1. Prediction and Prediction Intervals
Lecture 11
Review of Lecture 10
Prediction and Prediction Intervals
More Examples about Model Comparison
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2. Steps for Model Comparison :
RM H0: The RM is adequate vs FM H1: The FM is adequate
Step1: Fit the FM and get SSE (in the ANOVA table)
df (in the ANOVA table)
R_sq (under the Coefficient Table)
Step 2: Fit the RM and get SSE, df, and R_sq.
Step 3: Compute F-statistic:
Step 4: Conclusion: Reject H0 if F>F(r,df(SSE,F),alpha)
Can’t Reject H0 otherwise.
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ST3131, Lecture 11

3. Special Case: ANOVA Table (Analysis of Variance)
Source
Sum of Squares df
Mean Square
F-test
P-value
Regression
SSR p
MSR=SSR/p
F=MSR/MSE
Residuals
SSE
n-p-1
MSE=SSE/(n-p-1)
Total
SST
n-1
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4. Predictions: Recall the prediction for the SLR model:
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5. Prediction: for MLR Model
Standard Errors
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6. Problem 3. 5 (Page 76, textbook) Table 3
Problem 3.5 (Page 76, textbook) Table 3.11 shows the regression output, with some numbers erased, when a simple regression model relating a response variable Y to a predictor variable X1 is fitted based on 20 observations. Complete the 13 missing numbers, then compute Var(Y) and Var(X1).
ANOVA Table
Source
Sum of Squares df
Mean Square
F-test
Regression
Residual
Total
Coefficient Table
Variable
Coefficients
s.e.
T-test
P-value
Constant
12.74
.0824 X1
.1528
8.32
<.0001 n=
R^2=
Ra^2= S= df
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9. Sex (X1): An indicator variable(man=1, woman=0)
Problem 3.12 (Page 78, textbook) Table 3.14 shows the regression output of a MLR model relating the beginning salaries in dollars of employees in a given company to the following predictor variables:
Sex (X1): An indicator variable(man=1, woman=0)
Education(X2): Years of Schooling at the time of hire
Experience(X3): Number of months previous work experience
Months(X4): Number of months with the company
In (a)-(b) below, specify the null and alternative hypotheses the test used, and your conclusion using a 5% level of significance.
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ST3131, Lecture 11

10. Table 3.14 ANOVA Table Coefficient Table 11/21/2018 ST3131, Lecture 11
Source
Sum of Squares df
Mean Square
F-test
Regression 4
22.98
Residual 88
257477
Total 92
Coefficient Table
Variable
Coefficients
s.e.
T-test
P-value
Constant
3526.4
327.7
10.76
.000
Sex
722.5
117.8
6.13
Education
90.02
24.69
3.65
Experience
1.2690
.5877
2.16
.034
Month
23.406
5.201
4.50
n=93
R^2=.515
Ra^2=.489
S=507.4
Df=88
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11. Conclusion: H0, the overall fit is significant.
Conduct the F-test for the overall fit of the regression (F(4,88,.05)<2.53)
Test H0: vs H1:
Statistic F= df=( , )
Conclusion: H0, the overall fit is significant.
Is there a positive linear relationship between Salary and Experience, after accounting for the effect of the variables Sex, Education, and Months.
Test H0: vs H1:
Statistic T= P-value=
Conclusion: H0. The positive relationship is significant at 5% significance level.
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12. (c) What salary would you forecast for a man with 12 years of Education, 10 months of Experience, and 15 months with the company?
(d) What salary would you forecast, on average, for a man with 12 years of Education, 10 months of Experience, and 15 months with the company?
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13. (e) What salary would you forecast, on average, for a woman with 12 years of Education, 10 months of Experience, and 15 months with the company?
Problem 3.13 (Page 79, textbook) Consider the regression model that generated output in Table 31.4 to be a Full Model. Now consider the Reduced Model in which Salary is regression on only Education . The ANOVA table obtained when fitting this model is shown in Table Conduct a single test to compare the Full and Reduced Models. What conclusion can be drawn from the result of the test? (Use 5% significant level).
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14. Statistic SSE(R )= df(R )= SSE(F)= df(F)= F= df=( , ).
Table ANOVA Table
Source
Sum of Squares df
Mean Square
F-test
Regression 1
18.60
Residual 91
422646
Total 92
Test H0: vs H1:
Statistic SSE(R )= df(R )=
SSE(F)= df(F)= F=
df=( , ).
Conclusion: H0. The Reduced Model is significant
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15. After-class Questions:
Why ANOVA table can be used to test if R_sq=0?
Why F-test can be used to test if the effect of a predictor variable is significant or not?
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ST3131, Lecture 11