De indbyggede rutiner

Maple har indbyggede rutiner til at udf\303\270re chi2-test for s\303\245vel uafh\303\246ngighed som for Goodness of Fit (hvor det sidste endda findes i to udgaver, men vi vil kun kigge p\303\245 den simple udgave, hvor vi sammenligner en stikpr\303\270ve med en given population for at afg\303\270re om den er repr\303\246sentativ).

4a: Uafh\303\246ngighed (side 4 i kursusmaterialet)

K\303\270n\134Forbrug <1500 kr./m\303\245ned \342\211\245 1500 kr./m\303\245ned i alt

kvinder 98 102 200

m\303\246nd 60 100 160

i alt 158 202 360

L\303\270sning: Vi skal afg\303\270re om de oplyste data er i rimelig overensstemmelse med nulhypotesen om uafh\303\246ngighed mellem K\303\270n og Forbrug. Vi benytter det indbyggede test for uafh\303\246ngighed, der forventer at f\303\245 oplyst matricen for de observerede hyppigheder samt et signifikansniveau, som vi v\303\246lger at s\303\246tte til 5%:

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print(); # input placeholder

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print(); # input placeholder

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Vi f\303\245r da som vist en r\303\246kke koncentrerede oplysninger:

1) Nulhypotesen forkastes (p\303\245 5%-niveauet)

2) Den kritiske v\303\246rdi er 3.84, dvs. hvis testst\303\270rrelsen n\303\245r op over 3.84 er afvigelsen signifikant (p\303\245 5%-niveauet), og derfor forkastes nulhypotesen.

3) Testst\303\270rrelsen er chi2 fordelt med 1 frihedsgrad.

4) p-v\303\246rdien er 2.89%, dvs. sandsynligheden for at finde en tesst\303\270rrelse, der er mindst lige s\303\245 sk\303\246v som den observerede er 2.89%. Da den ligger under signifikansniveauet forkastes nulhypotesen.

5) testst\303\270rrelsen har v\303\246rdien 4.77

VI kan godt f\303\245 flere informationer: I s\303\245 fald skal vi blot s\303\246tte informationsniveauet op:

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print(); # input placeholder

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print(); # input placeholder

Chi-Square Test for Independence -------------------------------- Null Hypothesis: Two attributes within a population are independent of one another Alt. Hypothesis: Two attributes within a population are not independent of one another Dimensions: 2 Total Elements: 360 Distribution: ChiSquare(1) Computed statistic: 4.77353 Computed pvalue: 0.0289005 Critical value: criticalvalue Result: [Rejected] There exists statistical evidence against the null hypothesis

Eksempel 2: (side 24 i kursusmaterialet)

Indkomstfordelingen i stikpr\303\270ven var:

I = Indkomst i 1000 kr. I<50 50\342\211\244I<100 100\342\211\244I<150 150\342\211\244I<200 200\342\211\244I<300 300\342\211\244I<400 400\342\211\244I<500 500\342\211\244I

Observeret antal 98 88 199 136 210 179 52 38

Den forventede fordeling i stikpr\303\270ven baseret p\303\245 de ovenst\303\245ende procenter er tilsvarende givet ved:

I = Indkomst i 1000 kr. I<50 50\342\211\244I<100 100\342\211\244I<150 150\342\211\244I<200 200\342\211\244I<300 300\342\211\244I<400 400\342\211\244I<500 500\342\211\244I

Forventet antal 64 93 178 123 243 180 66 53

Sammenholder vi de observerede hyppigheder med de forventede f\303\270lges de s\303\245 nogenlunde ad. Men man kunne m\303\245ske v\303\246re bekymret for, om de laveste indkomster er overrepr\303\246senteret i stikpr\303\270ven. Her ligger den observerede hyppighed et godt stykke over den forventede.

L\303\270sning: Vi skal afg\303\270re om de oplyste data er i rimelig overensstemmelse med nulhypotesen om samme fordeling for stikpr\303\270ve og population. Vi benytter det indbyggede test for Goodness of fit, der forventer at f\303\245 oplyst lister for de observerede hyppigheder og de forventede hyppigheder, samt et signifikansniveau, som vi denne gang v\303\246lger at s\303\246tte til 1%:

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print(); # input placeholder

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JSFH

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print(); # input placeholder

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print(); # input placeholder

Chi-Square Test for Goodness-of-Fit ----------------------------------- Null Hypothesis: Observed sample does not differ from expected sample Alt. Hypothesis: Observed sample differs from expected sample Categories: 8 Distribution: ChiSquare(7) Computed statistic: 33.8848 Computed pvalue: 1.81006e-05 Critical value: 18.47530692 Result: [Rejected] There exists statistical evidence against the null hypothesis

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Vi f\303\245r da som vist en r\303\246kke oplysninger:

1) Nulhypotesen forkastes (p\303\245 5%-niveauet)

2) Den kritiske v\303\246rdi er 18.47, dvs. hvis testst\303\270rrelsen n\303\245r op over 18.47 er afvigelsen signifikant (p\303\245 1%-niveauet), og derfor forkastes nulhypotesen.

3) Testst\303\270rrelsen er chi2 fordelt med 7 frihedsgrad.

4) p-v\303\246rdien er 0.0018%, dvs. sandsynligheden for at finde en tesst\303\270rrelse, der er mindst lige s\303\245 sk\303\246v som den observerede er 0.0018%. Da den ligger under signifikansniveauet forkastes nulhypotesen.

5) testst\303\270rrelsen har v\303\246rdien 33.88

JSFH

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