13 regresi linier dan korelasi
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Pertemuan 13Regresi Linear dan Korelasi
Matakuliah : I0262 – Statistik Probabilitas
Tahun : 2007
Versi : Revisi
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Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat memilih statistik uji untuk koefisien regresi dan korelasi.
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Outline Materi
• Pengujian koefisien regresi dengan analisis varians
• Inferensia tentang koefisien korelasi
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Testing for Significance
• To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of 1 is zero.
• Two tests are commonly used– t Test– F Test
• Both tests require an estimate of 2, the variance of in the regression model.
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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:
210
2 )()ˆ(SSE iiii xbbyyy 210
2 )()ˆ(SSE iiii xbbyyy
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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.
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SSEMSE
n
s2
SSEMSE
n
s
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• Hypotheses
H0: 1 = 0
Ha: 1 = 0
• Test Statistic
• Rejection Rule
Reject H0 if t < -tor t > t
where t is based on a t distribution with
n - 2 degrees of freedom.
Testing for Significance: t Test
tbsb
1
1
tbsb
1
1
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• t Test – Hypotheses H0: 1 = 0
Ha: 1 = 0
– Rejection Rule
For = .05 and d.f. = 3, t.025 = 3.182
Reject H0 if t > 3.182
– Test Statistics
t = 5/1.08 = 4.63– Conclusions
Reject H0
Contoh Soal: Reed Auto Sales
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Confidence Interval for 1
• We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test.
• H0 is rejected if the hypothesized value of 1 is not included in the confidence interval for 1.
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Confidence Interval for 1
• The form of a confidence interval for 1 is:
where b1 is the point estimate
is the margin of error
is the t value providing an area
of /2 in the upper tail of a
t distribution with n - 2 degrees
of freedom
12/1 bstb 12/1 bstb
12/ bst 12/ bst2/t 2/t
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Contoh Soal: Reed Auto Sales
• Rejection Rule
Reject H0 if 0 is not included in the confidence interval for 1.
• 95% Confidence Interval for 1
= 5 +- 3.182(1.08) = 5 +- 3.44
/ or 1.56 to 8.44/• Conclusion
Reject H0
12/1 bstb 12/1 bstb
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Testing for Significance: Testing for Significance: FF Test Test
HypothesesHypotheses
HH00: : 11 = 0 = 0
HHaa: : 11 = 0 = 0 Test StatisticTest Statistic
FF = MSR/MSE = MSR/MSE Rejection RuleRejection Rule
Reject Reject HH00 if if FF > > FF
where where FF is based on an is based on an FF distribution with 1 distribution with 1 d.f. in d.f. in
the numerator and the numerator and nn - 2 d.f. in the - 2 d.f. in the denominator.denominator.
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F F Test Test
• HypothesesHypotheses H H00: : 11 = 0 = 0
HHaa: : 11 = 0 = 0
• Rejection RuleRejection Rule
For For = .05 and d.f. = 1, 3: = .05 and d.f. = 1, 3: FF.05.05 = = 10.1310.13
Reject Reject HH00 if F > 10.13. if F > 10.13.
• Test StatisticTest Statistic
FF = MSR/MSE = 100/4.667 = 21.43 = MSR/MSE = 100/4.667 = 21.43
• ConclusionConclusion
We can reject We can reject HH00..
Example: Reed Auto SalesExample: Reed Auto Sales
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Some Cautions about theInterpretation of Significance Tests
• Rejecting H0: 1 = 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: 1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.
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Confidence Interval Estimate of Confidence Interval Estimate of EE((yypp))
Prediction Interval Estimate of Prediction Interval Estimate of yypp
yypp ++ tt/2 /2 ssindind
where the confidence coefficient is 1 - where the confidence coefficient is 1 - and and
tt/2 /2 is based on ais based on a t t distribution with distribution with nn - 2 - 2 d.f.d.f.
Using the Estimated Regression Using the Estimated Regression EquationEquation
for Estimation and Predictionfor Estimation and Prediction
/ y t sp yp 2 / y t sp yp 2
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• 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
• Confidence Interval for E(yp)
95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is:
25 + 4.61 = 20.39 to 29.61 cars
• Prediction Interval for yp
95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: 25 + 8.28 = 16.72 to 33.28 cars
^̂
Contoh Soal: Reed Auto Sales
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• Residual for Observation i
yi – yi
• Standardized Residual for Observation i
where:
Residual Analysis
^̂y ysi i
y yi i
y ysi i
y yi i
^̂
^̂
s s hy y ii i 1s s hy y ii i 1
^̂
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Contoh Soal: Reed Auto Sales
• ResidualsObservation Predicted Cars Sold Residuals
1 15 -12 25 -13 20 -24 15 25 25 2
Observation Predicted Cars Sold Residuals1 15 -12 25 -13 20 -24 15 25 25 2
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Contoh Soal: Reed Auto Sales
• Residual Plot
TV Ads Residual Plot
-3
-2
-1
0
1
2
3
0 1 2 3 4TV Ads
Re
sid
ua
ls
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Residual Analysis• Detecting Outliers
– An outlier is an observation that is unusual in comparison with the other data.
– Minitab classifies an observation as an outlier if its standardized residual value is < -2 or > +2.
– This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier.
– This rule’s shortcoming can be circumvented by using studentized deleted residuals.
– The |i th studentized deleted residual| will be larger than the |i th standardized residual|.
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• Selamat Belajar Semoga Sukses.
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