Matakuliah
Tahun
: I0272 - STATISTIK PROBABILITAS
: 2009
INFERENSIA KORELASI DAN REGRESI LINIER
SEDERHANA
Pertemuan 12
Materi
• Pengujian koefisien korelasi
• Pengujian parameter regresi linier sederhana
• Koefisien determinasi dan peramalan
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Persamaan Regresi
• Persamaan matematika yang memungkinkan kita meramalkan nilainilai 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   x y 
 i
 i

i i
n
i

1
b
 i1
n 
2
2

n




 x - x
 x
 i

i 1
n 2  i  1 i 
 xi 
n
i1
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a  y - bx
4
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
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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
H 0: b1 = 0
H a: 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.
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Contoh Soal: Reed Auto Sales
• t Test
– Hypotheses
H 0: 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.
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Confidence Interval for b1
• The form of a confidence interval for b1 is:
where
b1  t / 2 sb1
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
 / 2 b1
t distribution with n - 2 degrees
 / 2 of freedom
t s
t
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Contoh Soal: Reed Auto Sales
b t s
• Rejection Rule
1
 / 2 b1
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.
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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.
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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.
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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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