Chapter 12
Simple Linear Regression
Simple Linear Regression Model
Least Squares Method
Coefficient of Determination
Model Assumptions
Testing for Significance
Using the Estimated Regression Equation
for Estimation and Prediction
Computer Solution
Residual Analysis: Validating Model Assumptions
1
Simple Linear Regression Model
The equation that describes how y is related to x and
an error term is called the regression model.
The simple linear regression model is:
y = b0 + b1x +e
– b0 and b1 are called parameters of the model.
– e is a random variable called the error term.
2
Simple Linear Regression Equation
n
The simple linear regression equation is:
E(y) = b0 + b1x
• Graph of the regression equation is a straight line.
• b0 is the y intercept of the regression line.
• b1 is the slope of the regression line.
• E(y) is the expected value of y for a given x value.
3
Simple Linear Regression Equation
n
Positive Linear Relationship
E(y)
Regression line
Intercept
b0
Slope b1
is positive
x
4
Simple Linear Regression Equation
n
Negative Linear Relationship
E(y)
Intercept
b0
Regression line
Slope b1
is negative
x
5
Simple Linear Regression Equation
n
No Relationship
E(y)
Regression line
Intercept
b0
Slope b1
is 0
x
6
Estimated Simple Linear Regression Equation
n
The estimated simple linear regression equation is:
ŷ  b0  b1 x
• The graph is called the estimated regression line.
• b0 is the y intercept of the line.
• b1 is the slope of the line.
• ŷ is the estimated value of y for a given x value.
7
Estimation Process
Regression Model
y = b0 + b1x +e
Regression Equation
E(y) = b0 + b1x
Unknown Parameters
b0, b1
b0 and b1
provide estimates of
b0 and b1
Sample Data:
x
y
x1
y1
.
.
.
.
xn yn
Estimated
Regression Equation
ŷ  b0  b1 x
Sample Statistics
b0, b1
8
Least Squares Method
Least Squares Criterion
min  (y i  y i ) 2
where:
yi = observed value of the dependent variable
for the ith observation
^
yi = estimated value of the dependent variable
for the ith observation
9
The Least Squares Method
Slope for the Estimated Regression Equation
 xi y i  (  xi  y i ) / n
b1 
2
2
 xi  (  xi ) / n
10
The Least Squares Method
n
y-Intercept for the Estimated Regression Equation
b0  y  b1 x
where:
xi = value of independent variable for ith observation
yi = value of dependent variable for ith observation
_
x = mean value for independent variable
_
y = mean value for dependent variable
n = total number of observations
11
Example: Reed Auto Sales
Simple Linear Regression
Reed Auto periodically has a special week-long
sale. As part of the advertising campaign Reed runs
one or more television commercials during the
weekend preceding the sale. Data from a sample of 5
previous sales are shown on the next slide.
12
Example: Reed Auto Sales
n
Simple Linear Regression
Number of TV Ads
1
3
2
1
3
Number of Cars Sold
14
24
18
17
27
13
Example: Reed Auto Sales
Slope for the Estimated Regression Equation
b1 = 220 - (10)(100)/5 = _____
24 - (10)2/5
y-Intercept for the Estimated Regression Equation
b0 = 20 - 5(2) = _____
Estimated Regression Equation
y^ = 10 + 5x
14
Example: Reed Auto Sales
Scatter Diagram
30
Cars Sold
25
20
^
y = 10 + 5x
15
10
5
0
0
1
2
TV Ads
3
4
15
The Coefficient of Determination
Relationship Among SST, SSR, SSE
SST = SSR + SSE
2
2
^ )2
 ( y i  y )   ( y^i  y )   ( y i  y
i
where:
SST = total sum of squares
SSR = sum of squares due to regression
SSE = sum of squares due to error
16
The Coefficient of Determination
n
The coefficient of determination is:
r2 = SSR/SST
where:
SST = total sum of squares
SSR = sum of squares due to regression
17
Example: Reed Auto Sales
Coefficient of Determination
r2 = SSR/SST = 100/114 =
The regression relationship is very strong because
88% of the variation in number of cars sold can be
explained by the linear relationship between the
number of TV ads and the number of cars sold.
18
The Correlation Coefficient
Sample Correlation Coefficient
rxy  (sign of b1 ) Coefficien t of Determinat ion
rxy  (sign of b1 ) r 2
where:
b1 = the slope of the estimated regression
equation yˆ  b0  b1 x
19
Example: Reed Auto Sales
Sample Correlation Coefficient
rxy  (sign of b1 ) r 2
The sign of b1 in the equation yˆ  10  5 x is “+”.
rxy = + .8772
rxy =
+.9366
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Model Assumptions
Assumptions About the Error Term e
1. The error e is a random variable with mean of
zero.
2. The variance of e , denoted by  2, is the same for
all values of the independent variable.
3. The values of e are independent.
4. The error e is a normally distributed random
variable.
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Testing for Significance
To test for a significant regression relationship, we
must conduct a hypothesis test to determine whether
the value of b1 is zero.
Two tests are commonly used
– t Test
– F Test
Both tests require an estimate of  2, the variance of e
in the regression model.
22
Testing for Significance
An Estimate of  2
The mean square error (MSE) provides the estimate
of  2, and the notation s2 is also used.
s2 = MSE = SSE/(n-2)
where:
SSE   (yi  yˆi ) 2   ( yi  b0  b1 xi ) 2
23
Testing for Significance
An Estimate of 
– To estimate  we take the square root of  2.
– The resulting s is called the standard error of the
estimate.
SSE
s  MSE 
n2
24
Testing for Significance: t Test
Hypotheses
H 0 : b1 = 0
H a : b1 = 0
Test Statistic
where sb 
1
b1
t
sb1
s
2
(
x

x
)
 i
25
Testing for Significance: t Test
n
Rejection Rule
Reject H0 if t < -t or t > t
where:
t is based on a t distribution
with n - 2 degrees of freedom
26
Example: Reed Auto Sales
t Test
– Hypotheses
H 0 : b1 = 0
H a : b1 = 0
– Rejection Rule
For  = .05 and d.f. = 3, t.025 = _____
Reject H0 if t > t.025 = _____
27
Example: Reed Auto Sales
n
t Test
• Test Statistics
t = _____/_____ = 4.63
• Conclusions
t = 4.63 > 3.182, so reject H0
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Confidence Interval for b1
We can use a 95% confidence interval for b1 to test
the hypotheses just used in the t test.
H0 is rejected if the hypothesized value of b1 is not
included in the confidence interval for b1.
29
Confidence Interval for b1
The form of a confidence interval for b1 is:
b1  t / 2 sb1
where
b1 is the point estimate
t / 2 sb1 is the margin of error
t / 2 is the t value providing an area
of /2 in the upper tail of a
t distribution with n - 2 degrees
of freedom
30
Example: Reed Auto Sales
Rejection Rule
Reject H0 if 0 is not included in
the confidence interval for b1.
95% Confidence Interval for b1
b1  t / 2 sb1= 5 +/- 3.182(1.08) = 5 +/- 3.44
or
____ to ____
Conclusion
0 is not included in the confidence interval.
Reject H0
31
Testing for Significance: F Test
n
Hypotheses
H 0 : b1 = 0
H a : b1 = 0
n
Test Statistic
F = MSR/MSE
32
Testing for Significance: F Test
n
Rejection Rule
Reject H0 if F > F
where:
F is based on an F distribution
with 1 d.f. in the numerator and
n - 2 d.f. in the denominator
33
Example: Reed Auto Sales
n
F Test
• Hypotheses
• Rejection Rule
H 0 : b1 = 0
H a : b1 = 0
For  = .05 and d.f. = 1, 3: F.05 = ______
Reject H0 if F > F.05 = ______.
34
Example: Reed Auto Sales
n
F Test
• Test Statistic
F = MSR/MSE = ____ / ______ = 21.43
• Conclusion
F = 21.43 > 10.13, so we reject H0.
35
Some Cautions about the
Interpretation of Significance Tests
Rejecting H0: b1 = 0 and concluding that the
relationship between x and y is significant does not
enable us to conclude that a cause-and-effect
relationship is present between x and y.
Just because we are able to reject H0: b1 = 0 and
demonstrate statistical significance does not enable us
to conclude that there is a linear relationship between
x and y.
36
Using the Estimated Regression Equation
for Estimation and Prediction
n
Confidence Interval Estimate of E(yp)
y p  t /2 s y p
n
Prediction Interval Estimate of yp
yp + t/2 sind
where:
confidence coefficient is 1 -  and
t/2 is based on a t distribution
with n - 2 degrees of freedom
37
Example: Reed Auto Sales
Point Estimation
If 3 TV ads are run prior to a sale, we expect the
mean number of cars sold to be:
y =^10 + 5(3) = ______ cars
38
Example: Reed Auto Sales
n
Confidence Interval for E(yp)
95% confidence interval estimate of the mean
number of cars sold when 3 TV ads are run is:
25 + 4.61 = ______ to _______ cars
39
Example: Reed Auto Sales
n
Prediction Interval for yp
95% prediction interval estimate of the number of
cars sold in one particular week when 3 TV ads are
run is:
25 + 8.28 = _____ to ______ cars
40
Residual Analysis
Residual for Observation i
yi – y^i
Standardized Residual for Observation i
where:
and
y i  y^i
syi  ^yi
syi  y^i  s 1  hi
1 ( xi  x ) 2
hi  
n  ( xi  x ) 2
41
Example: Reed Auto Sales
Residuals
Observation
Predicted Cars Sold
Residuals
1
15
-1
2
25
-1
3
20
-2
4
15
2
5
25
2
42
Example: Reed Auto Sales
Residual Plot
TV Ads Residual Plot
3
Residuals
2
1
0
-1
-2
-3
0
1
2
3
4
TV Ads
43
Residual Analysis
Residual Plot
Residual
y  yˆ
Good Pattern
0
x
44
Residual Analysis
Residual Plot
y  yˆ
Residual
n
Nonconstant Variance
0
x
45
Residual Analysis
Residual Plot
y  yˆ
Residual
n
Model Form Not Adequate
0
x
46
End of Chapter 12
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