{"id":3743,"date":"2025-04-15T23:24:06","date_gmt":"2025-04-15T21:24:06","guid":{"rendered":"https:\/\/mxth.dk\/?p=3743"},"modified":"2026-04-09T22:28:04","modified_gmt":"2026-04-09T20:28:04","slug":"test-af-uafhaengighed","status":"publish","type":"post","link":"https:\/\/mxth.dk\/?p=3743","title":{"rendered":"Test af uafh\u00e6ngighed"},"content":{"rendered":"\n\n\n\n\n<p class=\" eplus-wrapper\">Forestil dig at du har lavet en unders\u00f8gelse, hvor du har spurgt en r\u00e6kke mennesker om deres m\u00e5nedlige opsparing. Du har i den forbindelse ogs\u00e5 spurgt om andre ting her i blandt deres alder. Besvarelserne kan samles i f\u00f8lgende krydstabel.<\/p>\n\n\n\n<style type=\"text\/css\">\n  .tg  {border-collapse:collapse;border-spacing:0;}\n  .tg td{\n    border-color:black;border-style:solid;border-width:1px;\n    font-family:Arial, sans-serif;font-size:14px;\n    overflow:hidden;padding:10px 5px;word-break:normal;\n  }\n  .tg th{\n    border-color:black;border-style:solid;border-width:1px;\n    font-family:Arial, sans-serif;font-size:14px;\n    font-weight:normal;overflow:hidden;\n    padding:10px 5px;word-break:normal;\n  }\n  .tg .tg-abx8{background-color:#c0c0c0;font-weight:bold;text-align:left;vertical-align:top}\n  .tg .tg-y6fn{background-color:#c0c0c0;font-weight:bold;text-align:center;vertical-align:top}\n  .tg .tg-0lax{text-align:center;vertical-align:top}\n<\/style>\n\n<table class=\"tg\" align= \"center\" >\n  <thead>\n    <tr>\n      <th class=\"tg-abx8\">Aldersgruppe<\/th>\n      <th class=\"tg-y6fn\">0\u2013500 kr<\/th>\n      <th class=\"tg-y6fn\">501\u20132000 kr<\/th>\n      <th class=\"tg-y6fn\">2001\u20135000 kr<\/th>\n      <th class=\"tg-y6fn\">5001+ kr<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td class=\"tg-abx8\">18\u201329 \u00e5r<\/td>\n      <td class=\"tg-0lax\">40<\/td>\n      <td class=\"tg-0lax\">25<\/td>\n      <td class=\"tg-0lax\">10<\/td>\n      <td class=\"tg-0lax\">5<\/td>\n    <\/tr>\n    <tr>\n      <td class=\"tg-abx8\">30\u201344 \u00e5r<\/td>\n      <td class=\"tg-0lax\">20<\/td>\n      <td class=\"tg-0lax\">35<\/td>\n      <td class=\"tg-0lax\">25<\/td>\n      <td class=\"tg-0lax\">10<\/td>\n    <\/tr>\n    <tr>\n      <td class=\"tg-abx8\">45\u201360 \u00e5r<\/td>\n      <td class=\"tg-0lax\">15<\/td>\n      <td class=\"tg-0lax\">30<\/td>\n      <td class=\"tg-0lax\">30<\/td>\n      <td class=\"tg-0lax\">15<\/td>\n    <\/tr>\n    <tr>\n      <td class=\"tg-abx8\">60+ \u00e5r<\/td>\n      <td class=\"tg-0lax\">25<\/td>\n      <td class=\"tg-0lax\">20<\/td>\n      <td class=\"tg-0lax\">15<\/td>\n      <td class=\"tg-0lax\">10<\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<p class=\" eplus-wrapper\">Du er interesseret i at unders\u00f8ge, om der er en sammenh\u00e6ng. Om de to sp\u00f8rgsm\u00e5l om <strong>alder<\/strong> og <strong>m\u00e5nedlig opsparing<\/strong> er afh\u00e6ngige af hinanden. Men hvordan kan man afg\u00f8re, om forskellene bare skyldes tilf\u00e6ldigheder, eller om der faktisk er en sammenh\u00e6ng.<\/p>\n\n\n\n\n\n\n\n\n\n<p class=\" eplus-wrapper\">Til at unders\u00f8ge dette kan vi benytte en \u03c7<sup>2<\/sup>-test (chi-i-anden test). Denne test ser p\u00e5, hvor meget svarene i stikpr\u00f8ven afviger fra de forventede v\u00e6rdier. De forventede v\u00e6rdier er det antal svar vi ville forvente hvis der var uafh\u00e6ngighed mellem svarene, det vil sige, at det ikke har nogen betydning hvilken alder man har i forhold til den m\u00e5nedlige opsparing. De forventede v\u00e6rdier beregnes ud fra r\u00e6kke- og s\u00f8jlev\u00e6rdierne. Vi ville forvente at antallet af svar ville f\u00f8lge det samlede antal svar for en given kategori. Vi kan beregne de forventede v\u00e6rdier ud p\u00e5 f\u00f8lgende m\u00e5de<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(E_{ij}=\\frac{S_i\\cdot R_j}{n}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvor \\(S_i\\) og \\(R_j\\) er summen for den \\(i\\)\u2019te s\u00f8jle og \\(j\\)\u2019te r\u00e6kke.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Til at m\u00e5le hvor meget de observerede v\u00e6rdier afviger fra det vi ville forvente, hvis der var uafh\u00e6ngighed, beregner vi det der kaldes for testst\u00f8rrelsen<\/p>\n\n\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\[\\chi^2=Q=\\sum_{i=1}^{R}\\sum_{j=1}^{S} \\frac{(O_{ij}-E_{ij})^2}{E_{ij}}\\]<\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\">Hvis testst\u00f8rrelsen er pr\u00e6cis nul betyder det at vi observere pr\u00e6cis det vi ville hvis der var uafh\u00e6ngighed. Men det er ikke sikker at vi f\u00e5r pr\u00e6cis nul. Det kan v\u00e6re vi for testst\u00f8rrelsen f\u00e5r 0,01 eller 0,5 eller m\u00e5ske 4. Men er en testst\u00f8rrelse p\u00e5 0,01 nok til at afvise at der er uafh\u00e6ngighed? Eller skal den v\u00e6re 0,5 eller er 4 m\u00e5ske stadig bevis for at der er uafh\u00e6ngighed?<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi skal have et m\u00e5l som fort\u00e6ller os om der er uafh\u00e6ngighed eller ej. Dette m\u00e5l kaldes for den kritiske v\u00e6rdi og den kan vi sl\u00e5 op i en tabel.<\/p>\n\n\n\n<figure class=\" wp-block-image size-full eplus-wrapper\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"686\" height=\"338\" src=\"https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2025\/04\/IMG_1355.jpeg?resize=686%2C338&#038;ssl=1\" alt=\"\" class=\"wp-image-3830\" srcset=\"https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2025\/04\/IMG_1355.jpeg?w=686&amp;ssl=1 686w, https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2025\/04\/IMG_1355.jpeg?resize=300%2C148&amp;ssl=1 300w\" sizes=\"auto, (max-width: 686px) 100vw, 686px\" \/><\/figure>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\"><em> Tabel med kritiske v\u00e6rdier for \\(\\chi^2\\) (Butler, Christopher. 1985.&nbsp;Statistics in Linguistics. Oxford: Blackwell. s 176)<\/em><\/p>\n\n\n\n<p class=\" has-text-align-left eplus-wrapper\">For at finde den kritiske v\u00e6rdi som vi skal holde vores testst\u00f8rrelse op i mod skal vi kende to ting: frihedsgraden og signifikansniveauet. Vi vil som udgangspunkt altid v\u00e6lge et signifikansniveau p\u00e5 5% (eller 0,05). Man kan godt benytte andre signifikansniveauer, men for nu s\u00e5 er det 5%. Man kan se p\u00e5 de 5% som at vi ser bort fra de 5% h\u00f8jeste \\(\\chi^2\\) v\u00e6rdier. Sandsynligheden for at vi f\u00e5r en meget h\u00f8j v\u00e6rdi for testst\u00f8rrelsen (\\(\\chi^2\\)) er meget lille hvis der er uafh\u00e6ngighed. Vi v\u00e6lger derfor at se bort fra dem. Vi vil vende tilbage til dette p\u00e5 et senere tidspunkt n\u00e5r vi skal se p\u00e5 konfidensintervaller.<\/p>\n\n\n\n<p class=\" has-text-align-left eplus-wrapper\">Frihedgraden er bestem af vores krydstabel. Frihedsgraden (forkortet df for degrees of freedom) som er et m\u00e5l for hvor mange felter vi tilf\u00e6ldigt kan udfylde inden alle felter er l\u00e5st for en v\u00e6rdi. Vi kan beregne antallet af frihedsgrader som<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(df=(antal\\,r\u00e6kker\\,-\\,1)\\cdot(antal\\,kolonner\\,-\\,1)\\)<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n<p class=\" eplus-wrapper\">Vi vil altid teste for uafh\u00e6ngighed, s\u00e5 vores nulhypotese (H<sub>0<\/sub>) vil altid v\u00e6re at der er uafh\u00e6ngighed (alts\u00e5 ikke sammenh\u00e6ng) mellem svarene. Den alternative hypotese (H<sub>1<\/sub>) vil da v\u00e6re at der er afh\u00e6ngighed (alts\u00e5 sammenh\u00e6ng) mellem svarene.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">I dette tilf\u00e6lde med alder og m\u00e5nedlig opsparing kunne det en hypotese v\u00e6re. <\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\"><em>\u201cDer er ingen sammenh\u00e6ng mellem alder og m\u00e5nedlig opsparing\u201d<\/em><\/p>\n\n\n\n<p class=\" eplus-wrapper\">eller<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\"><em>\u201dDer er uafh\u00e6ngighed mellem alder og m\u00e5nedlig opsparing\u201d<\/em><\/p>\n\n\n\n<p class=\" eplus-wrapper\">eller <\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\"><em>\u201cHvor gammel man er har ingen betydning for hvor stor den m\u00e5nedlige opsparing er\u201d<\/em><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Man kan formulere ens hypotese p\u00e5 mange m\u00e5de, men det er vigtig at nulhypotesen fort\u00e6ller at der er uafh\u00e6ngighed mellem de to unders\u00f8gte kategorier.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Hvis der er uafh\u00e6ngighed mellem vores kategorier ville vi forvente en lille testst\u00f8rrelse (\\(\\chi^2\\)), da vores observationer ligger t\u00e6t p\u00e5 vores forventede v\u00e6rdier. Til geng\u00e6ld s\u00e5 vil vi have store testst\u00f8rrelser, hvis der er afh\u00e6ngighed, da de adspurgte vil have en tendens til at give et specifikt svar an p\u00e5 hvilken kategori de tilh\u00f8rer. Da vi fjerner de 5% st\u00f8rste testst\u00f8rrelser vil vi have, at hvis vores bidrag ligger over den kritiske v\u00e6rdi vil det ligge uden for hvad vi forventede som er vores nulhypotese og vi vil derfor forkaste vores nulhypotese om uafh\u00e6ngighed. <\/p>\n\n\n\n<p class=\" eplus-wrapper\">Det vil sige, at <\/p>\n\n\n<ul class=\" wp-block-list eplus-wrapper eplus-styles-uid-dcb501\">\n<li class=\" eplus-wrapper\">hvis testst\u00f8rrelsen \\(\\chi^2\\) &lt; kritisk v\u00e6rdi s\u00e5 beholder vi H<sub>0<\/sub><\/li>\n\n\n\n<li class=\" eplus-wrapper\">hvis testst\u00f8rrelsen \\(\\chi^2\\) &gt; kritisk v\u00e6rdi s\u00e5 forkaster vi H<sub>0<\/sub><\/li>\n<\/ul>\n\n\n<p class=\" eplus-wrapper\">Vi vil her ofte have en tendens til at acceptere den alternative hypotese om afh\u00e6ngighed. Dette vil ogs\u00e5 i de fleste tilf\u00e6lde v\u00e6re rigtig, men da vi ikke har testet for om der er afh\u00e6ngighed kan vi teknisk set ikke konkludere dette ud fra testen, selvom vi alligevel ofte g\u00f8r det.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi vil her se n\u00e6rmere p\u00e5 dette med et eksempel, men i kan l\u00e6se mere i bogen (<a href=\"https:\/\/matematikb-hhx.systime.dk\/?id=174\">https:\/\/matematikb-hhx.systime.dk\/?id=174<\/a>) b\u00e5de hvad ang\u00e5r teori og eksempler.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi starter med at se p\u00e5 et eksempel hvor vi regner det i h\u00e5nden for at f\u00e5 en forst\u00e5else for hvad der sker.<\/p>\n\n\n\n<figure class=\"wp-embed-aspect-16-9 wp-has-aspect-ratio wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube eplus-wrapper\"><div class=\"wp-block-embed__wrapper\">\n<span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/EtykwDCXuyk?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span>\n<\/div><\/figure>\n\n\n\n<p class=\" eplus-wrapper\">Vi kan nu beregne dette i h\u00e5nden, men hvad hvis vi gerne vil benytte os af CAS. Lad os se p\u00e5 hvorledes vi kan benytte Excel til at udregne en \u03c7<sup>2<\/sup>-test.<\/p>\n\n\n\n<figure class=\"wp-embed-aspect-16-9 wp-has-aspect-ratio wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube eplus-wrapper\"><div class=\"wp-block-embed__wrapper\">\n<span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/HqRT9kfw4i4?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span>\n<\/div><\/figure>\n\n\n\n<p class=\" eplus-wrapper\">Der er ogs\u00e5 mulighed for at benytte GeoGebra. Lad os se p\u00e5 hvorledes vi kan benytte GeoGebra til at beregne en \u03c7<sup>2<\/sup>-test.<\/p>\n\n\n\n<figure class=\"wp-embed-aspect-4-3 wp-has-aspect-ratio wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube eplus-wrapper\"><div class=\"wp-block-embed__wrapper\">\n<span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/qXkqV00_rck?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span>\n<\/div><\/figure>\n\n\n\n<p class=\" eplus-wrapper\"><a href=\"https:\/\/science-gym.dk\/cas-it\/it1213\/chi-i-anden-test_i_GeoGebra.pdf\">Denne pdf<\/a> fra Danske Science Gymnasier viser ogs\u00e5 hvorledes man kan g\u00f8re det i GeoGebra<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Der er ogs\u00e5 mulighed for at lave en \u03c7<sup>2<\/sup>-test i WordMat som denne video fra Mitfyns gymnasium viser<\/p>\n\n\n\n<figure class=\"wp-embed-aspect-4-3 wp-has-aspect-ratio wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube eplus-wrapper\"><div class=\"wp-block-embed__wrapper\">\n<span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/BH2p7ttqrzU?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span>\n<\/div><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Forestil dig at du har lavet en unders\u00f8gelse, hvor du har spurgt en r\u00e6kke mennesker om deres m\u00e5nedlige opsparing. Du har i den forbindelse ogs\u00e5 spurgt om andre ting her i blandt deres alder. Besvarelserne kan samles i f\u00f8lgende krydstabel. Aldersgruppe 0\u2013500 kr 501\u20132000 kr 2001\u20135000 kr 5001+ kr 18\u201329 \u00e5r 40 25 10 5 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3,68],"tags":[67],"class_list":["post-3743","post","type-post","status-publish","format-standard","hentry","category-matematik","category-sandsynlighedsregning","tag-hhx"],"featured_image_src":null,"author_info":{"display_name":"Henriksen","author_link":"https:\/\/mxth.dk\/?author=1"},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/3743","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3743"}],"version-history":[{"count":23,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/3743\/revisions"}],"predecessor-version":[{"id":4088,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/3743\/revisions\/4088"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3743"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3743"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3743"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}