1. ...Relax... 9/21/2018 ST3131, Lecture 3 ST5213 Semester II, 2000/2001
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9/21/2018
ST3131, Lecture 3

2. Lecture 3 Main Tasks Today! Review of Lecture 2
Statistical Inferences of the LS-Estimators
a). Properties of the LS-estimators
b). Accuracy of the LS-estimators
c). Significance Tests of the Parameters
9/21/2018
ST3131, Lecture 3

3. Review of Lecture 2 Standardization of Y: we have =0 and =1.
Covariance / Correlation of Y and X :
Properties of Cov(Y,X) / [Cor(Y,X)]
(1). Symmetric
(2). Scale-dependent /[invariant]
(3). Take values in the real line/[in [-1,1]]
(4). Measure of direction of the linear relationship between Y and X
(5). Not Robust statistics( affected by outliers)
9/21/2018
ST3131, Lecture 3

4. Review of Lecture 2 (cont.)
For linear regression equation Y= X ,
Cov(Y,X) or Cor(Y,X)
Linear Relationship between Y and X
Positive
Positive(Y increases as X increases)
zero
Uncorrelated (Y does not change as X changes)
negative
Negative (Y decreases as X increases)
Cor(Y,X)
Strength of Linear Relationship between Y and X
Close to –1 or 1
Very strong
Close to -.5 or .5
Strong
Close to 0
Very weak (near linearly uncorrelated)
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ST3131, Lecture 3

5. i 1 2 … n …. Review of Lecture 2 (cont.) Total
Note that Var(Y)=sum of squared deviations of Y divided by (n-1)
Var(X)=sum of squared deviations of X divided by (n-1)
Cov(Y,X)=sum of products of deviations of Y and X divided by (n-1).
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ST3131, Lecture 3

6. Review of Lecture 2 (cont.)
The Least Squares Estimators of and :
They are linear combinations of the observations That is
(Linearity)
(Linearity)
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ST3131, Lecture 3

7. Review of Lecture 2 (cont.)
Linear Regression Line:
Fitted Values:
Predicted Values
Linear Regression Line
For Computer Repair
Data
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ST3131, Lecture 3

8. Linear Estimator and Its Property
Linear Estimator = a linear combination of observations i.e.,
where c’s are any constants.
Notice: and are linear estimators of and
Property: when T= and are independently normally
distributed, we have
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ST3131, Lecture 3

9. Application to SLR Models
Simple Linear Regression Model:
Assumptions:
(1) Linearity between Y and X
(2) Normality for measurement errors
(3) Measurement Errors are independently identically distributed (iid)
Using the property of linear estimators, we have
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ST3131, Lecture 3

10. Application to Parametric Estimators of SLR Model
Using the property of linear estimators, we can show that
(Unbiased)
(Normality)
(Unbiased)
(Normality)
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ST3131, Lecture 3

11. Properties of the LS-Estimators
1.) Linearity
2). Normality
3). Unbiased
4). Best Linear Unbiased Estimators. That is , they have smallest variances among
all unbiased linear estimators. Let and be any other Unbiased Linear
Estimators of and respectively. Then
9/21/2018
ST3131, Lecture 3

12. Accuracy of the LS-Estimates
Noise Variance is usually unknown. It has a natural estimator using residuals, given by
SSE=Sum of Squared Errors (Residuals)
n-2= degrees of freedom of SSE=Sample size- # of Coefficients
Using the estimated noise variance, we can obtain the measures of the accuracy
(Standard Errors, s.e.) of the estimators and
9/21/2018
ST3131, Lecture 3

13. Significance Test of Parameters
In the SLR model, we often want to know if
(1) “ Y and X are linearly uncorrelated” (X has no effect on Y) ( )
(2) “ the linear regression line passes the origin” (no intercept is needed).( )
A hypothesis test has two hypotheses: H0 (Null) and H1 ( Alternative).
A general test about the slope (intercept ) can be written as
versus
(a)
versus
(b)
where ( or ) is a predetermined constant (belief of the experts). To test (1) [or (2)] is equivalent to test =0 [or =0]
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ST3131, Lecture 3

14. Test Statistics and 2-Sided Tests
To test the hypotheses, we need construct test statistics. Test statistic can usually be
constructed as:
For example, to test (1) and (2) , we have test statistics respectively
Which have t-distributions with n-2 degrees of freedom (dfs). Let be the Upper
- percentile of t-distribution with n-2 dfs. Then the level two- sided test for
test (1) or (2) is
can be found in the Appendix (Table A.2, Page 339)
9/21/2018
ST3131, Lecture 3

15. P-value of the Test
In another way, we can compute the p-value of the test statistic to perform the test:
If the p-value< , then the test is level significant. The smaller the p-value, the higher significant level the test is. In general, we say
1) when p-value<.05, the test is significant;
2) when p-value < .01, the test is very significant;
3) when p-value>.20, the test is not significant.
Example For computer repair data, to test
Since >2.18, the test is significant at 5%-level significance. The
< So the test is highly significant
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ST3131, Lecture 3

16. One-Sided Tests:
(a) (
(b) )
(c) (
(d) )
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ST3131, Lecture 3

17. Outputs for the computer repair data using Splus
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ST3131, Lecture 3

18. Results using Minitab Results for: P027.txt
Regression Analysis: Minutes versus Units
The regression equation is
Minutes = Units
Predictor Coef SE Coef T P-value
Constant
Units
S = R-Sq = 98.7% R-Sq(adj) = 98.6%
Note that the T-values calculated here are for testing`` the true parameters are zeros”.
9/21/2018
ST3131, Lecture 3

19. A Test Using Correlation Coefficient
Since =0 is equivalent to Cor(Y,X)=0. So the following two tests are equivalent to each other:
(3)
(4)
where denotes the true correlation between Y and X. To test (4), we can
use the following test statistic
We can also compute to perform the test
9/21/2018
ST3131, Lecture 3

20. Example: For the computer repair data, since Cor(Y,X)=.996, we have
Which is much larger than So the correlation is highly
Significantly different from 0. That is X has high impact on Y.
We reject H0 of test (4) or (3) at 5%-significant level. The corresponding
p-value much less than So the significance is very high.
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ST3131, Lecture 3

21. Review Sections 2.6-2.9 of Chapter 2.
Reading Assignment
Review Sections of Chapter 2.
Read Sections of Chapter 2. Consider problems:
a) What are special SLR models?
b) Why need we study the special SLR models?
c) How to do inference about the special SLR models?
9/21/2018
ST3131, Lecture 3