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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 untuk varians
PengujianHipotesisuntukVarians

PengujianHipotesis

untukVarians

SatuPopulasi

DuaPopulasi

Chi-Square test statistic

F test statistic

satu populasi
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
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
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
  • about a population variance.
slide6

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
slide7

Interval Estimation of 2

0.025

0.025

95% of the

possible 2 values

2

0

nilai kritis
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 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
Contoh
  • Sebuahmeriamharusmemilikiketepatanmenembakdenganvariasi yang minimum. Spesifikasidaripabriksenjatamenyebutkanbahwastandardeviasidariketepatanmenembakmeriamjenistersebutmaksimumadalah 4 meter. Untukmengujihaltersebut, diambilsampelsebanyak 16 meriamdandiperolehhasils2 = 24 meter. Ujilahstandardeviasidarispesifikasitersebut! Gunakan = 0.05
slide11

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

dua populasi
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 untuk perbedaan dua varians populasi
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 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

nilai kritis1
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
nilai kritis2
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
f test an example
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
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 solution1
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