Matakuliah : I0134/Metode Statistika
Tahun
: 2007
Pendugaan Parameter Varians dan Rasio Varians
Pertemuan 18
Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2002 South-Western/Thomson Learning
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Chapter 11
Inferences About Population Variances
• Inference about a Population Variance
• Inferences about the Variances of Two Populations
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Inferences About a Population Variance
• Chi-Square Distribution
• Interval Estimation of 2
• Hypothesis Testing
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Chi-Square Distribution
• The chi-square distribution is the sum of squared standardized
normal random variables such as
(z1)2+(z2)2+(z3)2 and so on.
• The chi-square distribution is based on sampling from a normal
population.
• The sampling distribution of (n - 1)s2/ 2 has a chi-square
distribution whenever a simple random sample of size n is
selected from a normal population.
• We can use the chi-square distribution to develop interval
estimates and conduct hypothesis tests about a population
variance.
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Interval Estimation of 2
• Interval Estimate of a Population Variance
( n  1) s 2
 /2
2
 
2
( n  1) s 2
 2(1 / 2)
where the  values are based on a chi-square distribution with
n - 1 degrees of freedom and where 1 -  is the confidence
coefficient.
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Interval Estimation of 
• Interval Estimate of a Population Standard Deviation
Taking the square root of the upper and lower limits of the
variance interval provides the confidence interval for the population
standard deviation.
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Interval Estimation of 2
• Chi-Square Distribution With Tail Areas of .025
.025
.025
95% of the
possible 2 values
0
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2
.975
2
.025
2
Example: Buyer’s Digest
Buyer’s Digest rates thermostats manufactured
for home temperature control. In a recent test, 10
thermostats manufactured by ThermoRite were selected
and placed in a test room that was maintained at a
temperature of 68oF. The temperature readings of the
ten thermostats are listed below.
We will use the 10 readings to develop a 95%
confidence interval estimate of the population variance.
Therm. 1
2
3 4
5
6
7
8
9 10
Temp. 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2
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Example: Buyer’s Digest
• Interval Estimation of 2
n - 1 = 10 - 1 = 9 degrees of freedom and  = .05
2
(
n

1)
s
2
2
.975



.025
2

.025
0
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2
.975
.025
2
.025
2
Example: Buyer’s Digest
• Interval Estimation of 2
n - 1 = 10 - 1 = 9 degrees of freedom and  = .05
2.70 
.025
(n  1)s 2
2
2
  .025
Area in
Upper Tail
= .975
2
0 2.70
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Example: Buyer’s Digest
• Interval Estimation of 2
n - 1 = 10 - 1 = 9 degrees of freedom and  =2.05
( n  1) s
2. 70 
 19 . 02
2

.025
Area in Upper
Tail = .025
2
0 2.70
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19.02
Example: Buyer’s Digest
• Interval Estimation of  2
Sample variance s2 provides a point estimate of  2.
2
(
x

x
)
6. 3

i
s2 

 . 70
n 1
9
A 95% confidence interval for the population variance is given
by:
(10  1). 70
(10  1). 70
 2 
19. 02
2. 70
.33 < 2 < 2.33
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Hypothesis Testing
About a Population Variance
• Left-Tailed Test
– Hypotheses
H 0 :  2   20
H a :  2   20
– Test Statistic
2 
– Rejection Rule
( n  1) s 2
 20
2

  2
Reject H0 if
(where
is based
on
2

a chi-square distribution with n - 1 d.f.) or
Reject H0 if p-value < 
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Hypothesis Testing
About a Population Variance

Right-Tailed Test
• Hypotheses
H 0 :  2   20
H a :  2   20
• Test Statistic
2 
• Rejection Rule
( n  1) s 2
 20
2
2
2
Reject H0 if    (1  ) (where  (1 ) is based on
a chi-square distribution with n - 1 d.f.) or
Reject H0 if p-value < 
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Hypothesis Testing
About a Population Variance

Two-Tailed Test
• Hypotheses
H 0 :  2   20
H a :  2   20
• Test Statistic
2 
( n  1) s 2
 20
• Rejection Rule
Reject H0 if  2   2(1  / 2 ) or  2   2 / 2 (where
(12  /2) and 2 /2 are based on a chi-square distribution with n - 1 d.f.) or Reject H0 if p-value < 
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Hypothesis Testing About the
Variances of Two Populations

One-Tailed Test
• Hypotheses
H 0 :  12   22
H a :  12   22
• Test Statistic
s12
F 2
s2
• Rejection Rule
Reject H0 if F > F where the value of F is based
on an F distribution with n1 - 1 (numerator) and
n2 - 1 (denominator) d.f.
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Hypothesis Testing About the
Variances of Two Populations

Two-Tailed Test
• Hypotheses
H 0 :  12   22
Ha : 12   22
• Test Statistic
s12
F 2
s2
• Rejection Rule
Reject H0 if F > F/2 where the value of F/2 is
based on an F distribution with n1 - 1
(numerator) and n2 - 1 (denominator) d.f.
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Example: Buyer’s Digest
Buyer’s Digest has conducted the same test, as was
described earlier, on another 10 thermostats, this time
manufactured by TempKing. The temperature readings
of the ten thermostats are listed below.
We will conduct a hypothesis test with  = .10 to see
if the variances are equal for ThermoRite’s thermostats
and TempKing’s thermostats.
Therm. 1
2
3
4
5
6
7
8
9 10
Temp. 66.4 67.8 68.2 70.3 69.5 68.0 68.1 68.6 67.9 66.2
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Example: Buyer’s Digest
• Hypothesis Testing About the Variances of Two Populations
–
–
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Hypotheses
H 0 :  12   22 (ThermoRite and TempKing thermostats have same temperature variance)
H a :  12   22 (Their variances are not equal)
Rejection Rule
The F distribution table shows that with  = .10,
9 d.f. (numerator), and 9 d.f. (denominator),
F.05 = 3.18.
Reject H0 if F > 3.18
Example: Buyer’s Digest
• Hypothesis Testing About the Variances of Two Populations
–
–
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Test Statistic
ThermoRite’s sample variance is .70.
TempKing’s sample variance is 1.52.
F = 1.52/.70 = 2.17
Conclusion
We cannot reject H0. There is insufficient
evidence to conclude that the population
variances differ for the two thermostat brands.
End of Chapter 11
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