Teoretiske udregninger h\303\270rende til hypotesetest 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 nu unders\303\270ge om der er afh\303\246ngighed mellem k\303\270n og t\303\270jforbrug! Vi f\303\245r da f\303\270rst og fremmest brug for at beregne de forventede v\303\246rdier og testst\303\270rrelsen. Derfor indskriver vi de oplyste data i passende lister (hvor vi fx som vist l\303\270ber tabellen igennem r\303\246kke for r\303\246kke). L\303\246g m\303\246rke til at vi indf\303\270rer et decimalpunktum i udregningen af de forventede st\303\270rrelser, idet vi dividerer med '360.'! Derved tvinger vi Maple til at regne numerisk - Maple kan sagtens udregne det hele symbolsk, men formlerne er ikke lige s\303\245 oplysende som tallene. QyQtSSV3aXRoRzYiNiNJK1N0YXRpc3RpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI= Pkkkb2JzRzYiNyYiIykqIiQtIiIjZyIkKyI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder NyYiIykqIiQtIiIjZyIkKyI= JSFH PkklZm9ydkc2IjcmKiYtSSIqRyUqcHJvdGVjdGVkRzYkIiRlIiIkKyMiIiItSSIvRzYkRilJKF9zeXNsaWJHRiQ2IyQiJGckIiIhRi0qJi1GKDYkIiQtI0YsRi1GLkYtKiYtRig2JEYrIiRnIkYtRi5GLSomLUYoNiRGOUY9Ri1GLkYt LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder NyYkIit5eHh4KCkhIikkIitBQUFBNiEiKCQiK0FBQUFxRiUkIit5eHh4KilGJQ== 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder JCIrNjBgdFohIio= Vi ved nu at tesst\303\270rrelsen er chi2-fordelt med 1 frihedsgrad. Vi kan derfor nemt finde p-v\303\246rdien, dvs. sandsynligheden for at vi rammer mindst lige s\303\245 sk\303\246vt som det observerede: PkkicEc2IiwmIiIiRiYtSSRDREZHRiQ2JC1JKkNoaVNxdWFyZUdGJDYjRiZJKWNoaTJ0ZXN0R0YkISIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder JCIqKnAvISpHISM1 JSFH Den kritiske sandsynlighed er alts\303\245 2.89%, hvorfor resultatet er signifikant p\303\245 5%-niveau, dvs. vi forkaster nulhypotesen om uafh\303\246ngighed p\303\245 5%-niveauet. Vi kunne ogs\303\245 have fundet den kritiske gr\303\246nse for testst\303\270rrelsen: JSFH PkkpY2hpMmtyaXRHNiItSSlRdWFudGlsZUdGJDYkLUkqQ2hpU3F1YXJlR0YkNiMiIiIkIiMmKiEiIw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder JCIrbWdYVFEhIio= Hvis testst\303\270rrelsen ligger over 3.84 er resultatet alts\303\245 kritisk, dvs. vi m\303\245 forkaste nulhypotesen (p\303\245 5%-niveauet). Endelig kan vi illustrere testen grafisk: PkkjRzFHNiItSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiUtSSRQREZHRiQ2JC1JKkNoaVNxdWFyZUdGJDYjIiIiSSJ4R0YkL0YyOyIiISIiKS9JInlHRiQ7RjVGMQ== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder 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PkkjRzJHNiItSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiYtSSRQREZHRiQ2JC1JKkNoaVNxdWFyZUdGJDYjIiIiSSJ4R0YkL0YyO0kpY2hpMnRlc3RHRiQiIikvSSJ5R0YkOyIiIUYxL0knZmlsbGVkR0YkSSV0cnVlR0Yo LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder 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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 unders\303\270ge om den observerede fordeling f\303\270lger den forventede. Vi overf\303\270rer derfor data til to lister og udregner testst\303\270rrelsen: QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK1N0YXRpc3RpY3NHRiUhIiI= PkknaHlwT2JzRzYiNyoiIykqIiMpKSIkKj4iJE8iIiQ1IyIkeiIiI18iI1E= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder NyoiIykqIiMpKSIkKj4iJE8iIiQ1IyIkeiIiI18iI1E= JSFH PkkoaHlwRm9ydkc2IjcqJCIjayIiISQiIyQqRigkIiR5IkYoJCIkQiJGKCQiJFYjRigkIiQhPUYoJCIjbUYoJCIjYEYo LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder NyokIiNrIiIhJCIjJCpGJSQiJHkiRiUkIiRCIkYlJCIkViNGJSQiJCE9RiUkIiNtRiUkIiNgRiU= JSFH PkkudGVzdFN0fGN6cnJlbHNlRzYiLUkkYWRkRyUqcHJvdGVjdGVkRzYkKiYsJiZJJ2h5cE9ic0dGJDYjSSJpR0YkIiIiJkkoaHlwRm9ydkdGJEYtISIiIiIjRjBGMi9GLjtGLyIiKQ== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder JCIrMVlbKVEkISIp JSFHJSFH Da testst\303\270rrelsen er chi2-fordelt med 7 frihedsgrader er testst\303\270rrelsen meget langt fra det forventede (antallet af frihedsgrader), hvorfor det p\303\245 forh\303\245nd er klart at vi m\303\245 forkaste nulhypotesen, men for illustrationens skyld udregner vi alligevel p-v\303\246rdien og den kritiske v\303\246rdi: PkkicEc2IiwmIiIiRiYtSSRDREZHRiQ2JC1JKkNoaVNxdWFyZUdGJDYjIiIoSS50ZXN0U3R8Y3pycmVsc2VHRiQhIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder JCInMTU9ISM1 Sandsynligheden under nulhypotesen for at finde en v\303\246rdi for tesst\303\270rrelsen, der er mindst lige s\303\245 sk\303\246v som den observerede er alts\303\245 helt nede p\303\245 0.0018% og afvigelsen er derfor signifikant for ethvert rimeligt signifikansniveau! PkksdGVzdEtyaXRpc2tHNiItSSlRdWFudGlsZUdGJDYkLUkqQ2hpU3F1YXJlR0YkNiMiIigkIiMqKiEiIw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder JCIrI3BJdiU9ISIp Hvis testst\303\270rrelsen ligger over 18.47 er afvigelsen alts\303\245 kritisk, dvs. vi m\303\245 forkaste nulhypotesen (p\303\245 1%-niveauet). Vi slutter som f\303\270r med at illustrere testet grafisk, men p\303\245 grund af den ekstremt lille p-v\303\246rdi viser vi grafen i to udgaver: en der illustrerer fordelingen, og en der illustrerer p-v\303\246rdien: PkkjRzFHNiItSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiUtSSRQREZHRiQ2JC1JKkNoaVNxdWFyZUdGJDYjIiIoSSJ4R0YkL0YyOyIiISIjUy9JInlHRiQ7RjUkIiIjISIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= = print(); # input placeholder 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