MATHHX B

MATHHX B

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15.4 Normalfordelte observationssæt

Lad os lege at vi plukker æbler fra en æbleplantage. Her mener jeg frugten æble og ikke computermærket ”Apple”. Vægten af et æble vil variere fra æble til æble, og derfor kan vægten af et tilfældigt æble beskrives med en stokastisk variable \(X\). Vi antager nu at vægten \(X\) målt i gram følger normalfordelingen:

\[X\sim N(100,20).\]

Lad os sige at vi plukker 500 æbler, og laver et histogram over deres vægt. Hvordan ville vi forvente at histogrammet så ud? Ja det må jo ligne tæthedsfunktionen for \(X\), fordi arealerne af søjlerne i histogrammet viser frekvenserne, og frekvenserne må jo svare til sandsynlighederne. Altså hvis der er \(15\%\) sandsynlighed for at et æble vejer mellem \(110\) og \(120\) gram, så vil vi jo forventer at \(15\%\), af de æbler vi plukker, vil veje mellem \(110\) og \(120\) gram osv. Jeg har simuleret 500 æbler ud fra normalfordelingen med \(\mu = 100\) og \(\sigma = 20\):

(-tikz- diagram)

Jeg tilføjer nu tæthedsfunktionen for \(X\sim N(100,20.\):

(-tikz- diagram)

Vi kan se at histogrammet følger tæthedsfunktionen som forventet.

I ovennævnte eksempel startede vi med at antage at æblerne var normalfordelte. Men typisk ved man ikke hvordan et givet observationssæt er fordelt. I sådan et tilfælde kan man undersøge om observationssættet er normalfordelt. Vi skal nu se hvordan. Vi vil tage udgangspunkt i det første observationssæt i filen: her. Vi skal også bruge WordMat-statistik (som I også kan åbne fra Word, hvis I har WordMat installeret – klik WordMat \(\rightarrow \) Statistik).

Vi starter med at åbne observationssættet og WordMat-statistik og copy-paste data fra det første observationssæt ind i WordMat-statistik som vist her:

(image)

Vi skal nu gruppere materialet. Vi kan se at værdierne (i hvert fald) ligger imellem 0 og 20 så vi vælger det som grænser. Vi sætter intervalbredden til 1, så vi får 20 intervaller. Generelt er det en god ide at lave histogrammet med 10-20 intervaller:

(image)

Efter vi er færdige med at gruppere klikker vi "Kopier til øvrige ark":

(image)

Vi klikker så fanen "Histograf- Fit"i bunden:

(image)

Vi ser at WordMat har tegnet en normalfordelingskurve oven i histogrammet. Den har ikke gjort det specielt godt som vi kan se

(image)

Heldigvis kan vi justere middelværdi og standardafvigelse på normalfordelingen. Vi prøver med \(\mu =10\) og \(\sigma =2\):

(image)

Det var meget bedre. Vi kan se at normalfordelingenskurven ligger nogenlunde pænt langs histogrammet og vi kan dermed sige at observationssættet er normalfordelt med \(\mu =10\) og \(\sigma =2\) – sådan ca. altså.

Vi kan finde et mere præcist bud på middelværdien og standardafvigelsen ved at klikke på "Normal-Plot":

(image)

Her har WordMat har lavet et såkaldt normalfordelingsplot af punkterne. Vi bemærker to ting:

  • 1. Punkterne ligger ca. på en linje. Det betyder at observationssættet er normalfordelt.

  • 2. Vi aflæser \(\mu =10{,}15\) og \(\sigma =1{,}95\) ude til venstre (markeret med rødt).

Øvelse 15.4.1

  • a) Afgør om de resterende observationssæt er er normalfordelte og angiv i givet fald et estimat for deres middelværdi og standardafvigelse.

Løsning 15.4.1

  • a) Udover det første er det kun "Observationssæt 4"som kan siges at være tilnærmelsesvis normalfordelt. Her har vi \(\mu =31\) og \(\sigma =1022\).