Chap 1424
Least Squares Criterion
b
0
and
b
1
are obtained by finding the values
of
b
0
and
b
1
that
minimize the sum of the
squared residuals
2
1
0
2
2
x))
b
(b
(y
)
y
ˆ
(y
e
+

=

=
∑
∑
∑
.
Chap 1425
The Least Squares Equation
The formulas for
b
1
and
b
0
are:
algebraic equivalent for b
1
:
∑
∑
∑
∑ ∑


=
n
)
x
(
x
n
y
x
xy
b
2
2
1
∑
∑



=
2
1
)
x
(x
)
y
)(y
x
(x
b
x
b
y
b
1
0

=
and
.
Chap 1426
b
0
is the estimated average value of
y
when the value of
x
is zero
b
1
is the estimated change in the
average value of
y
as a result of a
oneunit change in
x
Interpretation of the
Slope and the Intercept
.
Chap 1427
Finding the
Least Squares Equation
The coefficients
b
0
and
b
1
will usually be
found using computer software, such as
Excel or Minitab
Other regression measures will also be
computed as part of computerbased
regression analysis
.
Chap 1428
Simple Linear Regression
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
.
Chap 1429
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
.
Chap 1430
Regression Using Excel
Data / Data Analysis / Regression
.
Chap 1431
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
Pvalue
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
+
=
.
Chap 1432
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
.
Chap 1433
Interpretation of the
Intercept,
b
0
b
0
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 b
0
= 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
+
=
.
Chap 1434
Interpretation of the
Slope Coefficient,
b
1
b
1
measures the estimated change in the
average value of Y as a result of a one
unit change in X
Here, b
1
= .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
+
=
.
Chap 1435
Least Squares Regression
Properties
The sum of the residuals from the least squares
regression line is 0
(
)
The sum of the squared residuals is a minimum
(minimized
)
The simple regression line always passes through the
mean of the
y
variable and the mean of the
x
variable
The least squares coefficients are unbiased
estimates of
β
0
and
β
1
0
)
y
(y
=

∑
ˆ
2
)
y
(y
ˆ
∑

.
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 Spring '12
 Srinivas
 Statistics, Correlation, Linear Regression, Regression Analysis, R Square