425
255 1700

0
50
100
150
200
250
300
350
400
450
0 500 1000 1500 2000 2500 3000
Square Feet
House
Price
($1000s)
Graphical Presentation
§ House price model: scatter plot

Regression Using Excel
§ Tools / Data Analysis / Regression

Excel Output
Regression Statistics
Multiple R 0.76211
R Square 0.58082
Adjusted R Square 0.52842
Standard Error 41.33032
Observations 10
ANOVA
df SS MS F Significance F
Regression 1 18934.9348 18934.9348 11.0848 0.01039
Residual 8 13665.5652 1708.1957
Total 9 32600.5000
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386
Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580
The regression equation is:
feet)
(square
0.10977
98.24833
price
house +
=

0
50
100
150
200
250
300
350
400
450
0 500 1000 1500 2000 2500 3000
Square Feet
House
Price
($1000s)
Graphical Presentation
§ House price model: scatter plot and
regression line
feet)
(square
0.10977
98.24833
price
house +
=
Slope
= 0.10977
Intercept
= 98.248

Intercept, b0
§ b0 is the estimated average value of Y when the
value of X is zero (if X = 0 is in the range of
observed X values)
§ Here, no houses had 0 square feet, so b0 = 98.24833
just indicates that, for houses within the range of
sizes observed, $98,248.33 is the portion of the
house price not explained by square feet
feet)
(square
0.10977
98.24833
price
house +
=

Slope Coefficient, b1
§ b1 measures the estimated change in the
average value of Y as a result of a one-
unit change in X
§ Here, b1 = .10977 tells us that the average value of a
house increases by .10977($1000) = $109.77, on
average, for each additional one square foot of size
feet)
(square
0.10977
98.24833
price
house +
=

317.85
0)
0.1098(200
98.25
(sq.ft.)
0.1098
98.25
price
house
=
+
=
+
=
Predict the price for a house
with 2000 square feet:
The predicted price for a house with 2000
square feet is 317.85($1,000s) = $317,850
Predictions Using
Regression Analysis

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