Matakuliah
Tahun
Versi
: I0284 - Statistika
: 2008
: Revisi
Pertemuan 21 dan 22
Analisis Regresi dan Korelasi
Sederhana
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung
dugaan parameter regresi sederhana,
korelasi dan menguji keberartiannya.
2
Outline Materi
•
•
•
•
•
Estimasi koefisien regresi
Inferensia parameter regresi
Koefisien korelasi
Koefisien determinasi
Inferesia koefisien korelasi
3
Persamaan Regresi
• Persamaan matematika yang memungkinkan
kita meramalkan nilai-nilai peubah tak bebas
dari nilai-nilai satu atau lebih peubah bebas
disebut Persamaan Regresi
• Persamaan Regresi Sederhana:
ˆ  a  bx
Y
 n  n 
  x   y 
n 

i 
i


n
i

1
i

1








x
x
y
y



 xiyi 
 i
 i

n
b  i 1
 i1
n 
2
2

n


  x - x 
 x
 i

i 1
n 2  i  1 i 
 xi 
n
i1
dan
a  y - bx
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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 b1 is zero.
• Two tests are commonly used
– t Test
– F Test
• Both tests require an estimate of s 2, the
variance of e in the regression model.
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Testing for Significance
• An Estimate of s 2
The mean square error (MSE) provides the
estimate
of s 2, and the notation s2 is also used.
s2 = MSE = SSE/(n-2)
where:
SSE   (yi  yˆi ) 2   ( yi  b0  b1 xi ) 2
6
Testing for Significance
• An Estimate of s
– To estimate s we take the square root of s 2.
– The resulting s is called the standard error of
the estimate.
SSE
s  MSE 
n2
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Testing for Significance: t Test
• Hypotheses
H0: b1 = 0
Ha: b1 = 0
• Test Statistic
t 
b1
sb1
• Rejection Rule
Reject H0 if t < -t or t > t
where t is based on a t distribution with
n - 2 degrees of freedom.
8
Contoh Soal: Reed Auto Sales
• t Test
– Hypotheses
H0: b1 = 0
Ha: b1 = 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
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Confidence Interval for b1
• We can use a 95% confidence interval
for b1 to test the hypotheses just used in
the t test.
• H0 is rejected if the hypothesized value
of b1 is not included in the confidence
interval for b1.
10
Confidence Interval for b1
• The form of a confidence interval for b1 is:
b1  t / 2 sb1
where
b1 is the point estimate
t / 2 sb1 is the margin of error
t / 2 is the t value providing an area
of /2 in the upper tail of a
t distribution with n - 2 degrees
of freedom
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Contoh Soal: Reed Auto Sales
b1  t / 2 sb1
• Rejection Rule
Reject H0 if 0 is not included in the
confidence interval for b1.
• 95% Confidence Interval for b1
= 5 +- 3.182(1.08) = 5 +- 3.44
/
or 1.56 to 8.44/
• Conclusion
Reject H0
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Testing for Significance: F Test

Hypotheses
H 0 : b1 = 0
H a : b1 = 0

Test Statistic
F = MSR/MSE

Rejection Rule
Reject H0 if F > F
where F is based on an F distribution with 1 d.f. in
the numerator and n - 2 d.f. in the denominator.
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Example: Reed Auto Sales

F Test
• Hypotheses
• Rejection Rule
H 0 : b1 = 0
H a : b1 = 0
For  = .05 and d.f. = 1, 3: F.05 = 10.13
Reject H0 if F > 10.13.
• Test Statistic
F = MSR/MSE = 100/4.667 = 21.43
• Conclusion
We can reject H0.
14
Some Cautions about the
Interpretation of Significance Tests
• Rejecting H0: b1 = 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: b1 =
0 and demonstrate statistical significance
does not enable us to conclude that there
is a linear relationship between x and y.
15
Using the Estimated Regression Equation
for Estimation and Prediction

Confidence Interval Estimate of E(yp)
y p  t /2 s y p

Prediction Interval Estimate of yp
yp + t/2 sind
where the confidence coefficient is 1 -  and
t/2 is based on a t distribution with n - 2 d.f.
16
Contoh Soal: Reed Auto Sales
• 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
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Residual Analysis
• Residual for Observation i
yi – yi
• Standardized Residual for Observation i
y^i  y^i
sy^i  y^i
where:
syi  yi  s 1  hi
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Contoh Soal: Reed Auto Sales
• Residuals
Observation
1
2
3
4
5
Predicted Cars Sold
15
25
20
15
25
Residuals
-1
-1
-2
2
2
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Contoh Soal: Reed Auto Sales
• Residual Plot
TV Ads Residual Plot
3
Residuals
2
1
0
-1
-2
-3
0
1
2
3
4
TV Ads
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Korelasi Linear
• Koefisien korelasi linear didefiisikan sebagai ukuran
hubungan linear antara dua peubah X dan Y, dan
dilambangkan dengan r.
• Ukuran hubungan linear antara dua peubah X dan Y
diduga dengan koefisien korelasi contoh r yaitu
r
n xi yi   xi  yi 
n x  x  n y   y  
2
i
2
i
2
i
2
i
• Koefisien determinasi = r2
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Uji Korelasi Sederhana
• Hipotesis:
– Ho : r  0 tidak ada hubungan x dan y)
– Ha : r > 0, r < 0, atau r  0
• Statistik uji:
t
r n2
1 r 2
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• Selamat Belajar Semoga Sukses.
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