{"id":259,"date":"2023-08-03T11:07:43","date_gmt":"2023-08-03T11:07:43","guid":{"rendered":"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/"},"modified":"2023-08-03T11:07:43","modified_gmt":"2023-08-03T11:07:43","slug":"test-hipotezy-dotyczacy-proporcji","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/","title":{"rendered":"Testowanie hipotez dotycz\u0105cych proporcji"},"content":{"rendered":"<p>W tym artykule wyja\u015bniono, jaki udzia\u0142 w statystyce ma testowanie hipotez. Znajdziesz zatem wz\u00f3r na sprawdzenie hipotezy proporcji oraz dodatkowo \u0107wiczenie krok po kroku pozwalaj\u0105ce w pe\u0142ni zrozumie\u0107 jak to si\u0119 robi. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-una-prueba-de-hipotesis-para-la-proporcion\"><\/span> Na czym polega testowanie hipotez dotycz\u0105cych proporcji?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Testowanie hipotezy proporcji<\/strong> to metoda statystyczna stosowana do okre\u015blenia, czy odrzuci\u0107 hipotez\u0119 zerow\u0105 dotycz\u0105c\u0105 proporcji populacji.<\/p>\n<p> Zatem w zale\u017cno\u015bci od warto\u015bci statystyki testowej hipotezy dla proporcji i poziomu istotno\u015bci hipoteza zerowa zostaje odrzucona lub przyj\u0119ta.<\/p>\n<p> Nale\u017cy pami\u0119ta\u0107, \u017ce testowanie hipotez mo\u017cna r\u00f3wnie\u017c nazwa\u0107 kontrastami hipotez, testowaniem hipotez lub testowaniem istotno\u015bci. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-prueba-de-hipotesis-para-la-proporcion\"><\/span> Testowanie hipotezy Wz\u00f3r na proporcj\u0119<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Statystyka testu hipotezy dla proporcji jest r\u00f3wna r\u00f3\u017cnicy proporcji pr\u00f3bki pomniejszonej o proponowan\u0105 warto\u015b\u0107 proporcji podzielon\u0105 przez odchylenie standardowe proporcji.<\/p>\n<p> <strong>Wz\u00f3r hipotezy testowej dla proporcji<\/strong> jest zatem nast\u0119puj\u0105cy:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e2f5ad36dae72cd89279809ab3ea6d20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle Z=\\frac{\\widehat{p}-p}{\\displaystyle\\sqrt{\\frac{p(1-p)}{n}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"123\" style=\"vertical-align: -41px;\"><\/p>\n<\/p>\n<p style=\"margin-bottom:5px\"> Z\u0142oto:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0be116875001706f29a24434bd0d91c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest statystyk\u0105 testuj\u0105c\u0105 hipotez\u0119 dla proporcji.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ecd29d136a62fc6b274e1181e064e20e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\widehat{p}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"12\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest proporcj\u0105 pr\u00f3bki.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5faad0904f612a3fa5b27faafb8dc903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest warto\u015bci\u0105 proponowanej proporcji.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ec4217f4fa5fcd92a9edceba0e708cf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> to wielko\u015b\u0107 pr\u00f3bki.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-661a2bbf8dccd197bcf450330420cbc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\sqrt{\\frac{p(1-p)}{n}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"84\" style=\"vertical-align: -13px;\"><\/p>\n<p> jest odchyleniem standardowym proporcji.<\/li>\n<\/ul>\n<p> Nale\u017cy pami\u0119ta\u0107, \u017ce nie wystarczy obliczy\u0107 statystyk\u0119 testu hipotezy dla proporcji, ale wynik nale\u017cy nast\u0119pnie zinterpretowa\u0107:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Je\u017celi test hipotezy dla proporcji jest dwustronny, hipotez\u0119 zerow\u0105 odrzuca si\u0119, je\u015bli warto\u015b\u0107 bezwzgl\u0119dna statystyki jest wi\u0119ksza ni\u017c warto\u015b\u0107 krytyczna Z <sub>\u03b1\/2<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Je\u015bli test hipotezy dla proporcji odpowiada prawemu ogonowi, hipoteza zerowa jest odrzucana, je\u015bli statystyka jest wi\u0119ksza ni\u017c warto\u015b\u0107 krytyczna Z <sub>\u03b1<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Je\u015bli test hipotezy dla proporcji odpowiada lewemu ogonowi, hipoteza zerowa jest odrzucana, je\u015bli statystyka jest mniejsza ni\u017c warto\u015b\u0107 krytyczna -Z <sub>\u03b1<\/sub> .<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-7d5bd583532769e3014286e8ffd94c9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}H_1: p\\neq p_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } |Z|>Z_{\\alpha\/2} \\text{ se rechaza } H_0\\\\[3ex]H_1: p> p_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } Z>Z_{\\alpha} \\text{ se rechaza } H_0\\\\[3ex]H_1: p< p_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } Z<-Z_{\\alpha} \\text{ se rechaza } H_0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"437\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Pami\u0119taj, \u017ce warto\u015bci krytyczne mo\u017cna \u0142atwo uzyska\u0107 z tabeli rozk\u0142adu normalnego. <\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zobacz:<\/strong> <a href=\"https:\/\/statorials.org\/pl\/tablica-rozk\u0142adu-normalnego\/\">Tabela rozk\u0142adu normalnego<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-la-prueba-de-hipotesis-para-la-proporcion\"><\/span> Przyk\u0142ad testowania hipotezy dotycz\u0105cej proporcji<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Kiedy ju\u017c poznamy definicj\u0119 testowania hipotez pod k\u0105tem proporcji i jaki jest jej wz\u00f3r, rozwi\u0105\u017cemy przyk\u0142ad, aby lepiej zrozumie\u0107 t\u0119 koncepcj\u0119.<\/p>\n<ul>\n<li> Wed\u0142ug producenta lek na konkretn\u0105 chorob\u0119 ma 70% skuteczno\u015bci. W laboratorium testujemy skuteczno\u015b\u0107 tego leku, poniewa\u017c naukowcy uwa\u017caj\u0105, \u017ce proporcja jest inna. W tym celu lek jest testowany na pr\u00f3bie 1000 pacjent\u00f3w i 641 os\u00f3b zostaje wyleczonych. Wykonaj test hipotezy na temat proporcji populacji z poziomem istotno\u015bci 5%, aby odrzuci\u0107 lub nie hipotez\u0119 badaczy.<\/li>\n<\/ul>\n<p> W tym przypadku <a href=\"https:\/\/statorials.org\/pl\/hipoteza-zerowa-i-alternatywna\/\">hipoteza zerowa i hipoteza alternatywna<\/a> testu hipotezy dla proporcji populacji to:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f7da8281eeecc022e2ec7daea6a9756e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases}H_0: p=0,70\\\\[2ex] H_1:p\\neq 0,70 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"117\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Odsetek os\u00f3b w pr\u00f3bie, kt\u00f3re zosta\u0142y wyleczone dzi\u0119ki lekowi, wynosi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-93865dbb1097265b35967f0138a51d89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\widehat{p}=\\cfrac{641}{1000}=0,641\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"137\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Obliczamy statystyk\u0119 testu hipotezy dla proporcji, stosuj\u0105c wz\u00f3r pokazany powy\u017cej:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e689388b0a73e91c1e3d8812c2c4c42a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\displaystyle Z&amp;=\\frac{\\widehat{p}-p}{\\displaystyle\\sqrt{\\frac{p(1-p)}{n}}}\\\\[2ex]Z&amp;=\\frac{0,641-0,70}{\\displaystyle\\sqrt{\\frac{0,70\\cdot (1-0,70)}{1000}}}  \\\\[2ex] Z&amp;=-4,07\\end{aligned}}\" title=\"Rendered by QuickLaTeX.com\" height=\"195\" width=\"186\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Z drugiej strony, poniewa\u017c poziom istotno\u015bci wynosi 0,05 i jest to test hipotezy dwustronnej, warto\u015b\u0107 krytyczna testu wynosi 1,96. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f4aa318300c46315a36b7bbecbd57e02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z_{0,025}=1,96\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Patrz:<\/strong> <a href=\"https:\/\/statorials.org\/pl\/poziom-istotnosci\/\">Poziom istotno\u015bci 0,05<\/a><\/div>\n<p> Podsumowuj\u0105c, warto\u015b\u0107 bezwzgl\u0119dna statystyki testowej jest wi\u0119ksza od warto\u015bci krytycznej, dlatego odrzucamy hipotez\u0119 zerow\u0105 i przyjmujemy hipotez\u0119 alternatywn\u0105. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-06104fc4aa1cf5269aa8bc841cc05a9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|-4,07|=4,07>1,96 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Se rechaza } H_0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;424&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zobacz:<\/strong> <a href=\"https:\/\/statorials.org\/pl\/test-hipotezy-dla-sredniej\/\">Testowanie hipotez dla \u015bredniej<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"prueba-de-hipotesis-para-las-proporciones-de-dos-muestras\"><\/span> Testowanie hipotez dla dw\u00f3ch proporcji pr\u00f3bek<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Testowanie hipotez dla proporcji dw\u00f3ch pr\u00f3bek<\/strong> s\u0142u\u017cy do odrzucenia lub przyj\u0119cia hipotezy zerowej, \u017ce proporcje dw\u00f3ch r\u00f3\u017cnych populacji s\u0105 r\u00f3wne.<\/p>\n<p> Zatem hipoteza zerowa testu hipotezy dla proporcji dw\u00f3ch pr\u00f3bek wynosi zawsze:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5572e3ad602776be8957247415435ef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"H_0: p_1=p_2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Hipotez\u0105 alternatywn\u0105 mo\u017ce by\u0107 jedna z trzech opcji:<\/p>\n<pre class=\"ql-errors\"> *** QuickLaTeX cannot compile formula:\n\\begin{array}{l}H_1:p_1\\neq p_2\\\\[2ex]H_1:p_1&gt;p_2\\\\[2ex]H_1:p_1 The combined ratio of the two samples is calculated as follows:[latex]p=\\cfrac {x_1+x_2}{n_1+n_2}\n\n*** Error message:\nMissing $ inserted.\nleading text: \\begin{array}{l}\nPlease use \\mathaccent for accents in math mode.\nleading text: ...H_1:p_1&gt;p_2\\\\[2ex]H_1:p_1 The combined ratio\nPlease use \\mathaccent for accents in math mode.\nleading text: ...\\[2ex]H_1:p_1 The combined ratio of the two\nPlease use \\mathaccent for accents in math mode.\nleading text: ...combined of the two samples is calculated\n\\begin{array} on input line 8 ended by \\end{document}.\nleading text: \\end{document}\nImproper \\prevdepth.\nleading text: \\end{document}\nMissing $ inserted.\nleading text: \\end{document}\nMissing } inserted.\nleading text: \\end{document}\nMissing \\cr inserted.\nleading text: \\end{document}\nMissing $ inserted.\nleading text: \\end{document}\nYou can't use `\\end' in internal vertical mode.\n\n<\/pre>\n<p> Wz\u00f3r na obliczenie statystyki testu hipotezy dla dw\u00f3ch proporcji pr\u00f3bek to:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-27dc38fa4280149a55c46c6753204c9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle Z=\\frac{\\displaystyle \\frac{x_1}{n_1}-\\frac{x_2}{n_2}}{\\displaystyle \\sqrt{p(1-p)\\left(\\frac{1}{n_1}+\\frac{1}{n_2}\\right)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"94\" width=\"214\" style=\"vertical-align: -52px;\"><\/p>\n<\/p>\n<p style=\"margin-bottom:5px\"> Z\u0142oto:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0be116875001706f29a24434bd0d91c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest statystyk\u0105 testow\u0105 hipotezy dla proporcji dw\u00f3ch pr\u00f3bek.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-d7aa45c8899989487fb32dab51a8f7d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> to liczba wynik\u00f3w w pr\u00f3bce 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-de02251c2c969c17b8633e299d9a2244_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> to liczba wynik\u00f3w w pr\u00f3bce 2.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-cebc0a013985f2695aeb53ded9e7afb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> to pr\u00f3bka o wielko\u015bci 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bf1e42c248eee22a0911c24c95fe28f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> to pr\u00f3bka o wielko\u015bci 2.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5faad0904f612a3fa5b27faafb8dc903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u0142\u0105czn\u0105 proporcj\u0105 dw\u00f3ch pr\u00f3bek. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"prueba-de-hipotesis-para-las-proporciones-de-k-muestras\"><\/span> Testowanie hipotez dla k proporcji pr\u00f3bek<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Celem <strong>testu hipotezy dotycz\u0105cego proporcji k pr\u00f3bek<\/strong> jest ustalenie, czy wszystkie proporcje w r\u00f3\u017cnych populacjach s\u0105 r\u00f3wne, czy te\u017c przeciwnie, czy istniej\u0105 r\u00f3\u017cne proporcje. Zatem hipoteza zerowa i hipoteza alternatywna w tym przypadku to:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-77d7e13b427dd927953473a6bfbe9a55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases}H_0: \\text{Todas las proporciones son iguales}\\\\[2ex] H_1: \\text{No todas las proporciones son iguales} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"341\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> W takim przypadku \u0142\u0105czn\u0105 proporcj\u0119 wszystkich pr\u00f3bek oblicza si\u0119 w nast\u0119puj\u0105cy spos\u00f3b:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bd6eafe24ac6e33f95c11428cdf34b29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p=\\cfrac{\\displaystyle\\sum_{i=1}^k x_i}{\\displaystyle\\sum_{i=1}^k n_i}=\\cfrac{x_1+x_2+\\dots+x_k}{n_1+n_2+\\dots+n_k}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"250\" style=\"vertical-align: -52px;\"><\/p>\n<\/p>\n<p> Wz\u00f3r na znalezienie statystyki testu hipotezy dla k proporcji pr\u00f3bek to: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bf14ba17db281bb65549da86f59a0b65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\chi^2 =\\sum_{i=1}^k \\frac{(x_i-e_i)^2}{e_i}\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"146\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-7213d98c216b76fca752271158e4c7e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\chi^2 = \\frac{(x_1-e_1)^2}{e_1} +\\frac{(x_2-e_2)^2}{e_2} +\\dots+\\frac{(x_k-e_k)^2}{e_k}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"371\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p style=\"margin-bottom:5px\"> Z\u0142oto:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-984dc78529fc235b078a9f3b62d0f0c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\chi^2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest statystyk\u0105 testow\u0105 hipotezy dla k proporcji pr\u00f3bek. W tym przypadku statystyka ma rozk\u0142ad chi-kwadrat.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-dad27a9703483183e1afd245f5232b83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> to liczba wynik\u00f3w w pr\u00f3bce i.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5f087375b50e0b49186779714206626b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> to wielko\u015b\u0107 pr\u00f3bki, tj.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5faad0904f612a3fa5b27faafb8dc903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u0142\u0105czn\u0105 proporcj\u0105 wszystkich pr\u00f3bek.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-550848dfdd05b5c6464d4e22fbf52486_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"e_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"13\" style=\"vertical-align: -3px;\"><\/p>\n<p> to liczba oczekiwanych trafie\u0144 z pr\u00f3bki i. Oblicza si\u0119 go, mno\u017c\u0105c \u0142\u0105czn\u0105 proporcj\u0119<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5faad0904f612a3fa5b27faafb8dc903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> wed\u0142ug wielko\u015bci pr\u00f3bki<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5f087375b50e0b49186779714206626b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> .<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>W tym artykule wyja\u015bniono, jaki udzia\u0142 w statystyce ma testowanie hipotez. Znajdziesz zatem wz\u00f3r na sprawdzenie hipotezy proporcji oraz dodatkowo \u0107wiczenie krok po kroku pozwalaj\u0105ce w pe\u0142ni zrozumie\u0107 jak to si\u0119 robi. Na czym polega testowanie hipotez dotycz\u0105cych proporcji? Testowanie hipotezy proporcji to metoda statystyczna stosowana do okre\u015blenia, czy odrzuci\u0107 hipotez\u0119 zerow\u0105 dotycz\u0105c\u0105 proporcji populacji. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-259","post","type-post","status-publish","format-standard","hentry","category-statystyka"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Testowanie hipotez dotycz\u0105cych proporcji<\/title>\n<meta name=\"description\" content=\"Tutaj znajdziesz, czym jest test hipotezy proporcji, jego wz\u00f3r i konkretny przyk\u0142ad testu hipotezy proporcji.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Testowanie hipotez dotycz\u0105cych proporcji\" \/>\n<meta property=\"og:description\" content=\"Tutaj znajdziesz, czym jest test hipotezy proporcji, jego wz\u00f3r i konkretny przyk\u0142ad testu hipotezy proporcji.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-03T11:07:43+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e2f5ad36dae72cd89279809ab3ea6d20_l3.png\" \/>\n<meta name=\"author\" content=\"Benjamin Anderson\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Napisane przez\" \/>\n\t<meta name=\"twitter:data1\" content=\"Benjamin Anderson\" \/>\n\t<meta name=\"twitter:label2\" content=\"Szacowany czas czytania\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minuty\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/\",\"url\":\"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/\",\"name\":\"\u25b7 Testowanie hipotez dotycz\u0105cych proporcji\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pl\/#website\"},\"datePublished\":\"2023-08-03T11:07:43+00:00\",\"dateModified\":\"2023-08-03T11:07:43+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\"},\"description\":\"Tutaj znajdziesz, czym jest test hipotezy proporcji, jego wz\u00f3r i konkretny przyk\u0142ad testu hipotezy proporcji.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/#breadcrumb\"},\"inLanguage\":\"pl-PL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pl\/test-hipotezy-dotyczacy-proporcji\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Dom\",\"item\":\"https:\/\/statorials.org\/pl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Testowanie hipotez dotycz\u0105cych proporcji\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/statorials.org\/pl\/#website\",\"url\":\"https:\/\/statorials.org\/pl\/\",\"name\":\"Statorials\",\"description\":\"Tw\u00f3j przewodnik po kompetencjach statystycznych!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/statorials.org\/pl\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pl-PL\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\",\"name\":\"Benjamin Anderson\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pl-PL\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"contentUrl\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"caption\":\"Benjamin Anderson\"},\"description\":\"Cze\u015b\u0107, jestem Benjamin i jestem emerytowanym profesorem statystyki, kt\u00f3ry zosta\u0142 oddanym nauczycielem Statorials. 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