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16.2 Konfidensintervaller – binomialfordeling

Ligesom vi bestemte konfidensintervaller for middelværdien i en normalfordeling kan vi også bestemme konfidensintervaller for sandsynlighedsparameteren \(p\) i en binomialfordeling.

Vi husker at en binomialfordeling er noget der opstår, når vi gentager et forsøg med to muligheder flere gange. Hvis et meningsmålingsinstitut laver en meningsmåling, hvor de spørger 100 mennesker om de vil stemme blåt eller rødt til næste valg er resultatet binomialfordelt \(b(100,p)\), hvor \(p\) er sandsynligheden for at en tilfældig person vil stemme blåt. Antag nu, at en sådan meningsmåling viste at 51 personer stemte blåt. Ud fra resultatet kan vi estimere at \(\frac {51}{100}=51\%\) vil stemme blåt. Estimatet betegner vi med \(\hat {p}\). Vi har altså at \(\hat {p}=0,51\), men betyder det så at vi kan være sikre på blå blok vinder valget? Det kan de finde ud af ved at lave et konfidensinterval.

Vi laver konfidensintervaller for binomialfordelinger helt på tilsvarende måde som for normalfordelinger. Vi vælger bare "Z interval for andel"i Geogebra:

(image)

Vi indtaster nu data. Der var 51 personer der stemte blåt – det er vores succeser – og så havde vi 100 observationer:

(image)

Og vi kan så aflæse konfidensintervallet:

(image)

Vi får altså et \(95\%\)-procentkonfidensinterval på [0,41;0,61], hvilket betyder, at vi kan være \(95\%\) sikre på at blå bloks vælgertilslutning er mellem \(41\%\) og \(61\%\). Så den meningsmåling kan de ikke bruge til så meget.

Øvelse 16.2.1

En elev ville undersøge ungdomsarbejdsløsheden i Danmark og gik ned på Kultorvet og spurgte 200 unge om de var i arbejde. Der var 32 ud af de 200 som ikke var i arbejde.

  • a) Angiv et estimat af ungdomsarbejdsløsheden.

  • b) Bestem et \(90\%\)-konfidensinterval for ungdomsarbejdsløsheden og forklar, hvad det betyder.

Løsning 16.2.1

  • a) \(\hat {p}=0{,}16\)

  • b) Konfidensintervallet er \([0,12;0,20]\), hvilket betyder at vi med \(90\%\)-sikkerhed kan sige at ungdomsarbejdsløsheden ligger mellem \(12\%\) og \(20\%\)

Øvelse 16.2.2

Vi vender tilbage til meningsmålingen. For at blive klogere på om de vinder valget spørger de nu 1000 mennesker. Resultatet er at 527 ville stemme blåt.

  • a) Angiv et estimat af sansynligheden for at en tilfældig person vil stemme blåt

  • b) Kan blå blok nu være sikre på at vinde valget?

Løsning 16.2.2

  • a) \(\hat {p}=0{,}527\)

  • b) De kan være \(90\%\) sikre på at vinde det, men ikke \(95\%\).

Forudsætninger

Metoden virker kun for store stikprøver. En tommelfingerregel er et minimum på \(30\) observationer. Derudover må \(\hat {p}\) ikke være meget lille eller meget stor. Det kan udtrykkes ved uligheden:

\[n\cdot \hat {p}\cdot (1-\hat {p})>9\]

Vi ser at uligheden kun er opfyldt hvis \(\hat {p}\) ikke er meget lille eller meget stor. Uligheden er en tommelfingerregel ligesom kravet på minimum 30. Det er bedst hvis stikprøven er meget stor og \(\hat {p}\) ligger tæt på \(50\%\).