Pendugaan Parameter Varians dan Rasio Varians Pertemuan 18 Matakuliah: I0134/Metode Statistika Tahun: 2007.

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