{"id":290,"date":"2023-08-03T00:39:59","date_gmt":"2023-08-03T00:39:59","guid":{"rendered":"https:\/\/statorials.org\/pl\/test-t-ucznia\/"},"modified":"2023-08-03T00:39:59","modified_gmt":"2023-08-03T00:39:59","slug":"test-t-ucznia","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/test-t-ucznia\/","title":{"rendered":"Test t-studenta"},"content":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest test t-Studenta i do czego s\u0142u\u017cy w statystyce. W ten spos\u00f3b dowiesz si\u0119, jak przeprowadzany jest test t-Studenta, jakie s\u0105 r\u00f3\u017cne rodzaje test\u00f3w t-Studenta i jaki jest na nie wz\u00f3r. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-prueba-t-de-student\"><\/span> Co to jest test t-Studenta?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Test t-Studenta<\/strong> , zwany tak\u017ce <strong>testem T<\/strong> lub po prostu <strong>testem t<\/strong> , jest testem statystycznym, w kt\u00f3rym statystyka testowa jest zgodna z <a href=\"https:\/\/statorials.org\/pl\/dystrybucja-uczniow\/\">rozk\u0142adem t-Studenta<\/a> . Dlatego w statystyce test t-Studenta s\u0142u\u017cy do odrzucenia lub przyj\u0119cia hipotezy zerowej testu hipotezy.<\/p>\n<p> W szczeg\u00f3lno\u015bci test t-Studenta jest u\u017cywany do <a href=\"https:\/\/statorials.org\/pl\/testowanie-hipotez-1\/\">testowania hipotez<\/a> , w kt\u00f3rym badana populacja ma rozk\u0142ad normalny, ale wielko\u015b\u0107 pr\u00f3by jest zbyt ma\u0142a, aby pozna\u0107 wariancj\u0119 populacji.<\/p>\n<p> Kr\u00f3tko m\u00f3wi\u0105c, test t-Studenta s\u0142u\u017cy do odrzucenia lub przyj\u0119cia hipotezy badawczej niekt\u00f3rych test\u00f3w hipotez. Na przyk\u0142ad test t-Studenta s\u0142u\u017cy do testowania hipotez dla jednej pr\u00f3by, dla pr\u00f3b niezale\u017cnych lub dla pr\u00f3b pokrewnych. Nast\u0119pnie zobaczymy, jak w ka\u017cdym przypadku obliczany jest test t-Studenta. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"tipos-de-pruebas-t-de-student\"><\/span>Rodzaje test\u00f3w t-Studenta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Istniej\u0105 trzy <strong>rodzaje test\u00f3w t-Studenta<\/strong> :<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Test t-Studenta dla jednej pr\u00f3by<\/strong> \u2013 s\u0142u\u017cy do sprawdzenia hipotezy o warto\u015bci \u015bredniej z pr\u00f3by.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Test t-Studenta dla dw\u00f3ch niezale\u017cnych pr\u00f3b<\/strong> : pozwala sprawdzi\u0107 hipotez\u0119 o r\u00f3\u017cnicy \u015brednich z dw\u00f3ch niezale\u017cnych pr\u00f3bek.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Test t-Studenta dla dw\u00f3ch sparowanych pr\u00f3bek (lub pr\u00f3bek pokrewnych)<\/strong> \u2013 s\u0142u\u017cy do sprawdzenia hipotezy o \u015bredniej z pr\u00f3bki badanej dwukrotnie.<\/span> <\/li>\n<\/ul>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"prueba-t-de-student-para-una-muestra\"><\/span> Przyk\u0142ad testu t-Studenta<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Testy hipotez dla \u015bredniej pr\u00f3by to takie, w kt\u00f3rych hipoteza zerowa i hipoteza alternatywna testu m\u00f3wi\u0105 co\u015b o warto\u015bci \u015bredniej populacji.<\/p>\n<p> <strong>Wz\u00f3r na test t-Studenta dla jednej pr\u00f3by<\/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-f8277d83b2d28325f25f5d118486200f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\overline{x}-\\mu}{\\displaystyle \\frac{s}{\\sqrt{n}}} \" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"75\" style=\"vertical-align: -34px;\"><\/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 testu hipotezy dla \u015bredniej, kt\u00f3ra jest zdefiniowana przez rozk\u0142ad 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-a39858a792fb4fe9a3173e004701f2a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> to przyk\u0142adowe \u015brodki.<\/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-05d9eae892416bd34247a25207f8b718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest warto\u015bci\u0105 \u015bredniej zaproponowanej w te\u015bcie hipotezy.<\/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-1edc883862ceed1a21913f60358e31d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest odchyleniem standardowym 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-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<\/ul>\n<p> Po obliczeniu warto\u015bci testu t-Studenta wynik testu statystycznego z warto\u015bci\u0105 krytyczn\u0105 nale\u017cy zinterpretowa\u0107 tak, aby odrzuci\u0107 lub nie hipotez\u0119 zerow\u0105:<\/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 \u015bredniej jest dwustronny, hipotez\u0119 zerow\u0105 odrzuca si\u0119, je\u015bli warto\u015b\u0107 bezwzgl\u0119dna testu t-Studenta jest wi\u0119ksza od warto\u015bci krytycznej t <sub>\u03b1\/2|n-1<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Je\u017celi test hipotezy dla \u015bredniej odpowiada prawemu ogonowi, hipoteza zerowa zostaje odrzucona, je\u015bli warto\u015b\u0107 testu t-Studenta jest wi\u0119ksza ni\u017c warto\u015b\u0107 krytyczna t <sub>\u03b1|n-1<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Je\u015bli test hipotezy dla \u015bredniej odpowiada lewemu ogonowi, hipoteza zerowa zostaje odrzucona, je\u015bli warto\u015b\u0107 testu t-Studenta jest mniejsza ni\u017c warto\u015b\u0107 krytyczna -t <sub>\u03b1|n-1<\/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-31fb206b75a47181c7c673f54ba28ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}H_1: \\mu\\neq \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } |t|>t_{\\alpha\/2|n-1} \\text{ se rechaza } H_0\\\\[3ex]H_1: \\mu> \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } t>t_{\\alpha|n-1} \\text{ se rechaza } H_0\\\\[3ex]H_1: \\mu< \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } t<-t_{\\alpha|n-1} \\text{ se rechaza } H_0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"112\" width=\"457\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Nale\u017cy pami\u0119ta\u0107, \u017ce krytyczne warto\u015bci testowe uzyskuje si\u0119 z tabeli rozk\u0142adu Studenta. <\/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\/\">Konkretny przyk\u0142ad testowania hipotezy dla \u015bredniej<\/a> <\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"prueba-t-de-student-para-muestras-independientes\"><\/span> Test t-Studenta dla pr\u00f3b niezale\u017cnych<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Test t-Studenta dla pr\u00f3b niezale\u017cnych s\u0142u\u017cy do odrzucenia lub przyj\u0119cia hipotezy o zwi\u0105zku pomi\u0119dzy \u015brednimi dw\u00f3ch populacji, na przyk\u0142ad, \u017ce \u015brednie z dw\u00f3ch populacji s\u0105 r\u00f3\u017cne lub \u017ce \u015brednia populacji A jest wi\u0119ksza od \u015bredniej . populacja B.<\/p>\n<p> Jednak\u017ce w tym przypadku wz\u00f3r na test t-Studenta zmienia si\u0119 w zale\u017cno\u015bci od tego, czy mo\u017cna za\u0142o\u017cy\u0107, \u017ce wariancje populacji s\u0105 r\u00f3wne, czy nie. Zobaczymy w\u00f3wczas dwa mo\u017cliwe przypadki.<\/p>\n<h4 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><\/h4>\n<p> <strong>Wz\u00f3r na obliczenie testu t-Studenta dla pr\u00f3b niezale\u017cnych, gdy wariancje populacji s\u0105 nieznane, ale zak\u0142ada si\u0119, \u017ce s\u0105 r\u00f3wne,<\/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-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 zbiorczym 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 przy u\u017cyciu 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<h4 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><\/h4>\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 testu t-Studenta dla pr\u00f3b niezale\u017cnych<\/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\/test-hipotezy-na-roznice-srednich\/\">Konkretny przyk\u0142ad testowania hipotez pod k\u0105tem r\u00f3\u017cnicy \u015brednich<\/a> <\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"prueba-t-de-student-para-muestras-pareadas-o-relacionadas\"><\/span> Test t-Studenta dla pr\u00f3bek sparowanych lub powi\u0105zanych<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Test ten stosuje si\u0119, gdy dwie badane pr\u00f3bki s\u0105 ze sob\u0105 powi\u0105zane, tak \u017ce w rzeczywisto\u015bci jest to pojedyncza pr\u00f3bka osobnik\u00f3w, kt\u00f3ra zosta\u0142a poddana analizie dwukrotnie (za ka\u017cdym razem w innych warunkach).<\/p>\n<p> Mo\u017cna na przyk\u0142ad analizowa\u0107 oceny uczni\u00f3w z zaj\u0119\u0107 z matematyki i statystyki, aby sprawdzi\u0107, czy istnieje znacz\u0105ca r\u00f3\u017cnica mi\u0119dzy \u015brednimi z obu przedmiot\u00f3w. W tym przypadku ocena z matematyki ka\u017cdego ucznia jest powi\u0105zana z ocen\u0105 ze statystyki tego samego ucznia.<\/p>\n<p> <strong>Wz\u00f3r testu t-Studenta dla pr\u00f3bek sparowanych lub powi\u0105zanych<\/strong> 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-8dd7c22ea301c71dac72ee39ce2e1972_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\overline{x_D}-\\mu_D}{\\displaystyle \\frac{s_D}{\\sqrt{n}}} \" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"100\" style=\"vertical-align: -34px;\"><\/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 dla sparowanych \u015brednich, kt\u00f3ra jest zdefiniowana przez rozk\u0142ad 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-aa774a67ebe93af6717679d0fca9e8b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_D}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki utworzon\u0105 na podstawie r\u00f3\u017cnicy danych.<\/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-2df477ea5937645e4858aa3fdc9d699a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_D\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest warto\u015bci\u0105 \u015bredniej zaproponowanej w te\u015bcie hipotezy.<\/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-67e59a42de9fb7b544e3bbd8030feba5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s_D\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest odchyleniem standardowym pr\u00f3bki utworzonym przez r\u00f3\u017cnic\u0119 danych.<\/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<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"asunciones-de-la-prueba-t-de-student\"><\/span> Za\u0142o\u017cenia testu t-Studenta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Aby wykona\u0107 test t-Studenta musz\u0105 zosta\u0107 spe\u0142nione nast\u0119puj\u0105ce warunki:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Ci\u0105g\u0142o\u015b\u0107<\/strong> \u2013 przyk\u0142adowe dane s\u0105 ci\u0105g\u0142e.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Losowo\u015b\u0107<\/strong> : Pr\u00f3bki danych zosta\u0142y wybrane losowo.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Homogeniczno\u015b\u0107<\/strong> : wariancja pr\u00f3bki danych jest jednorodna.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Normalno\u015b\u0107<\/strong> \u2013 rozk\u0142ad definiuj\u0105cy pr\u00f3bk\u0119 danych jest w przybli\u017ceniu normalny.<\/span> <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hacer-una-prueba-t-de-student\"><\/span> Jak wykona\u0107 test t-Studenta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na koniec, podsumowuj\u0105c, szczeg\u00f3\u0142owo opisano kroki, kt\u00f3re nale\u017cy wykona\u0107, aby przeprowadzi\u0107 test t-Studenta.<\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">Zdefiniuj <a href=\"https:\/\/statorials.org\/pl\/hipoteza-zerowa-i-alternatywna\/\">hipotez\u0119 zerow\u0105 i alternatywn\u0105<\/a> podczas testowania hipotez.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">Ustal <a href=\"https:\/\/statorials.org\/pl\/poziom-istotnosci\/\">poziom istotno\u015bci (\u03b1)<\/a> testu hipotezy.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">Sprawd\u017a, czy spe\u0142nione s\u0105 za\u0142o\u017cenia testu t-Studenta.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">Zastosuj odpowiedni wz\u00f3r na test t-Studenta i oblicz statystyk\u0119 testow\u0105.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">Zinterpretuj wynik testu t-Studenta, por\u00f3wnuj\u0105c go z warto\u015bci\u0105 krytyczn\u0105 testu.<\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest test t-Studenta i do czego s\u0142u\u017cy w statystyce. W ten spos\u00f3b dowiesz si\u0119, jak przeprowadzany jest test t-Studenta, jakie s\u0105 r\u00f3\u017cne rodzaje test\u00f3w t-Studenta i jaki jest na nie wz\u00f3r. Co to jest test t-Studenta? Test t-Studenta , zwany tak\u017ce testem T lub po prostu testem t , jest [&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-290","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 t-Studenta<\/title>\n<meta name=\"description\" content=\"Tutaj znajdziesz czym jest test t-Studenta, jak jest wykonywany, jakie s\u0105 rodzaje test\u00f3w t-Studenta (wzory) oraz rozwi\u0105zane \u0107wiczenia.\" \/>\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-t-ucznia\/\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Test t-Studenta\" \/>\n<meta property=\"og:description\" content=\"Tutaj znajdziesz czym jest test t-Studenta, jak jest wykonywany, jakie s\u0105 rodzaje test\u00f3w t-Studenta (wzory) oraz rozwi\u0105zane \u0107wiczenia.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pl\/test-t-ucznia\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-03T00:39:59+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f8277d83b2d28325f25f5d118486200f_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=\"5 minut\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pl\/test-t-ucznia\/\",\"url\":\"https:\/\/statorials.org\/pl\/test-t-ucznia\/\",\"name\":\"\u25b7 Test t-Studenta\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pl\/#website\"},\"datePublished\":\"2023-08-03T00:39:59+00:00\",\"dateModified\":\"2023-08-03T00:39:59+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\"},\"description\":\"Tutaj znajdziesz czym jest test t-Studenta, jak jest wykonywany, jakie s\u0105 rodzaje test\u00f3w t-Studenta (wzory) oraz rozwi\u0105zane \u0107wiczenia.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pl\/test-t-ucznia\/#breadcrumb\"},\"inLanguage\":\"pl-PL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pl\/test-t-ucznia\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pl\/test-t-ucznia\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Dom\",\"item\":\"https:\/\/statorials.org\/pl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Test t-studenta\"}]},{\"@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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