Review of Lecture 10

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Lecture 11
Review of Lecture 10 Prediction and Prediction
Intervals More Examples about Model Comparison
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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
FgtF(r,df(SSE,F),alpha)
Cant Reject H0 otherwise.
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Special Case ANOVA Table (Analysis of Variance)
Source Sum of Squares df Mean Square F-test P-value
Regression SSR p MSRSSR/p FMSR/MSE
Residuals SSE n-p-1 MSESSE/(n-p-1)
Total SST n-1
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Predictions Recall the prediction for the SLR
model
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Prediction for MLR Model

Standard Errors
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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 1848.76
Residual
Total
Coefficient Table
Variable Coefficients s.e. T-test P-value
Constant -23.4325 12.74 .0824
X1 .1528 8.32 lt.0001
n R2 Ra2 S df
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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(man1, woman0) 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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Table 3.14 ANOVA Table
Source Sum of Squares df Mean Square F-test
Regression 23665352 4 5916338 22.98
Residual 22657938 88 257477
Total 46323290 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 .000
Education 90.02 24.69 3.65 .000
Experience 1.2690 .5877 2.16 .034
Month 23.406 5.201 4.50 .000
n93 R2.515 Ra2.489 S507.4 Df88
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• Conduct the F-test for the overall fit of the
  regression (F(4,88,.05)lt2.53)
• 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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(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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(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 3.15. 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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Table 3.15 ANOVA Table
Source Sum of Squares df Mean Square F-test
Regression 7862535 1 7862535 18.60
Residual 38460756 91 422646
Total 46323291 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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• After-class Questions
• Why ANOVA table can be used to test if R_sq0?
• Why F-test can be used to test if the effect of
  a predictor variable is significant or not?