{"id":243,"date":"2023-08-03T17:38:49","date_gmt":"2023-08-03T17:38:49","guid":{"rendered":"https:\/\/statorials.org\/tr\/ortalama-icin-guven-araligi\/"},"modified":"2023-08-03T17:38:49","modified_gmt":"2023-08-03T17:38:49","slug":"ortalama-icin-guven-araligi","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/ortalama-icin-guven-araligi\/","title":{"rendered":"Ortalama i\u00e7in g\u00fcven aral\u0131\u011f\u0131"},"content":{"rendered":"<p>Bu makalede istatistikte ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131n ne oldu\u011fu ve ne i\u00e7in kullan\u0131ld\u0131\u011f\u0131 a\u00e7\u0131klanmaktad\u0131r. Benzer \u015fekilde, ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131 nas\u0131l hesaplayaca\u011f\u0131n\u0131z\u0131 ve ad\u0131m ad\u0131m bir al\u0131\u015ft\u0131rmay\u0131 ke\u015ffedeceksiniz. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-el-intervalo-de-confianza-para-la-media\"><\/span> Ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131 nedir?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131,<\/strong> bir pop\u00fclasyonun ortalamas\u0131 i\u00e7in izin verilen de\u011ferlerin aral\u0131\u011f\u0131n\u0131 sa\u011flayan bir aral\u0131kt\u0131r. Ba\u015fka bir deyi\u015fle, ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131 bize pop\u00fclasyon ortalamas\u0131n\u0131n de\u011ferini bir hata pay\u0131 ile ba\u011flayan bir maksimum de\u011fer ve bir minimum de\u011fer verir.<\/p>\n<p> \u00d6rne\u011fin, bir pop\u00fclasyon ortalamas\u0131 i\u00e7in %95 g\u00fcven aral\u0131\u011f\u0131 (6,10) ise bu, pop\u00fclasyon ortalamas\u0131n\u0131n %95&#8217;inin 6 ile 10 aras\u0131nda olaca\u011f\u0131 anlam\u0131na gelir.<\/p>\n<p> Bu nedenle ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131, bir pop\u00fclasyon ortalamas\u0131n\u0131n aras\u0131nda yer ald\u0131\u011f\u0131 iki de\u011feri tahmin etmek i\u00e7in kullan\u0131l\u0131r. Bu nedenle ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131, t\u00fcm de\u011ferleri bilinmedi\u011finde bir pop\u00fclasyonun ortalamas\u0131na yakla\u015fmak i\u00e7in \u00e7ok kullan\u0131\u015fl\u0131d\u0131r. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-intervalo-de-confianza-para-la-media\"><\/span> Ortalama i\u00e7in g\u00fcven aral\u0131\u011f\u0131 form\u00fcl\u00fc<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <a href=\"https:\/\/statorials.org\/tr\/tiplendirme\/\">Bir de\u011fi\u015fken girme<\/a> s\u00fcrecinin \u015fu \u015fekilde ger\u00e7ekle\u015fti\u011fini varsayarsak:<\/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-f93a5e76c38b2e2556bd62ba66db5947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{X-\\mu}{\\displaystyle\\frac{\\sigma}{\\sqrt{n}}} \\sim N(0,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"165\" style=\"vertical-align: -34px;\"><\/p>\n<\/p>\n<p> Ortalama i\u00e7in g\u00fcven aral\u0131\u011f\u0131, Z <sub>\u03b1\/2<\/sub> de\u011ferinin standart sapma (\u03c3) ile \u00e7arp\u0131lmas\u0131 ve numune boyutunun (n) karek\u00f6k\u00fcne b\u00f6l\u00fcnmesiyle numune ortalamas\u0131ndan toplan\u0131p \u00e7\u0131kar\u0131larak hesaplan\u0131r. Bu nedenle ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131 hesaplama form\u00fcl\u00fc \u015f\u00f6yledir:<\/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-985a374c69b468deb3d6d159fcbe6de5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-z_{\\alpha\/2}\\cdot \\frac{\\sigma}{\\sqrt{n}} \\ , \\ \\overline{x}+z_{\\alpha\/2}\\cdot \\frac{\\sigma}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"251\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> B\u00fcy\u00fck \u00f6rneklem boyutlar\u0131 ve %95 g\u00fcven d\u00fczeyi i\u00e7in kritik de\u011fer Z <sub>\u03b1\/2<\/sub> = 1,96 ve %99 g\u00fcven d\u00fczeyi i\u00e7in kritik de\u011fer Z <sub>\u03b1\/2<\/sub> = 2,576&#8217;d\u0131r.<\/p>\n<p> Pop\u00fclasyon varyans\u0131 bilindi\u011finde yukar\u0131daki form\u00fcl kullan\u0131l\u0131r. Bununla birlikte, e\u011fer pop\u00fclasyon varyans\u0131 bilinmiyorsa (ki \u00e7o\u011fu zaman b\u00f6yledir), <strong>ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131 a\u015fa\u011f\u0131daki form\u00fcl kullan\u0131larak hesaplan\u0131r:<\/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-e40e2c40fbfe4f5aa96c21ff979aae5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\ , \\ \\overline{x}+t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/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-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> \u00f6rnek anlam\u0131na gelir.<\/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-33bd5d525ddf37600aef97b0c8c08f94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t_{\\alpha\/2}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"29\" style=\"vertical-align: -8px;\"><\/p>\n<p> \u03b1\/2 olas\u0131l\u0131\u011f\u0131 ile n-1 serbestlik derecesinin \u00d6\u011frenci t da\u011f\u0131l\u0131m\u0131n\u0131n de\u011feridir.<\/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> \u00f6rnek standart sapmas\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-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> \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fcd\u00fcr. <\/li>\n<\/ul>\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\/intervalle-de-confiance.png\" alt=\"g\u00fcven aral\u0131\u011f\u0131\" class=\"wp-image-5249\" width=\"351\" height=\"339\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-del-calculo-de-un-intervalo-de-confianza-para-la-media\"><\/span> Ortalama i\u00e7in g\u00fcven aral\u0131\u011f\u0131 hesaplama \u00f6rne\u011fi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Bir pop\u00fclasyon ortalamas\u0131n\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131n nas\u0131l hesapland\u0131\u011f\u0131n\u0131 g\u00f6rebilmeniz i\u00e7in, a\u015fa\u011f\u0131da ad\u0131m ad\u0131m \u00e7\u00f6z\u00fclm\u00fc\u015f bir \u00f6rnek b\u0131rak\u0131yoruz.<\/p>\n<ul>\n<li> A\u015fa\u011f\u0131da g\u00f6sterilen de\u011ferlere sahip 8 g\u00f6zlemlik bir \u00f6rne\u011fimiz var. %95 g\u00fcven seviyesinde pop\u00fclasyonun g\u00fcven aral\u0131\u011f\u0131 ne anlama gelir?<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"> 206 203 201 212<br \/> 194 176 208 201<\/p>\n<p> \u00d6nceki b\u00f6l\u00fcmde g\u00f6rd\u00fc\u011f\u00fcm\u00fcz gibi, pop\u00fclasyon standart sapmas\u0131n\u0131 bilmedi\u011fimizde pop\u00fclasyon ortalamas\u0131n\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131 elde etmenin form\u00fcl\u00fc a\u015fa\u011f\u0131daki gibidir:<\/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-e40e2c40fbfe4f5aa96c21ff979aae5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\ , \\ \\overline{x}+t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Bu nedenle ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131 belirlemek i\u00e7in \u00f6ncelikle \u00f6rneklem ortalamas\u0131n\u0131 ve standart sapmay\u0131 hesaplamam\u0131z gerekir. <\/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-e4db3aa9e4a141ef4ac81e1816a243ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\mu =200,13 \\\\[4ex]s=11,13\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"87\" style=\"vertical-align: 0px;\"><\/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>Bak\u0131n\u0131z:<\/strong> <a href=\"https:\/\/statorials.org\/tr\/aritmetik-ortalama\/\">Aritmetik Ortalama Hesaplay\u0131c\u0131<\/a><br \/> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Bak\u0131n\u0131z:<\/strong> <a href=\"https:\/\/statorials.org\/tr\/standart-sapma-veya-standart-sapma\/\">Standart sapma hesaplay\u0131c\u0131s\u0131<\/a><\/div>\n<p> G\u00fcven aral\u0131\u011f\u0131n\u0131 1-\u03b1=%95 g\u00fcven d\u00fczeyiyle ve \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc 8 ile bulmak istedi\u011fimiz i\u00e7in, \u00d6\u011frenci t da\u011f\u0131l\u0131m tablosuna eri\u015fmemiz ve hangi de\u011ferin t <sub>0.025|7&#8217;ye<\/sub> kar\u015f\u0131l\u0131k geldi\u011fini g\u00f6rmemiz gerekiyor. <\/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-776a555edd1d064cf29a3a354caa5325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1-\\alpha=0,95 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\alpha=0,05 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black}\\ \\alpha\/2=0,025\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"534\" 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-2caa8c4576ebc261acfdb1341dc4567d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}t_{\\alpha\/2| n-1}= \\ \\color{orange}\\bm{?}\\\\[4ex]t_{0,025| 7}=2,365\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"72\" width=\"152\" style=\"vertical-align: 0px;\"><\/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>Bak\u0131n\u0131z:<\/strong> <a href=\"https:\/\/statorials.org\/tr\/ogrenci-t-dagilim-tablosu\/\">\u00d6\u011frenci t da\u011f\u0131l\u0131m tablosunun de\u011ferleri<\/a><\/div>\n<p> Bu nedenle ortalamaya g\u00fcven aral\u0131\u011f\u0131 form\u00fcl\u00fcn\u00fc uyguluyoruz ve aral\u0131\u011f\u0131n s\u0131n\u0131rlar\u0131n\u0131 bulmak i\u00e7in hesaplamalar yap\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-e40e2c40fbfe4f5aa96c21ff979aae5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\ , \\ \\overline{x}+t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/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-7c9bc920ed22b19f5b15528eef268878_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(200,13-2,365\\cdot \\frac{11,13}{\\sqrt{8}} \\ , \\ 200,13+2,365\\cdot \\frac{11,13}{\\sqrt{8}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"394\" style=\"vertical-align: -17px;\"><\/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-322fdcebe92ea2e3868440918d39d431_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(190,82 \\ , \\ 209,43 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sonu\u00e7 olarak hesaplanan g\u00fcven aral\u0131\u011f\u0131 bize %95 g\u00fcven d\u00fczeyi ile n\u00fcfus ortalamas\u0131n\u0131n 190,82 ile 209,43 aras\u0131nda olaca\u011f\u0131n\u0131 s\u00f6yl\u00fcyor.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bu makalede istatistikte ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131n ne oldu\u011fu ve ne i\u00e7in kullan\u0131ld\u0131\u011f\u0131 a\u00e7\u0131klanmaktad\u0131r. Benzer \u015fekilde, ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131 nas\u0131l hesaplayaca\u011f\u0131n\u0131z\u0131 ve ad\u0131m ad\u0131m bir al\u0131\u015ft\u0131rmay\u0131 ke\u015ffedeceksiniz. Ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131 nedir? Ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131, bir pop\u00fclasyonun ortalamas\u0131 i\u00e7in izin verilen de\u011ferlerin aral\u0131\u011f\u0131n\u0131 sa\u011flayan bir aral\u0131kt\u0131r. Ba\u015fka bir deyi\u015fle, ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131 bize pop\u00fclasyon ortalamas\u0131n\u0131n de\u011ferini bir [&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-243","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 Ortalama i\u00e7in g\u00fcven aral\u0131\u011f\u0131 - Olas\u0131l\u0131k ve istatistik<\/title>\n<meta name=\"description\" content=\"Bu makalede istatistikte ortalaman\u0131n g\u00fcven aral\u0131\u011f\u0131n\u0131n ne oldu\u011fu ve ne i\u00e7in kullan\u0131ld\u0131\u011f\u0131 a\u00e7\u0131klanmaktad\u0131r. 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