{"id":280,"date":"2023-08-03T05:03:01","date_gmt":"2023-08-03T05:03:01","guid":{"rendered":"https:\/\/statorials.org\/tr\/ortalamalar-farkinin-ornekleme-dagilimi\/"},"modified":"2023-08-03T05:03:01","modified_gmt":"2023-08-03T05:03:01","slug":"ortalamalar-farkinin-ornekleme-dagilimi","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/ortalamalar-farkinin-ornekleme-dagilimi\/","title":{"rendered":"Ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131"},"content":{"rendered":"<p>Bu makalede istatistikte iki ortalama aras\u0131ndaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ne oldu\u011fu a\u00e7\u0131klanmaktad\u0131r. Ayr\u0131ca ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131na ili\u015fkin form\u00fcl\u00fc ve buna ek olarak ad\u0131m ad\u0131m \u00e7\u00f6z\u00fclm\u00fc\u015f bir al\u0131\u015ft\u0131rmay\u0131 da bulacaks\u0131n\u0131z. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-distribucion-muestral-de-la-diferencia-de-medias\"><\/span> Ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 nedir?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Ortalamalardaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131,<\/strong> iki farkl\u0131 pop\u00fclasyondan olas\u0131 t\u00fcm \u00f6rneklerin ortalamalar\u0131 aras\u0131ndaki farklar\u0131n hesaplanmas\u0131ndan kaynaklanan da\u011f\u0131l\u0131md\u0131r.<\/p>\n<p> Yani, ortalamalar aras\u0131ndaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131 elde etmek i\u00e7in, iki \u00e7al\u0131\u015fma pop\u00fclasyonundan olas\u0131 t\u00fcm \u00f6rneklerin se\u00e7ilmesi gerekir, ard\u0131ndan se\u00e7ilen her \u00f6rne\u011fin ortalamas\u0131 hesaplan\u0131r ve son olarak iki pop\u00fclasyondan hesaplanan t\u00fcm ortalamalar aras\u0131ndaki fark. B\u00f6ylece t\u00fcm bu i\u015flemlerin uygulanmas\u0131ndan sonra elde edilen de\u011ferler k\u00fcmesi, ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131 olu\u015fturur. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/distribution-de-lechantillon-de-la-difference-de-moyennes.png\" alt=\"ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131\" class=\"wp-image-5927\" width=\"561\" height=\"237\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Ortalama \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n fark\u0131, iki farkl\u0131 pop\u00fclasyondan <a href=\"https:\/\/statorials.org\/tr\/ornek-ortalama\/\">rastgele se\u00e7ilen \u00f6rneklerin ortalamalar\u0131<\/a> aras\u0131ndaki fark\u0131n, pop\u00fclasyon ortalamalar\u0131ndaki farka yak\u0131n olma olas\u0131l\u0131\u011f\u0131n\u0131 hesaplamak i\u00e7in kullan\u0131l\u0131r. <\/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>Bak\u0131n\u0131z:<\/strong> <a href=\"https:\/\/statorials.org\/tr\/ortalamanin-ornekleme-dagilimi\/\">Ortalaman\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-distribucion-muestral-de-la-diferencia-de-medias\"><\/span> Ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 i\u00e7in form\u00fcl<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \u00d6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc yeterince b\u00fcy\u00fckse (n <sub>1<\/sub> \u226530 ve n <sub>2<\/sub> \u226530), ortalamalardaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 normal bir da\u011f\u0131l\u0131m izler. Daha do\u011frusu s\u00f6z konusu da\u011f\u0131l\u0131m\u0131n parametreleri \u015fu \u015fekilde hesaplan\u0131r:<\/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>Not:<\/strong> Her iki pop\u00fclasyon da <a href=\"https:\/\/statorials.org\/tr\/normal-dagilim\/\">normal da\u011f\u0131l\u0131ma<\/a> sahipse, ortalamalardaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131, \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fcnden ba\u011f\u0131ms\u0131z olarak normal bir da\u011f\u0131l\u0131m izler.<\/p>\n<p> Bu nedenle, ortalamalardaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 normal bir da\u011f\u0131l\u0131mla tan\u0131mland\u0131\u011f\u0131ndan, <strong>ortalamalardaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n istatisti\u011fini hesaplamak i\u00e7in form\u00fcl<\/strong> \u015fu \u015fekildedir:<\/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\"> Alt\u0131n:<\/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> \u00f6rnek i&#8217;nin ortalamas\u0131d\u0131r.<\/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> n\u00fcfusun ortalamas\u0131d\u0131r 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-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> pop\u00fclasyonun standart sapmas\u0131 i&#8217;dir.<\/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> \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc i&#8217;dir.<\/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> standart normal da\u011f\u0131l\u0131m N(0,1) taraf\u0131ndan tan\u0131mlanan bir de\u011fi\u015fkendir.<\/li>\n<\/ul>\n<p> Farkl\u0131 pop\u00fclasyonlardan al\u0131nan numunelerin farkl\u0131 numune boyutlar\u0131na sahip olabilece\u011fini unutmay\u0131n. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-resuelto-de-la-distribucion-muestral-de-la-diferencia-de-medias\"><\/span> Ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n somut \u00f6rne\u011fi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Ortalama \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n fark\u0131n\u0131n tan\u0131m\u0131n\u0131 ve form\u00fcl\u00fcn\u00fcn ne oldu\u011funu g\u00f6rd\u00fckten sonra, ortalama \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n fark\u0131 kavram\u0131n\u0131 anlamay\u0131 tamamlamak i\u00e7in ad\u0131m ad\u0131m bir \u00f6rnek g\u00f6rece\u011fiz.<\/p>\n<ul>\n<li> \u0130statistiksel bir \u00e7al\u0131\u015fmada belirli bir ya\u015fa gelen k\u0131z ve erkek \u00e7ocuklar\u0131n boy fark\u0131n\u0131 analiz etmek istiyoruz. Bu ya\u015ftaki erkek n\u00fcfusunu tan\u0131mlayan da\u011f\u0131l\u0131m\u0131n ortalamas\u0131n\u0131n 157 cm, standart sapmas\u0131n\u0131n 9 cm, buna kar\u015f\u0131l\u0131k bu ya\u015ftaki k\u0131z n\u00fcfusunu tan\u0131mlayan da\u011f\u0131l\u0131m\u0131n ortalamas\u0131n\u0131n 148 cm oldu\u011funu biliyoruz. cm ve standart sapmas\u0131 7 cm&#8217;dir. Bu ya\u015ftan 30 erkek \u00e7ocuktan ve bu ya\u015ftan 35 k\u0131zdan olu\u015fan bir \u00f6rneklem se\u00e7ilirse, erkek \u00f6rnekleminin ortalama boyunun k\u0131z \u00f6rnekleminin ortalama boyundan 12 cm daha b\u00fcy\u00fck olma olas\u0131l\u0131\u011f\u0131 nedir?<\/li>\n<\/ul>\n<p> Bu sorunu \u00e7\u00f6zmek i\u00e7in yap\u0131lacak ilk \u015fey, ortalama \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 fark\u0131n\u0131n istatisti\u011fini hesaplamakt\u0131r. Bu nedenle yukar\u0131da g\u00f6r\u00fclen form\u00fcl\u00fc uyguluyoruz:<\/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-3897bf456851ccd8de2c501bd779dd02_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}}\\right)}=\\cfrac{12-(157-148)}{\\displaystyle\\sqrt{\\frac{9^2}{30}+\\frac{7^2}{35}}}=1,48\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"415\" style=\"vertical-align: -52px;\"><\/p>\n<\/p>\n<p> Dolay\u0131s\u0131yla erkeklerin ortalama \u00f6rnek boylar\u0131n\u0131n k\u0131zlar\u0131n ortalama boylar\u0131ndan 12 cm daha b\u00fcy\u00fck olma olas\u0131l\u0131\u011f\u0131, Z de\u011fi\u015fkeninin 1,48&#8217;den b\u00fcy\u00fck olma olas\u0131l\u0131\u011f\u0131na e\u015fde\u011ferdir.<\/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-683cd2a72980115d8d18c906d619370a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P[(\\overline{x_1}-\\overline{x_2})>12]=P[Z>1,48]&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;253&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<p> Bu nedenle <a href=\"https:\/\/statorials.org\/tr\/normal-dagilim-tablosu\/\">Z tablosunda<\/a> Z&gt;1,48 olas\u0131l\u0131\u011f\u0131n\u0131 ar\u0131yoruz:<\/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-5b83fdc11bd2320249679c4752660451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P[(\\overline{x_1}-\\overline{x_2})>12]=P[Z>1,48]=0,0694&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;331&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<p> K\u0131sacas\u0131 erkek \u00e7ocuklar\u0131n ortalama boy uzunlu\u011funun k\u0131zlar\u0131n ortalama boyundan 12 cm fazla olma olas\u0131l\u0131\u011f\u0131 %6,94&#8217;t\u00fcr.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bu makalede istatistikte iki ortalama aras\u0131ndaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ne oldu\u011fu a\u00e7\u0131klanmaktad\u0131r. Ayr\u0131ca ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131na ili\u015fkin form\u00fcl\u00fc ve buna ek olarak ad\u0131m ad\u0131m \u00e7\u00f6z\u00fclm\u00fc\u015f bir al\u0131\u015ft\u0131rmay\u0131 da bulacaks\u0131n\u0131z. Ortalamalar fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 nedir? Ortalamalardaki fark\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131, iki farkl\u0131 pop\u00fclasyondan olas\u0131 t\u00fcm \u00f6rneklerin ortalamalar\u0131 aras\u0131ndaki farklar\u0131n hesaplanmas\u0131ndan kaynaklanan da\u011f\u0131l\u0131md\u0131r. Yani, ortalamalar aras\u0131ndaki fark\u0131n [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-280","post","type-post","status-publish","format-standard","hentry","category-istatistik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Ortalamalar\u0131n fark\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131<\/title>\n<meta name=\"description\" content=\"Burada ortalama \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ne kadar farkl\u0131 oldu\u011funu, form\u00fcl\u00fcn\u00fc ve ad\u0131m ad\u0131m \u00f6rne\u011fini bulacaks\u0131n\u0131z.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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