2. Regresi dan Korelasi Pertemuan 10
Matakuliah : D Statistika dan Aplikasinya
Tahun : 2010
Regresi dan Korelasi Pertemuan 10

3. Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu :
menghubungkan dua variabel dalam analisis regresi dan korelasi linier sederhana
dapat menunjukkan hubungan antara variabel berdasarkan hasil uji hipotesis

4. The Simple Linear Regression Model
The population simple linear regression model:
Y= 0 + 1 X 
Nonrandom or Random
Systematic Component
Component
where
Y is the dependent variable, the variable we wish to explain or predict
X is the independent variable, also called the predictor variable
 is the error term, the only random component in the model, and thus, the only source of randomness in Y.
0 is the intercept of the systematic component of the regression relationship.
1 is the slope of the systematic component.
The conditional mean of Y:

5. Picturing the Simple Linear Regression ModelX Y
E[Y]=0 + 1 X Xi }
1 = Slope 1
0 = Intercept Yi {
Error: i
Regression Plot
The simple linear regression model gives an exact linear relationship between the expected or average value of Y, the dependent variable, and X, the independent or predictor variable:
E[Yi]=0 + 1 Xi
Actual observed values of Y differ from the expected value by an unexplained or random error:
Yi = E[Yi] + i
= 0 + 1 Xi + i

6. 10-3 Estimation: The Method of Least Squares
Estimation of a simple linear regression relationship involves finding estimated or predicted values of the intercept and slope of the linear regression line.
The estimated regression equation:
Y = b0 + b1X + e
where b0 estimates the intercept of the population regression line, 0 ;
b1 estimates the slope of the population regression line, 1;
and e stands for the observed errors - the residuals from fitting the estimated regression line b0 + b1X to a set of n points.

7. Fitting a Regression LineY Y
Data
Three errors from the least squares regression line X X Y
Errors from the least squares regression line are minimized
Three errors from a fitted line X X

8. Errors in Regression Y . { X Xi

9. Least Squares Regressionb0
SSE b1
Least squares b0
Least squares b1
At this point SSE is minimized with respect to b0 and b1

10. Sums of Squares, Cross Products, and Least Squares Estimators

11. Error Variance and the Standard Errors of Regression EstimatorsX Y
Square and sum all regression errors to find SSE.

12. Standard Errors of Estimates in Regression

13. Confidence Intervals for the Regression Parameters
Length = 1
Height = Slope
Least-squares point estimate:
b1=
Upper 95% bound on slope:
Lower 95% bound:
(not a possible value of the
regression slope at 95%)

14. Correlation
The correlation between two random variables, X and Y, is a measure of the degree of linear association between the two variables.
The population correlation, denoted by, can take on any value from -1 to 1.
   indicates a perfect negative linear relationship
-1 <  < 0 indicates a negative linear relationship
   indicates no linear relationship
0 <  < 1 indicates a positive linear relationship
   indicates a perfect positive linear relationship
The absolute value of  indicates the strength or exactness of the relationship.

15. Illustrations of CorrelationY X
 = 0 Y X
 = 1 Y X
 = -1 Y X
 = -.8 Y X
 = 0 Y X
 = .8

16. Covariance and Correlation
Example 10 - 1: = r SS XY X Y 4 84 29
9824 ( )( ) .
*Note: If  < 0, b1 < 0 If  = 0, b1 = 0 If  > 0, b1 >0

17. Hypothesis Tests for the Correlation Coefficient
H0:  = 0 (No linear relationship)
H1:   0 (Some linear relationship)
Test Statistic:

18. Hypothesis Tests about the Regression RelationshipX
Constant Y
Unsystematic Variation
Nonlinear Relationship
A hypothes
is test fo
r the exis
tence of a
linear re
lationship
between X
and Y: H 1
Test stati
stic for t
he existen
ce of a li
near relat
ionship be
tween X an
d Y: ( - )
where
is the le
ast
squares es
timate of
the regres
sion slope
and
) is the s
tandard er
ror of .
When the
null hypot
hesis is t
rue,
the stati
stic has a
distribu
tion with
degrees o f
freedom. : b 2 = t n s

19. Hypothesis Tests for the Regression Slope

20. How Good is the Regression?
The coefficient of determination, r2, is a descriptive measure of the strength of the regression relationship, a measure of how well the regression line fits the data. . { Y X }
Total Deviation
Explained Deviation
Unexplained Deviation
Percentage of total variation explained by the regression.

21. The Coefficient of DeterminationY Y Y X X X
SST
SST
SST S E
r2=0
SSE
r2=0.50
SSE
SSR
r2=0.90
SSR 5 4 3 2 1 7 6 M i l e s D o a r

22. Analysis of Variance and an F Test of the Regression Model

23. Use of the Regression Model for Prediction
Point Prediction
A single-valued estimate of Y for a given value of X obtained by inserting the value of X in the estimated regression equation.
Prediction Interval
For a value of Y given a value of X
Variation in regression line estimate
Variation of points around regression line
For an average value of Y given a value of X

24. Prediction Interval for a Value of Y

25. Prediction Interval for the Average Value of Y

26. RINGKASAN
Regresi :
Bentuk hubungan anatara variabel bebas dengan variabel tak bebas
Korelasi:
Keeratan dan arah hubungan antara dua variabel
Uji hipotesis parameter regresi
Uji hipotesis korelasi