Download Chapter 12 Simple Linear Regression

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Time series wikipedia, lookup

Instrumental variables estimation wikipedia, lookup

Choice modelling wikipedia, lookup

Regression analysis wikipedia, lookup

Linear regression wikipedia, lookup

Coefficient of determination wikipedia, lookup

Transcript
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
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.
Simple Linear Regression Equation

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.
Simple Linear Regression Equation

Positive Linear Relationship
E(y)
Regression line
Intercept
b0
Slope b1
is positive
x
Simple Linear Regression Equation

Negative Linear Relationship
E(y)
Intercept
b0
Regression line
Slope b1
is negative
x
Simple Linear Regression Equation

No Relationship
E(y)
Regression line
Intercept
b0
Slope b1
is 0
x
Estimated Simple Linear Regression Equation

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.
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
Least Squares Method

Least Squares Criterion
min  (y i  y i ) 2
where:
yi = observed value of the dependent variable
for the ith observation
y^i = estimated value of the dependent variable
for the ith observation
The Least Squares Method

Slope for the Estimated Regression Equation
 xi y i  (  xi  y i ) / n
b1 
2
2
 xi  (  xi ) / n
where:
xi = value of independent variable for ith observation
yi = value of dependent variable for ith observation
n = total number of observations
The Least Squares Method

y-Intercept for the Estimated Regression Equation
b0  y  b1 x
where:
_
x = mean value for independent variable
_
y = mean value for dependent variable
n = total number of observations
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.
Example: Reed Auto Sales

Simple Linear Regression
Number of TV Ads
1
3
2
1
3
Number of Cars Sold
14
24
18
17
27
Example: Reed Auto Sales

Slope for the Estimated Regression Equation
b1 = 220 - (10)(100)/5 = 5
24 - (10)2/5

y-Intercept for the Estimated Regression Equation
b0 = 20 - 5(2) = 10

Estimated Regression Equation
y^ = 10 + 5x
Example: Reed Auto Sales
Scatter Diagram
30
25
Cars Sold

20
^ = 10 + 5x
y
15
10
5
0
0
1
2
TV Ads
3
4
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
The Coefficient of Determination

The coefficient of determination is:
r2 = SSR/SST
where:
SST = total sum of squares
SSR = sum of squares due to regression
Example: Reed Auto Sales

Coefficient of Determination
r2 = SSR/SST = 100/114 = .8772
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.
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
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
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.
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.
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
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
Testing for Significance: t Test

Hypotheses
H 0 : b1 = 0
H a : b1 = 0

Test Statistic
b1
t
sb1
Testing for Significance: t Test

Rejection Rule
Reject H0 if t < -t or t > t
where:
t is based on a t distribution
with n - 2 degrees of freedom
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 = 3.182
Reject H0 if t > 3.182
Example: Reed Auto Sales

t Test
• Test Statistics
t = 5/1.08 = 4.63
• Conclusions
t = 4.63 > 3.182, so reject H0
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.
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
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 1.56 to 8.44

Conclusion
0 is not included in the confidence interval.
Reject H0
Testing for Significance: F Test

Hypotheses
H 0 : b1 = 0
H a : b1 = 0

Test Statistic
F = MSR/MSE
Testing for Significance: F Test

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
Example: Reed Auto Sales

F Test
• Hypotheses
• Rejection Rule
H 0 : b1 = 0
H a : b1 = 0
For  = .05 and d.f. = 1, 3: F.05 = 10.13
Reject H0 if F > 10.13.
Example: Reed Auto Sales

F Test
• Test Statistic
F = MSR/MSE = 100/4.667 = 21.43
• Conclusion
F = 21.43 > 10.13, so we reject H0.
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.
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) = 25 cars