Pengujian Hipotesis untuk Satu dan Dua Varians Populasi

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Pengujian Hipotesis untuk Satu dan Dua Varians Populasi. Pengujian Hipotesis untuk Varians. Pengujian Hipotesis untuk Varians. Satu Populasi. Dua Populasi. Chi-Square test statistic. F test statistic. Satu Populasi. Pengujian Hipotesis untuk Varians. *. Satu Populasi.

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### Pengujian Hipotesis untukSatu dan Dua Varians Populasi

PengujianHipotesisuntukVarians

PengujianHipotesis

untukVarians

SatuPopulasi

DuaPopulasi

Chi-Square test statistic

F test statistic

SatuPopulasi

PengujianHipotesisuntukVarians

*

SatuPopulasi

H0: σ2 = σ02

HA: σ2≠σ02

Two tailed test

H0: σ2σ02

HA: σ2<σ02

Lower tail test

Chi-Square test statistic

H0: σ2≤σ02

HA: σ2>σ02

Upper tail test

Chi-Square Test Statistic

PengujianHipotesisuntukVarians

StatistikUji:

SatuPopulasi

*

Chi-Square test statistic

Dimana:

2 = variabel standardized chi-square

n = jumlahsampel

s2 = varianssampel

σ2 = varians yang dihipotesiskan

Chi-Square Distribution
• The chi-square distributionis 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

Examples of Sampling Distribution of (n - 1)s2/ 2

With 2 degrees

of freedom

With 5 degrees

of freedom

With 10 degrees

of freedom

0

• Distribusichi-squaretergantungdariderajatbebasnya: d.f. = n – 1

Interval Estimation of 2

0.025

0.025

95% of the

possible 2 values

2

0

NilaiKritis
• Nilaikritis, ,dapatdilihatdaritabel chi-square

2

Upper tail test:

H0: σ2≤σ02

HA: σ2>σ02

2

Do not reject H0

Reject H0

2

Lower Tail or Two Tailed Chi-square Tests

Lower tail test:

Two tail test:

H0: σ2σ02

HA: σ2<σ02

H0: σ2 = σ02

HA: σ2≠σ02

/2

/2

2

2

Reject

Do not reject H0

Do not

reject H0

Reject

Reject

21-

21-/2

2/2

Contoh
• Sebuahmeriamharusmemilikiketepatanmenembakdenganvariasi yang minimum. Spesifikasidaripabriksenjatamenyebutkanbahwastandardeviasidariketepatanmenembakmeriamjenistersebutmaksimumadalah 4 meter. Untukmengujihaltersebut, diambilsampelsebanyak 16 meriamdandiperolehhasils2 = 24 meter. Ujilahstandardeviasidarispesifikasitersebut! Gunakan = 0.05

Hipotesis:

H0: σ2≤16

HA: σ2>16

• Nilaikritisdaritabel chi-square :

2

= 24.9958 ( = 0.05 dand.f. = 16 – 1 = 15)

StatistikUji:

Karena 22.5 < 24.9958,

Tidakdapatmenolak H0

 = .05

2

Do not reject H0

Reject H0

2

= 24.9958

DuaPopulasi

PengujianHipotesisuntukVarians

*

DuaPopulasi

H0: σ12 – σ22 = 0

HA: σ12 – σ22 ≠ 0

Two tailed test

H0: σ12 – σ22 0

HA: σ12 – σ22 < 0

Lower tail test

F test statistic

H0: σ12 – σ22≤ 0

HA: σ12 – σ22 > 0

Upper tail test

F Test untukPerbedaanDuaVariansPopulasi

PengujianHipotesisuntukVarians

F test statistic :

DuaPopulasi

*

F test statistic

= Variance of Sample 1

n1 - 1 = numerator degrees of freedom

= Variance of Sample 2

n2 - 1 = denominator degrees of freedom

The F Distribution
• The F critical valueis found from the F table
• The are two appropriate degrees of freedom: numerator and denominator
• In the F table,
• numerator degrees of freedom determine the row
• denominator degrees of freedom determine the column

where df1 = n1 – 1 ; df2 = n2 – 1

NilaiKritis

H0: σ12 – σ22 0

HA: σ12 – σ22 < 0

H0: σ12 – σ22≤ 0

HA: σ12 – σ22 > 0

0

F

0

F

F

Reject

Do not reject H0

Do not

reject H0

Reject H0

F1-

• rejection region
• rejection region
NilaiKritis

H0: σ12 – σ22 = 0

HA: σ12 – σ22 ≠ 0

/2

/2

0

F

F/2

Do not

reject H0

Reject

Reject

F1-/2

• rejection region for a two-tailed test is
You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data:

NYSENASDAQNumber 21 25

Mean 3.27 2.53

Std dev 1.30 1.16

Is there a difference in the variances between the NYSE & NASDAQ at the =0.1 level?

F Test: An Example
F Test: Example Solution
• Form the hypothesis test:

H0: σ21 – σ22 = 0 (there is no difference between variances)

HA: σ21 – σ22 ≠ 0 (there is a difference between variances)

• Find the F critical value for  = 0.1:
• Numerator:
• df1 = n1 – 1 = 21 – 1 = 20
• Denominator:
• df2 = n2 – 1 = 25 – 1 = 24

F0.05, 20, 24 = 2.03

F0.95, 20, 24 = 0.48

F Test: Example Solution

(continued)

• The test statistic is:

H0: σ12 – σ22 = 0

HA: σ12 – σ22 ≠ 0

/2 = 0.05

/2 = 0.05

0

• F = 1.256 is not greater than the critical F value of 2.327 or not less than the critical F value of 0.48, so we do not reject H0

Reject H0

Do not

reject H0

Reject H0

F1-α/2=0.48

F/2=2.03

• Conclusion: There is no evidence of a difference in variances at  = .05