{"id":285,"date":"2023-08-03T02:21:11","date_gmt":"2023-08-03T02:21:11","guid":{"rendered":"https:\/\/statorials.org\/pl\/rozklad-probkowania-1\/"},"modified":"2023-08-03T02:21:11","modified_gmt":"2023-08-03T02:21:11","slug":"rozklad-probkowania-1","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/rozklad-probkowania-1\/","title":{"rendered":"Dystrybucja pr\u00f3bek"},"content":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest rozk\u0142ad pr\u00f3bkowania w statystyce i do czego si\u0119 go u\u017cywa. W ten spos\u00f3b znajdziesz znaczenie rozk\u0142adu pr\u00f3bkowania, konkretny przyk\u0142ad rozk\u0142adu pr\u00f3bkowania, a ponadto wzory na najcz\u0119stsze typy rozk\u0142ad\u00f3w pr\u00f3bkowania. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-distribucion-muestral\"><\/span> Jaki jest rozk\u0142ad pr\u00f3bkowania?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Rozk\u0142ad pr\u00f3bkowania<\/strong> lub <strong>rozk\u0142ad pr\u00f3bkowania<\/strong> to rozk\u0142ad wynikaj\u0105cy z uwzgl\u0119dnienia wszystkich mo\u017cliwych pr\u00f3bek z populacji. Innymi s\u0142owy, rozk\u0142ad pr\u00f3bkowania to rozk\u0142ad uzyskany poprzez obliczenie parametru pr\u00f3bkowania wszystkich mo\u017cliwych pr\u00f3bek z populacji.<\/p>\n<p> Na przyk\u0142ad, je\u015bli wyodr\u0119bnimy wszystkie mo\u017cliwe pr\u00f3bki z populacji statystycznej i obliczymy \u015bredni\u0105 ka\u017cdej pr\u00f3bki, zbi\u00f3r \u015brednich z pr\u00f3by tworzy rozk\u0142ad pr\u00f3bkowania. Dok\u0142adniej, poniewa\u017c obliczony parametr jest \u015bredni\u0105 arytmetyczn\u0105, jest to rozk\u0142ad \u015bredniej z pr\u00f3bkowania.<\/p>\n<p> W statystyce rozk\u0142ad pr\u00f3bkowania s\u0142u\u017cy do obliczenia prawdopodobie\u0144stwa zbli\u017cenia si\u0119 do warto\u015bci parametru populacji podczas badania pojedynczej pr\u00f3by. Podobnie rozk\u0142ad pr\u00f3bkowania pozwala oszacowa\u0107 b\u0142\u0105d pr\u00f3bkowania dla danej liczebno\u015bci pr\u00f3by. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-la-distribucion-muestral\"><\/span> Przyk\u0142ad rozk\u0142adu pr\u00f3bkowania<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Teraz, gdy znamy definicj\u0119 rozk\u0142adu pr\u00f3bkowania, sp\u00f3jrzmy na prosty przyk\u0142ad, aby w pe\u0142ni zrozumie\u0107 t\u0119 koncepcj\u0119.<\/p>\n<ul>\n<li> Do pude\u0142ka wk\u0142adamy trzy kule i na ka\u017cdej z nich zapisana jest liczba od 1 do 3, tak aby jedna kula mia\u0142a numer 1, druga kula mia\u0142a numer 2, a ostatnia kula mia\u0142a numer 3. Dla pr\u00f3bki o rozmiarze n = 2, oblicza prawdopodobie\u0144stwa rozk\u0142adu \u015bredniej w przypadku wybrania pr\u00f3bek z zast\u0119powaniem.<\/li>\n<\/ul>\n<p> Pr\u00f3bki wybierane s\u0105 z wymian\u0105, czyli kula zebrana w celu wybrania pierwszego elementu pr\u00f3bki wraca do pude\u0142ka i mo\u017ce zosta\u0107 ponownie wybrana podczas drugiej ekstrakcji. Dlatego wszystkie mo\u017cliwe pr\u00f3bki z populacji to:<\/p>\n<p class=\"has-text-align-center\"> 1,1 1,2 1,3<br \/> 2.1 2.2 2.3<br \/> 3.1 3.2 3.3<\/p>\n<p> W ten spos\u00f3b obliczamy \u015bredni\u0105 arytmetyczn\u0105 ka\u017cdej mo\u017cliwej pr\u00f3bki: <\/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-36b3a1e0bbb1be6eddc1a5d9899c5643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,1) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{11}=\\cfrac{1+1}{2}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"295\" style=\"vertical-align: -12px;\"><\/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-ff446066e6102f75d2d5435ad9dc46d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,2) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{12}=\\cfrac{1+2}{2}=1,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"311\" style=\"vertical-align: -12px;\"><\/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-904c1ee161fd7214c2c20ef15a038ea2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,3) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{13}=\\cfrac{1+3}{2}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"295\" style=\"vertical-align: -12px;\"><\/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-4669b3b3d3ca07f035456cc50110134f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2,1) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{21}=\\cfrac{2+1}{2}=1,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"311\" style=\"vertical-align: -12px;\"><\/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-9fae1b39671fce31802b9ff66c8c1b9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2,2) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{22}=\\cfrac{2+2}{2}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"295\" style=\"vertical-align: -12px;\"><\/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-fb036db8580a0d7389daddf6a938c541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2,3) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{23}=\\cfrac{2+3}{2}=2,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"311\" style=\"vertical-align: -12px;\"><\/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-7b77da5430a43732726cc62fb0fffe78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,1) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{31}=\\cfrac{3+1}{2}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"295\" style=\"vertical-align: -12px;\"><\/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-5edbfa4f2b8752676ceba5ad0cd34d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,2) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{32}=\\cfrac{3+2}{2}=2,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"311\" style=\"vertical-align: -12px;\"><\/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-bb60dfff5e3c5c090e019253ee84b198_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,3) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\overline{x}_{33}=\\cfrac{3+3}{2}=3\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"296\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Zatem prawdopodobie\u0144stwa uzyskania ka\u017cdej warto\u015bci \u015bredniej pr\u00f3by przy wyborze pr\u00f3by losowej z populacji s\u0105 nast\u0119puj\u0105ce: <\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/table-de-distribution-dechantillonnage.png\" alt=\"przyk\u0142adowa tabela rozk\u0142adu\" class=\"wp-image-6145\" width=\"166\" height=\"195\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Prawdopodobie\u0144stwa rozk\u0142adu pr\u00f3bkowania pokazane w powy\u017cszej tabeli obliczono dziel\u0105c liczb\u0119 pr\u00f3bek posiadaj\u0105cych wspomnian\u0105 warto\u015b\u0107 \u015bredni\u0105 przez ca\u0142kowit\u0105 liczb\u0119 mo\u017cliwych przypadk\u00f3w. Na przyk\u0142ad: \u015brednia pr\u00f3bki wynosi 1,5 w dw\u00f3ch przypadkach na dziewi\u0119\u0107 mo\u017cliwych, zatem P(1,5)=2\/9. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"tipos-de-distribuciones-muestrales\"><\/span> Rodzaje rozk\u0142ad\u00f3w pr\u00f3bkowania<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Rozk\u0142ady pr\u00f3bkowania (lub rozk\u0142ady pr\u00f3bkowania) mo\u017cna klasyfikowa\u0107 na podstawie parametru pr\u00f3bkowania, z kt\u00f3rego zosta\u0142y uzyskane. Zatem najcz\u0119stsze typy dystrybucji s\u0105 nast\u0119puj\u0105ce:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:18px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Rozk\u0142ad \u015bredniej pr\u00f3by<\/strong> : Jest to rozk\u0142ad pr\u00f3by wynikaj\u0105cy z obliczenia \u015bredniej arytmetycznej ka\u017cdej pr\u00f3bki.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Proporcjonalny rozk\u0142ad pr\u00f3bkowania<\/strong> : Jest to rozk\u0142ad pr\u00f3bkowania uzyskany poprzez obliczenie proporcji wszystkich pr\u00f3bek.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Rozk\u0142ad wariancji pr\u00f3bkowania<\/strong> : jest to rozk\u0142ad pr\u00f3bkowania, kt\u00f3ry tworzy zbi\u00f3r wszystkich wariancji w pr\u00f3bie.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>R\u00f3\u017cnica rozk\u0142adu<\/strong> \u015brednich pr\u00f3b z dw\u00f3ch r\u00f3\u017cnych populacji.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>R\u00f3\u017cnica w proporcjonalnym rozk\u0142adzie pr\u00f3bkowania<\/strong> : jest rozk\u0142adem pr\u00f3bkowania uzyskanym przez odj\u0119cie wszystkich mo\u017cliwych proporcji pr\u00f3bkowania z dw\u00f3ch populacji.<\/span><\/li>\n<\/ul>\n<p> Ka\u017cdy rodzaj rozk\u0142adu pr\u00f3bkowania wyja\u015bniono bardziej szczeg\u00f3\u0142owo poni\u017cej. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"distribucion-muestral-de-la-media\"><\/span> Pr\u00f3bkowanie rozk\u0142adu \u015bredniej<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Bior\u0105c pod uwag\u0119 populacj\u0119, kt\u00f3ra ma normalny rozk\u0142ad prawdopodobie\u0144stwa ze \u015bredni\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-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> i odchylenie standardowe<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-eaaf379fee5e67946f3fedf5631047b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> i pobierane s\u0105 pr\u00f3bki wielko\u015bci<\/p>\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> , rozk\u0142ad \u015bredniej pr\u00f3by b\u0119dzie r\u00f3wnie\u017c okre\u015blony rozk\u0142adem normalnym maj\u0105cym nast\u0119puj\u0105ce cechy:<\/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-44571aa7337b095ab9c9fa1f746e93a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\mu_{\\overline{x}}=\\mu \\qquad \\sigma_{\\overline{x}}=\\cfrac{\\sigma}{\\sqrt{n}}\\\\[4ex]\\displaystyle N_{\\overline{x}}\\left(\\mu, \\frac{\\sigma}{\\sqrt{n}}\\right) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Z\u0142oto<\/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-8ed084decbdfb365889aae767cf63e81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_{\\overline{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"20\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 rozk\u0142adu pr\u00f3by \u015bredniej i<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-9067e28d896c7e5278763081c6cc40d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_{\\overline{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest jego odchyleniem standardowym. Ponadto,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-7d78e2a2f2fae99a53eb087263cbb478_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\sigma}{\\sqrt{n}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"26\" style=\"vertical-align: -16px;\"><\/p>\n<p> jest b\u0142\u0119dem standardowym rozk\u0142adu pr\u00f3bkowania.<\/p>\n<p> <strong>Uwaga:<\/strong> Je\u017celi populacja nie ma rozk\u0142adu normalnego, ale wielko\u015b\u0107 pr\u00f3by jest du\u017ca (n&gt;30), rozk\u0142ad \u015bredniej pr\u00f3by mo\u017cna r\u00f3wnie\u017c przybli\u017cy\u0107 do powy\u017cszego rozk\u0142adu normalnego za pomoc\u0105 granicy centralnego twierdzenia.<\/p>\n<p> Dlatego te\u017c, poniewa\u017c rozk\u0142ad \u015bredniej z pr\u00f3by ma rozk\u0142ad normalny, <strong>wz\u00f3r na obliczenie prawdopodobie\u0144stwa zwi\u0105zanego ze \u015bredni\u0105 z pr\u00f3bki<\/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-ecd8bcb78b739c50d01b8bad563e5cb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{\\overline{x}-\\mu}{\\displaystyle\\frac{\\sigma}{\\sqrt{n}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"81\" 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 style=\"margin-bottom:5px\">\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 style=\"margin-bottom:5px\">\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> To jest \u015brednia populacji.<\/li>\n<li style=\"margin-bottom:5px\">\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 populacji.<\/li>\n<li style=\"margin-bottom:5px\">\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 style=\"margin-bottom:5px\">\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 zmienn\u0105 zdefiniowan\u0105 przez standardowy rozk\u0142ad normalny N(0,1). <\/li>\n<\/ul>\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-sredniej\/\">Rozwi\u0105zane \u0107wiczenie dotycz\u0105ce rozk\u0142adu pr\u00f3bkowania \u015bredniej<\/a> <\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"distribucion-muestral-de-la-proporcion\"><\/span> Pr\u00f3bkowanie rozk\u0142adu proporcji<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Tak naprawd\u0119, badaj\u0105c cz\u0119\u015b\u0107 pr\u00f3by, analizujemy przypadki sukcesu. Dlatego zmienna losowa w badaniu ma dwumianowy rozk\u0142ad prawdopodobie\u0144stwa.<\/p>\n<p> Zgodnie z centralnym twierdzeniem granicznym, dla du\u017cych rozmiar\u00f3w (n&gt;30) mo\u017cemy przybli\u017cy\u0107 rozk\u0142ad dwumianowy do rozk\u0142adu normalnego. Dlatego <strong>rozk\u0142ad pr\u00f3bkowania proporcji jest zbli\u017cony do rozk\u0142adu normalnego z nast\u0119puj\u0105cymi parametrami:<\/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-f3408076893f390bb65baecfe38e6eff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\displaystyle\\mu_{p}=p \\qquad \\sigma_{p}=\\sqrt{\\frac{pq}{n}}\\\\[4ex]\\displaystyle N_{p}\\left(p, \\sqrt{\\frac{pq}{n}}\\right) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"168\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Z\u0142oto<\/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-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 prawdopodobie\u0144stwem sukcesu i<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-420eca7b6df080cc5f01773d1978f44a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest prawdopodobie\u0144stwo niepowodzenia<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e1d214c21abe0d79fa453d635a025865_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q=1-p\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -4px;\"><\/p>\n<p> .<\/p>\n<p> <strong>Uwaga:<\/strong> Rozk\u0142ad dwumianowy mo\u017cna przybli\u017cy\u0107 do rozk\u0142adu normalnego tylko wtedy, gdy:<\/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-88e275a965c091eb810599a07b0f8d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n>30&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;14&#8243; width=&#8221;52&#8243; style=&#8221;vertical-align: -2px;&#8221;><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-41130f51ca4b83f1bf25b9dde90ecbfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"np\\ge 5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"51\" style=\"vertical-align: -4px;\"><\/p>\n<p> I<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0af2ac5b6eb874f65b406b3bc39f0c7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"nq\\ge 5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"51\" style=\"vertical-align: -4px;\"><\/p>\n<p> .<\/p>\n<p> Poniewa\u017c zatem rozk\u0142ad pr\u00f3bkowania tej proporcji mo\u017cna przybli\u017cy\u0107 do rozk\u0142adu normalnego, <strong>wz\u00f3r na obliczenie prawdopodobie\u0144stwa zwi\u0105zanego z proporcj\u0105 pr\u00f3bki<\/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-5b7a4224240587268d0dd7865a33ac31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{\\widehat{p}-p}{\\displaystyle\\sqrt{\\frac{pq}{n}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"78\" 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 style=\"margin-bottom:5px\">\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 style=\"margin-bottom:5px\">\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 odsetek populacji.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-420eca7b6df080cc5f01773d1978f44a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest prawdopodobie\u0144stwem niepowodzenia populacji,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e1d214c21abe0d79fa453d635a025865_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q=1-p\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -4px;\"><\/p>\n<p> .<\/li>\n<li style=\"margin-bottom:5px\">\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 style=\"margin-bottom:5px\">\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 zmienn\u0105 zdefiniowan\u0105 przez standardowy rozk\u0142ad normalny N(0,1). <\/li>\n<\/ul>\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-proporcji-probkowania\/\">Rozwi\u0105zane \u0107wiczenie dotycz\u0105ce rozk\u0142adu pr\u00f3bkowania proporcji<\/a> <\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"distribucion-muestral-de-la-varianza\"><\/span> Pr\u00f3bkowanie Rozk\u0142ad wariancji<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Rozk\u0142ad wariancji pr\u00f3bkowania jest zdefiniowany przez rozk\u0142ad prawdopodobie\u0144stwa chi-kwadrat. Dlatego <strong>wz\u00f3r na statystyk\u0119 rozk\u0142adu wariancji z 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-3917636d4c911eeaad1a005195204d08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\chi^2=\\cfrac{(n-1)s^2}{\\sigma^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"115\" style=\"vertical-align: -12px;\"><\/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 rozk\u0142adu wariancji z pr\u00f3bkowania, kt\u00f3ra jest zgodna z rozk\u0142adem 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-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-3ab572e85f9cb7cb6f495387f2a6ab0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s^2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest wariancj\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-c6d52162ef1ec2e8130fb00687aca707_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma^2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<p> jest wariancj\u0105 populacji. <\/li>\n<\/ul>\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-wariancji-probkowania\/\">Rozwi\u0105zane \u0107wiczenie dotycz\u0105ce rozk\u0142adu pr\u00f3bkowania wariancji<\/a> <\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"distribucion-muestral-de-la-diferencia-de-medias\"><\/span> Pr\u00f3bkowanie rozk\u0142adu r\u00f3\u017cnicy \u015brednich<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Je\u017celi wielko\u015b\u0107 pr\u00f3by jest wystarczaj\u0105co du\u017ca (n <sub>1<\/sub> \u226530 i n <sub>2<\/sub> \u226530), rozk\u0142ad \u015bredniej r\u00f3\u017cnicy w pr\u00f3bce jest zgodny z rozk\u0142adem normalnym. Dok\u0142adniej, parametry wspomnianego rozk\u0142adu 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-90c67b74b4e9326b7869d641a59725d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\displaystyle \\mu_{\\overline{x_1}-\\overline{x_2}}=\\mu_1-\\mu_2 \\qquad \\sigma_{\\overline{x_1}-\\overline{x_2}}=\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}\\\\[6ex]\\displaystyle N_{\\overline{x_1}-\\overline{x_2}}\\left(\\mu_1-\\mu_2, \\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}\\right) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"151\" width=\"328\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong>Uwaga:<\/strong> Je\u017celi obie populacje maj\u0105 rozk\u0142ady normalne, w\u00f3wczas rozk\u0142ad r\u00f3\u017cnicy \u015brednich w pr\u00f3bce jest zgodny z rozk\u0142adem normalnym niezale\u017cnie od wielko\u015bci pr\u00f3by.<\/p>\n<p> Dlatego te\u017c, poniewa\u017c rozk\u0142ad pr\u00f3bkowania r\u00f3\u017cnicy \u015brednich jest okre\u015blony przez rozk\u0142ad normalny, <strong>wz\u00f3r na obliczenie statystyki rozk\u0142adu pr\u00f3bkowania 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-767964a3f07b303178ee08ec191eef43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{(\\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=\"80\" 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 style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-a4071a38558726a684ed069430c89fe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_i}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest \u015bredni\u0105 pr\u00f3bki i.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-78e04dacbf6a47efcbdcc0417020dcbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_i\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest \u015bredni\u0105 populacji, tj.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-dc8c8f782c0ed8b7925012b60e174fa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> jest odchyleniem standardowym populacji, tj.<\/li>\n<li style=\"margin-bottom:5px\">\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 style=\"margin-bottom:5px\">\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 zmienn\u0105 zdefiniowan\u0105 przez standardowy rozk\u0142ad normalny N(0,1).<\/li>\n<\/ul>\n<p> Nale\u017cy pami\u0119ta\u0107, \u017ce pr\u00f3bki z r\u00f3\u017cnych populacji mog\u0105 mie\u0107 r\u00f3\u017cn\u0105 wielko\u015b\u0107 pr\u00f3by. <\/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\/\">Rozwi\u0105zane \u0107wiczenie dotycz\u0105ce rozk\u0142adu pr\u00f3bkowania r\u00f3\u017cnicy \u015brednich<\/a> <\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"distribucion-muestral-de-la-diferencia-de-proporciones\"><\/span> Pr\u00f3bkowanie rozk\u0142adu r\u00f3\u017cnicy proporcji<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Pr\u00f3by wybrane do r\u00f3\u017cnicy proporcji rozk\u0142adu pr\u00f3bkowania s\u0105 definiowane przez rozk\u0142ady dwumianowe, poniewa\u017c dla cel\u00f3w praktycznych proporcja jest stosunkiem przypadk\u00f3w zako\u0144czonych sukcesem do ca\u0142kowitej liczby obserwacji.<\/p>\n<p> Jednak\u017ce, dzi\u0119ki centralnemu twierdzeniu granicznemu, rozk\u0142ady dwumianowe mo\u017cna aproksymowa\u0107 do normalnych rozk\u0142ad\u00f3w prawdopodobie\u0144stwa. Dlatego rozk\u0142ad pr\u00f3bkowania r\u00f3\u017cnicy proporcji mo\u017cna przybli\u017cy\u0107 do rozk\u0142adu normalnego o nast\u0119puj\u0105cych cechach:<\/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-a1ce359b5dd6d80f8d27b0b9a1034bed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\displaystyle\\mu_{\\widehat{p_1}-\\widehat{p_2}}=p_1-p_2 \\qquad \\sigma_{\\widehat{p_1}-\\widehat{p_2}}=\\sqrt{\\frac{p_1q_1}{n_1}+\\frac{p_2q_2}{n_2}}\\\\[6ex]\\displaystyle N_{p}\\left(p_1-p_2, \\sqrt{\\frac{p_1q_1}{n_1}+\\frac{p_2q_2}{n_2}}\\right) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"122\" width=\"348\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong>Uwaga:<\/strong> Rozk\u0142ad pr\u00f3bkowania r\u00f3\u017cnicy proporcji mo\u017cna przybli\u017cy\u0107 do rozk\u0142adu normalnego tylko wtedy, gdy:<\/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-6a7ebccb76a4ee9bbf44bb0f41ffee53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_1\\geq30\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -3px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-93c3febf2679c77d41d7b319e262f298_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_2\\geq 30\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -3px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-19a35b2095afa5133c32d92de163adaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_1p_1\\geq5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"67\" style=\"vertical-align: -4px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-a89c44bd89266e2fba37bf5211a6e30e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_2p_2\\geq5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"67\" style=\"vertical-align: -4px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-08c0b04a830a0062f4e7f25801c45fa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_1q_1\\geq5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -4px;\"><\/p>\n<p> I<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-789e8bfde9b6a18c7ff9b1390feca142_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_2q_2\\geq5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -4px;\"><\/p>\n<p> .<\/p>\n<p> Poniewa\u017c zatem rozk\u0142ad pr\u00f3bkowania r\u00f3\u017cnicy proporcji mo\u017cna przybli\u017cy\u0107 do rozk\u0142adu normalnego, <strong>wz\u00f3r na obliczenie statystyki rozk\u0142adu pr\u00f3bkowania r\u00f3\u017cnicy proporcji<\/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-a6b74dafd0599052a453e77646e5a77a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{(\\widehat{p_1}-\\widehat{p_2})-(p_1-p_2)}{\\displaystyle\\sqrt{\\frac{p_1q_1}{n_1}+\\frac{p_2q_2}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"198\" 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 style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ad10d8ae9a51401d94ca9742249d6d15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\widehat{p_i}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest proporcj\u0105 pr\u00f3bki, tj.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-a5db80b23c0dc6e4f21c509cb298856a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p_i\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\"><\/p>\n<p> to odsetek populacji, tj.<\/li>\n<li style=\"margin-bottom:5px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-4b2d0075b0f4fd8e4e14194b33ed0fe8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q_i\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\"><\/p>\n<p> jest prawdopodobie\u0144stwem niepowodzenia populacji i,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-917f2422b9b0d7d99ec3de548cc6bba3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q_i=1-p_i\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<p> .<\/li>\n<li style=\"margin-bottom:5px\">\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 style=\"margin-bottom:5px\">\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 zmienn\u0105 zdefiniowan\u0105 przez standardowy rozk\u0142ad normalny N(0,1). <\/li>\n<\/ul>\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-proporcji\/\">Rozwi\u0105zane \u0107wiczenie dotycz\u0105ce rozk\u0142adu pr\u00f3bkowania r\u00f3\u017cnicy proporcji<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest rozk\u0142ad pr\u00f3bkowania w statystyce i do czego si\u0119 go u\u017cywa. W ten spos\u00f3b znajdziesz znaczenie rozk\u0142adu pr\u00f3bkowania, konkretny przyk\u0142ad rozk\u0142adu pr\u00f3bkowania, a ponadto wzory na najcz\u0119stsze typy rozk\u0142ad\u00f3w pr\u00f3bkowania. Jaki jest rozk\u0142ad pr\u00f3bkowania? Rozk\u0142ad pr\u00f3bkowania lub rozk\u0142ad pr\u00f3bkowania to rozk\u0142ad wynikaj\u0105cy z uwzgl\u0119dnienia wszystkich mo\u017cliwych pr\u00f3bek z populacji. Innymi [&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-285","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>Co to jest rozk\u0142ad pr\u00f3bkowania? 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