{"id":282,"date":"2023-08-03T03:11:14","date_gmt":"2023-08-03T03:11:14","guid":{"rendered":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/"},"modified":"2023-08-03T03:11:14","modified_gmt":"2023-08-03T03:11:14","slug":"test-hipotezy-na-roznice-srednich","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/","title":{"rendered":"Testowanie hipotez pod k\u0105tem r\u00f3\u017cnicy \u015brednich"},"content":{"rendered":"<p>W artykule wyja\u015bniono, na czym polega testowanie hipotez r\u00f3\u017cnicy \u015brednich w statystyce i do czego s\u0142u\u017cy. W ten sam spos\u00f3b dowiesz si\u0119, jak przeprowadzi\u0107 test hipotezy na temat r\u00f3\u017cnicy \u015brednich, oraz dowiesz si\u0119, jak wykona\u0107 \u0107wiczenie rozwi\u0105zane krok po kroku. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-prueba-de-hipotesis-para-la-diferencia-de-medias\"><\/span> Na czym polega testowanie hipotez pod k\u0105tem \u015bredniej r\u00f3\u017cnicy?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Testowanie hipotez pod k\u0105tem r\u00f3\u017cnicy \u015brednich<\/strong> to test statystyczny u\u017cywany do odrzucenia lub przyj\u0119cia hipotezy, \u017ce \u015brednie w dw\u00f3ch populacjach s\u0105 r\u00f3\u017cne. Oznacza to, \u017ce test hipotezy r\u00f3\u017cnicy \u015brednich s\u0142u\u017cy do ustalenia, czy \u015brednie dw\u00f3ch populacji s\u0105 takie same, czy r\u00f3\u017cne.<\/p>\n<p> Nale\u017cy pami\u0119ta\u0107, \u017ce decyzje podejmowane podczas testowania hipotez opieraj\u0105 si\u0119 na wcze\u015bniej ustalonym <a href=\"https:\/\/statorials.org\/pl\/poziom-pewnosci\/\">poziomie ufno\u015bci<\/a> , dlatego nie mo\u017cna zagwarantowa\u0107, \u017ce wynik testu hipotezy b\u0119dzie zawsze prawid\u0142owy, ale raczej, \u017ce jest to najbardziej prawdopodobny wynik, kt\u00f3ry jest prawdziwy.<\/p>\n<p> Testowanie hipotez pod k\u0105tem r\u00f3\u017cnicy dw\u00f3ch \u015brednich polega na obliczeniu statystyki testowej i por\u00f3wnaniu jej z warto\u015bci\u0105 krytyczn\u0105, aby odrzuci\u0107 hipotez\u0119 zerow\u0105 lub nie. Poni\u017cej zobaczymy, jak przeprowadzi\u0107 test hipotezy dla r\u00f3\u017cnicy \u015brednich.<\/p>\n<p> Na koniec pami\u0119taj, \u017ce w statystyce testowanie hipotez mo\u017cna r\u00f3wnie\u017c nazwa\u0107 kontrastami hipotez, testowaniem hipotez lub testowaniem istotno\u015bci. <\/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\/rozk\u0142ad-probkowania-roznicy-srednich\/\">Pr\u00f3bkowanie rozk\u0142adu r\u00f3\u017cnicy \u015brednich<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-prueba-de-hipotesis-para-la-diferencia-de-medias\"><\/span> Testowanie hipotez Wz\u00f3r na r\u00f3\u017cnic\u0119 \u015brednich<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Wz\u00f3r, kt\u00f3rego nale\u017cy u\u017cywa\u0107 do testowania hipotez dotycz\u0105cych r\u00f3\u017cnicy \u015brednich, r\u00f3\u017cni si\u0119 w zale\u017cno\u015bci od tego, czy znane s\u0105 wariancje populacji, a je\u015bli nie, to czy mo\u017cna za\u0142o\u017cy\u0107, \u017ce s\u0105 takie same lub r\u00f3\u017cne. Zatem w tej sekcji zobaczymy, jakiej formu\u0142y u\u017cy\u0107 w zale\u017cno\u015bci od przypadku.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"varianzas-conocidas\"><\/span> Znane odmiany<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> <strong>Wz\u00f3r na obliczenie statystyki testu hipotezy dla r\u00f3\u017cnicy \u015brednich, gdy znane s\u0105 wariancje,<\/strong> jest 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-0b7968d96046c1f555a7c95e7c3c408e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle Z=\\frac{\\displaystyle (\\overline{x_1}-\\overline{x_2})-(\\mu_1-\\mu_2)}{\\displaystyle\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"78\" width=\"203\" 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 testuj\u0105c\u0105 hipotez\u0119 dotycz\u0105c\u0105 r\u00f3\u017cnicy dw\u00f3ch \u015brednich o znanej wariancji, kt\u00f3ra jest zgodna ze standardowym rozk\u0142adem normalnym.<\/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-776e8d37515f607d16b6ea33cf27cb8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 populacji 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-eed6e51baac07f2aaf1c82b4b328ded8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 populacji 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-1436bb55d331d1efe559fd1e55245851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki 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-f06b28a75570ecb4df1371dea2d9f2b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki 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-802bfc799b8f222d7b47270e1569770b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest odchyleniem standardowym populacji 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-063ad2fadbc81bd1df35589566b0bd16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest odchyleniem standardowym populacji 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<\/ul>\n<p> Nale\u017cy pami\u0119ta\u0107, \u017ce jest to najrzadziej spotykany przypadek, wi\u0119c tej formu\u0142y u\u017cywa si\u0119 tylko w niekt\u00f3rych okre\u015blonych przypadkach.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"varianzas-desconocidas-e-iguales\"><\/span> Nieznane i r\u00f3wne odchylenia<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> <strong>Wz\u00f3r na obliczenie statystyki testu hipotezy dla r\u00f3\u017cnicy \u015brednich, gdy wariancje populacji s\u0105 nieznane, ale zak\u0142ada si\u0119, \u017ce s\u0105 r\u00f3wne<\/strong> :<\/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-b879974c3d05ebcb049125dbaf9f0a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\displaystyle (\\overline{x_1}-\\overline{x_2})-(\\mu_1-\\mu_2)}{\\displaystyle s_p\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"197\" 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-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest statystyk\u0105 testuj\u0105c\u0105 hipotez\u0119 dotycz\u0105c\u0105 r\u00f3\u017cnicy \u015brednich o nieznanych wariancjach, kt\u00f3ra jest zgodna z rozk\u0142adem t-Studenta z n <sub>1<\/sub> + n <sub>2<\/sub> -2 stopniami swobody.<\/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-776e8d37515f607d16b6ea33cf27cb8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 populacji 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-eed6e51baac07f2aaf1c82b4b328ded8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 populacji 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-1436bb55d331d1efe559fd1e55245851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki 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-f06b28a75570ecb4df1371dea2d9f2b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki 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-db24109a40d5779feee64c7bccc11371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s_p\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"15\" style=\"vertical-align: -6px;\"><\/p>\n<p> jest po\u0142\u0105czonym odchyleniem standardowym.<\/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<\/ul>\n<p> \u0141\u0105czne odchylenie standardowe dw\u00f3ch pr\u00f3bek oblicza si\u0119 za pomoc\u0105 nast\u0119puj\u0105cego wzoru: <\/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-a8fb43d179e453d19f5781319aa9e1ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle s_p=\\sqrt{\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"240\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"varianzas-desconocidas-y-diferentes\"><\/span> Nieznane i r\u00f3\u017cne odmiany<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> <strong>Je\u017celi wariancje populacji nie s\u0105 znane, a ponadto zak\u0142ada si\u0119, \u017ce s\u0105 r\u00f3\u017cne, wz\u00f3r na obliczenie statystyki testu hipotezy dla r\u00f3\u017cnicy \u015brednich<\/strong> jest 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-01c79415ff9c173fa4517d4b1a859a8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\displaystyle (\\overline{x_1}-\\overline{x_2})-(\\mu_1-\\mu_2)}{\\displaystyle \\sqrt{\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"78\" width=\"197\" 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-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest statystyk\u0105 testuj\u0105c\u0105 hipotez\u0119 dotycz\u0105c\u0105 r\u00f3\u017cnicy \u015brednich o nieznanych wariancjach, kt\u00f3ra jest zgodna z rozk\u0142adem t-Studenta.<\/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-776e8d37515f607d16b6ea33cf27cb8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 populacji 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-eed6e51baac07f2aaf1c82b4b328ded8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 populacji 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-1436bb55d331d1efe559fd1e55245851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki 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-f06b28a75570ecb4df1371dea2d9f2b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki 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-802bfc799b8f222d7b47270e1569770b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest odchyleniem standardowym populacji 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-063ad2fadbc81bd1df35589566b0bd16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest odchyleniem standardowym populacji 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<\/ul>\n<p> Jednak\u017ce w tym przypadku stopnie swobody rozk\u0142adu t-Studenta obliczane s\u0105 ze wzoru: <\/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-ecf1ccfa1ecb47b889c569e5faa65a78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle GL=\\frac{\\displaystyle\\left(\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}\\right)^2}{\\displaystyle\\frac{\\displaystyle\\frac{s_1^2}{n_1}}{n_1-1}+\\frac{\\displaystyle\\frac{s_2^2}{n_2}}{n_2-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"115\" width=\"180\" style=\"vertical-align: -61px;\"><\/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\/przedzia\u0142-ufnosci-dla-roznicy-srednich\/\">Wz\u00f3r na przedzia\u0142 ufno\u015bci dla r\u00f3\u017cnicy \u015brednich<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-resuelto-de-la-prueba-de-hipotesis-para-la-diferencia-de-medias\"><\/span> Konkretny przyk\u0142ad testowania hipotezy pod k\u0105tem r\u00f3\u017cnicy \u015brednich<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Aby zako\u0144czy\u0107 przyswajanie koncepcji testowania hipotez na podstawie r\u00f3\u017cnicy \u015brednich, zobaczymy konkretny przyk\u0142ad tego typu testowania hipotez.<\/p>\n<ul>\n<li> Chcesz przeprowadzi\u0107 badanie statystyczne dotycz\u0105ce wynagrodze\u0144 w dw\u00f3ch konkurencyjnych firmach, a dok\u0142adniej chcesz ustali\u0107, czy \u015brednie wynagrodzenie w obu firmach jest r\u00f3\u017cne. W tym celu pobiera si\u0119 pr\u00f3b\u0119 47 pracownik\u00f3w z jednego przedsi\u0119biorstwa i drug\u0105 pr\u00f3b\u0119 55 pracownik\u00f3w z drugiego przedsi\u0119biorstwa. Z pierwszej pr\u00f3by uzyskano \u015brednie wynagrodzenie w wysoko\u015bci 40 000 USD i odchylenie standardowe w wysoko\u015bci 12 000 USD, natomiast w drugiej pr\u00f3bie uzyskano \u015brednie wynagrodzenie w wysoko\u015bci 46 000 USD i odchylenie standardowe w wysoko\u015bci 18 000 USD. Wykonaj test hipotezy z poziomem istotno\u015bci 5%, aby okre\u015bli\u0107, czy \u015brednie wynagrodzenia r\u00f3\u017cni\u0105 si\u0119, czy nie.<\/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 r\u00f3\u017cnicy dw\u00f3ch \u015brednich s\u0105 nast\u0119puj\u0105ce:<\/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-4dd9e80c0a87987d689d7b8d99be9d90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases}H_0: \\mu_1-\\mu_2=0\\\\[2ex] H_1:\\mu_1-\\mu_2\\neq 0 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"141\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> W tym przypadku luki populacyjne nie s\u0105 znane, ale mo\u017cna za\u0142o\u017cy\u0107, \u017ce s\u0105 r\u00f3wne, gdy\u017c s\u0105 to firmy konkurencyjne, a warunki pracy na rynku, na kt\u00f3rym dzia\u0142aj\u0105, s\u0105 bardzo podobne. Zatem wz\u00f3r na statystyk\u0119 testuj\u0105c\u0105 hipotez\u0119 dotycz\u0105c\u0105 r\u00f3\u017cnicy \u015brednich, kt\u00f3r\u0105 powinni\u015bmy zastosowa\u0107, wygl\u0105da nast\u0119puj\u0105co:<\/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-b879974c3d05ebcb049125dbaf9f0a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\displaystyle (\\overline{x_1}-\\overline{x_2})-(\\mu_1-\\mu_2)}{\\displaystyle s_p\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"197\" style=\"vertical-align: -41px;\"><\/p>\n<\/p>\n<p> Dlatego obliczamy zbiorcze odchylenie standardowe dw\u00f3ch pr\u00f3bek:<\/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-0308363d68c4677761d44b214b1609d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\displaystyle s_p&amp;=\\sqrt{\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}\\\\[2ex]\\displaystyle s_p&amp;=\\sqrt{\\frac{(47-1)\\cdot 12000^2+(55-1)\\cdot 18000^2}{47+55-2}}\\\\[2ex]s_p&amp;=15530,61\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"161\" width=\"337\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Zastosujemy teraz wz\u00f3r na testowanie hipotezy dla r\u00f3\u017cnicy \u015brednich:<\/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-c5bf7791d156d1d7593182a7ffc64417_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\displaystyle (\\overline{x_1}-\\overline{x_2})-(\\mu_1-\\mu_2)}{\\displaystyle s_p\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}=\\cfrac{(40000-46000)-0}{\\displaystyle 15530,61\\sqrt{\\frac{1}{47}+\\frac{1}{55}}}=-1,94\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"451\" style=\"vertical-align: -41px;\"><\/p>\n<\/p>\n<p> Natomiast warto\u015bci krytycznej testu hipotezy dla r\u00f3\u017cnicy \u015brednich szukamy w <a href=\"https:\/\/statorials.org\/pl\/tabela-rozk\u0142adu-t-ucznia\/\">tabeli t-Studenta<\/a> :<\/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-8154fa6e5bfee35698c6dc26928cd98c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha=0,05 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black}\\ \\alpha\/2=0,025\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"300\" style=\"vertical-align: -5px;\"><\/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-d5f11ecd0336e50c63b0ab2fab4bd530_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}t_{\\alpha\/2| n_1+n_2-2}= \\ \\color{orange}\\bm{?}\\\\[4ex]t_{0,025| 100}=1,984\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"72\" width=\"184\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Nast\u0119pnie, poniewa\u017c warto\u015b\u0107 bezwzgl\u0119dna statystyki testowej jest mniejsza ni\u017c krytyczna warto\u015b\u0107 testowa, przyjmuje si\u0119 hipotez\u0119 zerow\u0105, a hipotez\u0119 alternatywn\u0105 odrzuca si\u0119.<\/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-ca4a2499b93203ce31a36a1cbacbcc8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|-1,94|=1,94<1,984 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Se rechaza } H_1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"432\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>W artykule wyja\u015bniono, na czym polega testowanie hipotez r\u00f3\u017cnicy \u015brednich w statystyce i do czego s\u0142u\u017cy. W ten sam spos\u00f3b dowiesz si\u0119, jak przeprowadzi\u0107 test hipotezy na temat r\u00f3\u017cnicy \u015brednich, oraz dowiesz si\u0119, jak wykona\u0107 \u0107wiczenie rozwi\u0105zane krok po kroku. Na czym polega testowanie hipotez pod k\u0105tem \u015bredniej r\u00f3\u017cnicy? Testowanie hipotez pod k\u0105tem r\u00f3\u017cnicy \u015brednich [&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-282","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 Test hipotez dla r\u00f3\u017cnicy \u015brednich<\/title>\n<meta name=\"description\" content=\"Tutaj dowiesz si\u0119 na czym polega testowanie hipotezy na r\u00f3\u017cnic\u0119 \u015brednich, jak si\u0119 je przeprowadza (wz\u00f3r) oraz przyk\u0142ad rozwi\u0105zany krok po kroku.\" \/>\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-na-roznice-srednich\/\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Test hipotez dla r\u00f3\u017cnicy \u015brednich\" \/>\n<meta property=\"og:description\" content=\"Tutaj dowiesz si\u0119 na czym polega testowanie hipotezy na r\u00f3\u017cnic\u0119 \u015brednich, jak si\u0119 je przeprowadza (wz\u00f3r) oraz przyk\u0142ad rozwi\u0105zany krok po kroku.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-03T03:11:14+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0b7968d96046c1f555a7c95e7c3c408e_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-na-roznice-srednich\/\",\"url\":\"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/\",\"name\":\"\u25b7 Test hipotez dla r\u00f3\u017cnicy \u015brednich\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pl\/#website\"},\"datePublished\":\"2023-08-03T03:11:14+00:00\",\"dateModified\":\"2023-08-03T03:11:14+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\"},\"description\":\"Tutaj dowiesz si\u0119 na czym polega testowanie hipotezy na r\u00f3\u017cnic\u0119 \u015brednich, jak si\u0119 je przeprowadza (wz\u00f3r) oraz przyk\u0142ad rozwi\u0105zany krok po kroku.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/#breadcrumb\"},\"inLanguage\":\"pl-PL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Dom\",\"item\":\"https:\/\/statorials.org\/pl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Testowanie hipotez pod k\u0105tem r\u00f3\u017cnicy \u015brednich\"}]},{\"@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. Dzi\u0119ki bogatemu do\u015bwiadczeniu i wiedzy specjalistycznej w dziedzinie statystyki ch\u0119tnie dziel\u0119 si\u0119 swoj\u0105 wiedz\u0105, aby wzmocni\u0107 pozycj\u0119 uczni\u00f3w za po\u015brednictwem Statorials. Wiedzie\u0107 wi\u0119cej\",\"sameAs\":[\"https:\/\/statorials.org\/pl\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u25b7 Test hipotez dla r\u00f3\u017cnicy \u015brednich","description":"Tutaj dowiesz si\u0119 na czym polega testowanie hipotezy na r\u00f3\u017cnic\u0119 \u015brednich, jak si\u0119 je przeprowadza (wz\u00f3r) oraz przyk\u0142ad rozwi\u0105zany krok po kroku.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/","og_locale":"pl_PL","og_type":"article","og_title":"\u25b7 Test hipotez dla r\u00f3\u017cnicy \u015brednich","og_description":"Tutaj dowiesz si\u0119 na czym polega testowanie hipotezy na r\u00f3\u017cnic\u0119 \u015brednich, jak si\u0119 je przeprowadza (wz\u00f3r) oraz przyk\u0142ad rozwi\u0105zany krok po kroku.","og_url":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/","og_site_name":"Statorials","article_published_time":"2023-08-03T03:11:14+00:00","og_image":[{"url":"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0b7968d96046c1f555a7c95e7c3c408e_l3.png"}],"author":"Benjamin Anderson","twitter_card":"summary_large_image","twitter_misc":{"Napisane przez":"Benjamin Anderson","Szacowany czas czytania":"4 minuty"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/","url":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/","name":"\u25b7 Test hipotez dla r\u00f3\u017cnicy \u015brednich","isPartOf":{"@id":"https:\/\/statorials.org\/pl\/#website"},"datePublished":"2023-08-03T03:11:14+00:00","dateModified":"2023-08-03T03:11:14+00:00","author":{"@id":"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965"},"description":"Tutaj dowiesz si\u0119 na czym polega testowanie hipotezy na r\u00f3\u017cnic\u0119 \u015brednich, jak si\u0119 je przeprowadza (wz\u00f3r) oraz przyk\u0142ad rozwi\u0105zany krok po kroku.","breadcrumb":{"@id":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/#breadcrumb"},"inLanguage":"pl-PL","potentialAction":[{"@type":"ReadAction","target":["https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/statorials.org\/pl\/test-hipotezy-na-roznice-srednich\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Dom","item":"https:\/\/statorials.org\/pl\/"},{"@type":"ListItem","position":2,"name":"Testowanie hipotez pod k\u0105tem r\u00f3\u017cnicy \u015brednich"}]},{"@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. Dzi\u0119ki bogatemu do\u015bwiadczeniu i wiedzy specjalistycznej w dziedzinie statystyki ch\u0119tnie dziel\u0119 si\u0119 swoj\u0105 wiedz\u0105, aby wzmocni\u0107 pozycj\u0119 uczni\u00f3w za po\u015brednictwem Statorials. Wiedzie\u0107 wi\u0119cej","sameAs":["https:\/\/statorials.org\/pl"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/posts\/282","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/comments?post=282"}],"version-history":[{"count":0,"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/posts\/282\/revisions"}],"wp:attachment":[{"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/media?parent=282"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/categories?post=282"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statorials.org\/pl\/wp-json\/wp\/v2\/tags?post=282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}