MATHHX B

MATHHX B

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17.3 Chi i anden-test med Excel og GeoGebra

I de forgående afsnit har vi taget udgangspunkt i følgende tabel over observerede værdier:

.
Dreng Pige Total
Drømmekage
\(13\) \(5\) \(18\)
Chokoladekage
\(14\) \(11\) \(25\)
Andet
\(6\) \(6\) \(12\)
Total
\(33\) \(22\) \(\textbf {55}\)

En sådan tabel kaldes en pivottabel og vi skal først se på hvordan man opstiller den ud fra rå data.

Pivottabeller i Excel

Normal tager vi udgangspunkt i data som er i rå form. I forhold til kagetesten kunne det se sådan ud:

(image)

Typisk har vi mange hundrede af observationer, så det er ikke praktisk at tælle dem op en ad gangen. I stedet får vi Excel til at lave en pivottabel for os.

(image)

Min var allerede indstillet korrekt, men her skal I vælge data. I skal bare vælge det hele (inklusiv titlerne på søjlerne):

(image)

Så er det tid til drag and drop.

(image)

Og vi har vores tabel:

(image)

Chi i anden-test i GeoGebra

Vi er nu klar til at lave chi i anden-test I GeoGebra. Det er nemt. Vi åbner sandsynlighedslommeregneren i GeoGebra, vælger statistik, vælger ”Chi\(^2\)-Test” og indtaster de observerede værdier:

(image)

Vi ser (markeret med grønt), at der er \(2\) frihedsgrader, \(\chi ^2=1{,}79\) og \(p=0{,}41\) Vi holder lige øje med de totaler værdier (her altså \(33\) og \(22\)), så vi kan se at vi har tastet rigtigt.

Vi kan også finde de forventede værdier:

(image)

…og bidragene til \(\chi ^2\)-teststørrelsen:

(image)

Øvelse 17.3.1

Disclaimer: Følgende er en tænkt situation.

En ridelære havde lagt mærke til at der var nogle hestenavne var særligt populære i Danmark. For hopperne var det Beauty, Freja, Lady og Musse. Ridelæreren havde en mistanke om at ens alder måske kunne sige noget om, hvilke navne man bedst kunne lide. For at undersøge dette lavede hun en undersøgelse, hvor hun spurgte rideskolens medlemmer, hvilke af de 4 navne de bedst kunne lide.

Du skal nu vha. en \(\chi ^2\)-test undersøge om der er nogen sammenhæng mellem alder (junior, ungdom og senior) og yndlingsnavn. Brug \(\alpha =5\%\) og regn det hele i Excel og Geogebra.

Data kan ses her her

  • a) Opstil hypoteser

  • b) Lav en pivottabel

  • c) Bestem de forventede værdier

  • d) Bestem bidragene til teststørrelsen.

  • e) Bestem antallet af frihedsgrader, teststørrelsen og \(p\)-værdien.

  • f) Afgør om vi skal forkaste \(H_0\).

  • g) Afgør om der er sammenhæng mellem holding til navn og alderskategori.

  • h) Forklar mere om sammenhængen mellem holdning navn og alderskategori. Tag udgangspunkt i bidragene til \(\chi ^2\)-teststørrelsen og ud fra dem så sammenlign de observerede og de forventede værdier.

Løsning 17.3.1

  • a) \(H_0\): Der er uafhængighed mellem alderskategori og navn.
    \(H_1\): Der er ikke uafhængighed mellem alderskategori og navn.

  • b)

    .
    Junior Senior Ungdom Total
    Beauty
    \(11\) \(19\) \(16\) \(46\)
    Freja
    \(19\) \(11\) \(22\) \(52\)
    Lady
    \(17\) \(10\) \(14\) \(41\)
    Musse
    \(36\) \(12\) \(12\) \(60\)
    Total
    \(83\) \(52\) \(64\) \(\textbf {199}\)
  • c)

    .
    Junior Senior Ungdom
    Beauty
    \(19{,}19\) \(12{,}02\) \(14{,}79\)
    Freja
    \(21{,}69\) \(13{,}59\) \(16{,}72\)
    Lady
    \(17{,}10\) \(10{,}71\) \(13{,}19\)
    Musse
    \(25{,}03\) \(15{,}68\) \(19{,}30\)
  • d)

    .
    Junior Senior Ungdom
    Beauty
    \(3{,}49\) \(4{,}05\) \(0{,}10\)
    Freja
    \(0{,}33\) \(0{,}49\) \(1{,}67\)
    Lady
    \(0{,}00\) \(0{,}05\) \(0{,}05\)
    Musse
    \(4{,}81\) \(0{,}86\) \(2{,}76\)
  • e) \(f=6\), \(\chi ^2=18{,}67\), og \(p=0{,}48\%\)

  • f) Jepper, da \(p<5\%\)

  • g) Det er der. Vi forkastede jo \(H_0\) og accepterede \(H_1\).

  • h) Der var især holdningsforskelle til navnene Beauty og Musse. Juniorgruppen kunne ikke lide navnet Beauty, mens seniorerne var ville med det. Musse var til gengæld mægtig populært blandt juniorerne.