Pendugaan Parameter Varians dan Rasio Varians Pertemuan 18 Matakuliah: I0134/Metode Statistika...
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Transcript of Pendugaan Parameter Varians dan Rasio Varians Pertemuan 18 Matakuliah: I0134/Metode Statistika...
Pendugaan Parameter Varians dan Rasio Varians Pertemuan 18
Matakuliah : I0134/Metode StatistikaTahun : 2007
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Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS
St. Edward’s UniversitySt. Edward’s University
© 2002 South-Western/Thomson Learning© 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
where the values are based on a chi-square distribution with n - 1 degrees of freedom and where 1 - is the confidence coefficient.
( ) ( )
/ ( / )
n s n s
1 12
22
22
1 22
( ) ( )
/ ( / )
n s n s
1 12
22
22
1 22
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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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• Chi-Square Distribution With Tail Areas of .025
95% of thepossible 2 values 95% of thepossible 2 values
22
00
.025.025.025.025
.9752.9752 .025
2.0252
Interval Estimation of 2
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Example: Buyer’s Digest
Buyer’s Digest rates thermostats manufacturedfor home temperature control. In a recent test, 10thermostats manufactured by ThermoRite were selected and placed in a test room that was maintained at atemperature 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
22
00
.025.025.025.025
22 2.975 .0252
( 1)n s
2
2 2.975 .0252
( 1)n s
2.9752.975 2
.0252
.025
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• Interval Estimation of 2
n - 1 = 10 - 1 = 9 degrees of freedom and = .05
22
00
.025.025
2.702.70
22.0252
( 1)2.70
n s
2
2.0252
( 1)2.70
n s
Example: Buyer’s Digest
Area inArea inUpper TailUpper Tail
= .975= .975
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Example: Buyer’s Digest
• Interval Estimation of 2
n - 1 = 10 - 1 = 9 degrees of freedom and = .05 2 70
119 02
2
2.( )
. n s
2 70
119 02
2
2.( )
. n s
22
00
Area in UpperArea in UpperTail = .025Tail = .025Area in UpperArea in UpperTail = .025Tail = .025
.025.025
2.702.70 19.0219.02
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• Interval Estimation of 2
Sample variance s2 provides a point estimate of 2.
A 95% confidence interval for the population variance is given by:
.33 < 2 < 2.33
sx xni2
2
16 39
70
( ) .
.sx xni2
2
16 39
70
( ) .
.
( )..
( )..
10 1 7019 02
10 1 702 70
2 ( ).
.( ).
.10 1 70
19 0210 1 70
2 702
Example: Buyer’s Digest
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• Left-Tailed Test– Hypotheses
– Test Statistic
– Rejection Rule
Reject H0 if (where is based on
a chi-square distribution with n - 1 d.f.) or
Reject H0 if p-value <
Hypothesis TestingAbout a Population Variance
H02
02: H0
202:
Ha : 202Ha : 202
22
02
1 ( )n s
22
02
1 ( )n s
2 2 2 2
22
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Right-Tailed TestRight-Tailed Test
• HypothesesHypotheses
• Test StatisticTest Statistic
• Rejection RuleRejection Rule
Reject Reject HH00 if (where is if (where is based onbased on
a chi-square distribution with a chi-square distribution with nn - 1 d.f.) - 1 d.f.) oror
Reject Reject HH00 if if pp-value < -value <
Hypothesis TestingHypothesis TestingAbout a Population VarianceAbout a Population Variance
22
02
1 ( )n s
22
02
1 ( )n s
2(1 ) 2(1 )
Ha : 202Ha : 202
H02
02: H0
202:
2
12 ( )
21
2 ( )
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Two-Tailed TestTwo-Tailed Test
• HypothesesHypotheses
• Test StatisticTest Statistic
• Rejection RuleRejection Rule
Reject Reject HH00 if if (where (where are based on a are based on a chi-square distribu-chi-square distribu- tion with tion with nn - 1 d.f.) or - 1 d.f.) or Reject Reject HH00 if if pp-value < -value <
Hypothesis TestingHypothesis TestingAbout a Population VarianceAbout a Population Variance
22
02
1 ( )n s
22
02
1 ( )n s
Ha : 202Ha : 202
H02
02: H0
202:
2
1 22 ( / )
21 2
2 ( / ) or 2
22 /or
22
2 /2 2(1 / 2) / 2 and 2 2(1 / 2) / 2 and
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One-Tailed TestOne-Tailed Test
• HypothesesHypotheses
• Test StatisticTest Statistic
• Rejection RuleRejection Rule
Reject Reject HH00 if if FF > > FF where the value of where the value of FF is based is based on an on an FF distribution with distribution with nn11 - 1 - 1 (numerator) and (numerator) and nn2 2 - 1 (denominator) d.f. - 1 (denominator) d.f.
Hypothesis Testing About theHypothesis Testing About theVariances of Two PopulationsVariances of Two Populations
Fs
s 1
2
22
Fs
s 1
2
22
H0 12
22: H0 1
222:
Ha : 12
22Ha : 1
222
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Two-Tailed TestTwo-Tailed Test
• HypothesesHypotheses
• Test StatisticTest Statistic
• Rejection RuleRejection Rule
Reject Reject HH00 if if FF > > FF/2 /2 where the value of where the value of FF/2 /2 is is based on an based on an FF distribution with distribution with nn11 - 1 - 1
(numerator) and (numerator) and nn2 2 - 1 (denominator) d.f. - 1 (denominator) d.f.
Hypothesis Testing About theHypothesis Testing About theVariances of Two PopulationsVariances of Two Populations
H0 12
22: H0 1
222:
Ha : 12
22Ha : 1
222
Fs
s 1
2
22
Fs
s 1
2
22
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Buyer’s Digest has conducted the same test, as wasdescribed earlier, on another 10 thermostats, this time manufactured by TempKing. The temperature readingsof the ten thermostats are listed below.
We will conduct a hypothesis test with = .10 to see if the variances are equal for ThermoRite’s thermostatsand TempKing’s thermostats.
Therm. 1 2 3 4 5 6 7 8 9 10 Temp. 66.467.868.270.3 69.5 68.0 68.1 68.6 67.9
66.2
Example: Buyer’s Digest
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• Hypothesis Testing About the Variances of Two Populations– Hypotheses
(ThermoRite and TempKing thermo- stats have same temperature variance)
(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
H0 12
22: H0 1
222:
Ha : 12
22Ha : 1
222
Example: Buyer’s Digest
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• Hypothesis Testing About the Variances of Two Populations– 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 populationvariances differ for the two thermostat
brands.
Example: Buyer’s Digest
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End of Chapter 11