{"id":429,"date":"2023-07-29T23:33:30","date_gmt":"2023-07-29T23:33:30","guid":{"rendered":"https:\/\/statorials.org\/tr\/merkezi-limit-teoremi\/"},"modified":"2023-07-29T23:33:30","modified_gmt":"2023-07-29T23:33:30","slug":"merkezi-limit-teoremi","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/merkezi-limit-teoremi\/","title":{"rendered":"Merkezi limit teoremi: tan\u0131m + \u00f6rnekler"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\"><strong>Merkezi limit teoremi,<\/strong><a href=\"https:\/\/statorials.org\/tr\/ornekleme-dagilimi-1\/\" target=\"_blank\" rel=\"noopener noreferrer\">pop\u00fclasyon da\u011f\u0131l\u0131m\u0131<\/a> normal <em>olmasa bile,<\/em> \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc yeterince b\u00fcy\u00fckse, bir \u00f6rneklem ortalamas\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n yakla\u015f\u0131k olarak normal oldu\u011funu belirtir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Merkezi limit teoremi ayr\u0131ca \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n a\u015fa\u011f\u0131daki \u00f6zelliklere sahip olaca\u011f\u0131n\u0131 belirtir:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>1.<\/strong> \u00d6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131 n\u00fcfus da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131na e\u015fit olacakt\u0131r:<\/span><\/p>\n<p style=\"text-align: center;\"> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u00b5<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\"><strong>2.<\/strong> \u00d6rneklem da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131, ana k\u00fctle da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131n\u0131n \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fcne b\u00f6l\u00fcnmesine e\u015fit olacakt\u0131r:<\/span><\/p>\n<p style=\"text-align: center;\"> <strong><span style=\"color: #000000;\"><sup>s2<\/sup> = <sup>\u03c32<\/sup><\/span> <span style=\"color: #000000;\">\/n<\/span><\/strong><\/p>\n<h2> <strong><span style=\"color: #000000;\">Merkezi Limit Teoremine \u00d6rnekler<\/span><\/strong><\/h2>\n<p> <span style=\"color: #000000;\">Burada merkezi limit teoremini pratikte g\u00f6stermek i\u00e7in baz\u0131 \u00f6rnekler verilmi\u015ftir.<\/span><\/p>\n<h3> <strong><span style=\"color: #000000;\">\u00dcniforma da\u011f\u0131t\u0131m\u0131<\/span><\/strong><\/h3>\n<p> <span style=\"color: #000000;\">Bir kaplumba\u011fan\u0131n kabu\u011funun geni\u015fli\u011finin minimum 2 in\u00e7 ve maksimum 6 in\u00e7 geni\u015flikte d\u00fczg\u00fcn bir da\u011f\u0131l\u0131m izledi\u011fini varsayal\u0131m. Yani rastgele bir kaplumba\u011fa se\u00e7ip kabu\u011funun geni\u015fli\u011fini \u00f6l\u00e7ersek, <em>geni\u015fli\u011finin<\/em> de muhtemelen 2 ile 6 in\u00e7 aras\u0131nda olmas\u0131 muhtemeldir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Kaplumba\u011fa kabu\u011fu geni\u015fliklerinin da\u011f\u0131l\u0131m\u0131n\u0131 temsil edecek bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1472 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_uniforme1.jpg\" alt=\"Merkezi Limit Teoreminin D\u00fczg\u00fcn Da\u011f\u0131l\u0131m \u00d6rne\u011fi\" width=\"430\" height=\"264\" srcset=\"\" sizes=\"auto, \"><br \/> <span style=\"color: #000000;\">D\u00fczg\u00fcn bir da\u011f\u0131l\u0131m\u0131n ortalamas\u0131 <strong>\u03bc<\/strong> = (b+a) \/ 2&#8217;dir; burada <em>b<\/em> m\u00fcmk\u00fcn olan en b\u00fcy\u00fck de\u011fer ve <em>a<\/em> m\u00fcmk\u00fcn olan en k\u00fc\u00e7\u00fck de\u011ferdir. Bu durumda (6+2) \/ 2 = 4 olur.<\/span><\/p>\n<p> <span style=\"color: #000000;\">D\u00fczg\u00fcn bir da\u011f\u0131l\u0131m\u0131n varyans\u0131 <strong><sup>\u03c32<\/sup><\/strong> = (ba) <sup>2\/12&#8217;dir<\/sup> . Bu durumda (6-2) <sup>2\/12<\/sup> = <strong>1,33<\/strong> olur<\/span><\/p>\n<h4> <span style=\"color: #000000;\"><strong>D\u00fczg\u00fcn da\u011f\u0131l\u0131mdan rastgele 2 \u00f6rnek alma<\/strong><\/span><\/h4>\n<p> <span style=\"color: #000000;\">\u015eimdi bu pop\u00fclasyondan rastgele 2 kaplumba\u011fa \u00f6rne\u011fi ald\u0131\u011f\u0131m\u0131z\u0131 ve her bir kaplumba\u011fan\u0131n kabu\u011funun geni\u015fli\u011fini \u00f6l\u00e7t\u00fc\u011f\u00fcm\u00fcz\u00fc hayal edin. \u0130lk kaplumba\u011fan\u0131n kabu\u011funun 3 in\u00e7, ikincisinin ise 6 in\u00e7 geni\u015fli\u011finde oldu\u011funu varsayal\u0131m. 2 kaplumba\u011fadan olu\u015fan bu \u00f6rne\u011fin ortalama geni\u015fli\u011fi 4,5 in\u00e7tir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Daha sonra, bu pop\u00fclasyondan rastgele 2 kaplumba\u011fadan olu\u015fan ba\u015fka bir \u00f6rnek ald\u0131\u011f\u0131m\u0131z\u0131 ve her bir kaplumba\u011fan\u0131n kabuk geni\u015fli\u011fini tekrar \u00f6l\u00e7t\u00fc\u011f\u00fcm\u00fcz\u00fc hayal edin. \u0130lk kaplumba\u011fan\u0131n kabu\u011funun 2,5 in\u00e7, ikincisinin de 2,5 in\u00e7 geni\u015fli\u011finde oldu\u011funu varsayal\u0131m. 2 kaplumba\u011fadan olu\u015fan bu \u00f6rne\u011fin ortalama geni\u015fli\u011fi 2,5 in\u00e7tir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Tekrar tekrar 2 kaplumba\u011fadan rastgele \u00f6rnekler ald\u0131\u011f\u0131m\u0131z\u0131 ve her seferinde ortalama kabuk geni\u015fli\u011fini buldu\u011fumuzu hayal edin.<\/span><\/p>\n<p> <span style=\"color: #000000;\">2 kaplumba\u011fadan al\u0131nan t\u00fcm bu \u00f6rneklerin ortalama kabuk geni\u015fli\u011fini temsil edecek bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1485 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_uniforme2-2.jpg\" alt=\"D\u00fczg\u00fcn da\u011f\u0131l\u0131m i\u00e7in \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc 2 i\u00e7in merkezi limit teoremi\" width=\"431\" height=\"271\" srcset=\"\" sizes=\"auto, \"><br \/> <span style=\"color: #000000;\">Buna <strong>\u00f6rnek ortalamalar\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131<\/strong> denir \u00e7\u00fcnk\u00fc \u00f6rnek ortalamalar\u0131n da\u011f\u0131l\u0131m\u0131n\u0131 g\u00f6sterir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131<\/span> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u03bc = 4&#8217;t\u00fcr<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131: <strong><sup>s2<\/sup> = <sup>\u03c32<\/sup> \/ n = 1,33 \/ 2 = 0,665<\/strong><\/span><\/p>\n<h4> <span style=\"color: #000000;\"><strong>D\u00fczg\u00fcn da\u011f\u0131l\u0131mdan rastgele 5 \u00f6rnek alma<\/strong><\/span><\/h4>\n<p> <span style=\"color: #000000;\">\u015eimdi ayn\u0131 deneyi tekrarlad\u0131\u011f\u0131m\u0131z\u0131 hayal edin, ancak bu sefer 5 kaplumba\u011fadan rastgele \u00f6rnekler al\u0131p her seferinde ortalama kabuk geni\u015fli\u011fini buluyoruz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">5 kaplumba\u011fan\u0131n t\u00fcm \u00f6rneklerinin ortalama kabuk geni\u015fli\u011fini temsil edecek bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1484 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_uniform4-2.jpg\" alt=\"Tekd\u00fcze \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc da\u011f\u0131l\u0131m\u0131 i\u00e7in merkezi limit teoremi 5\" width=\"431\" height=\"275\" srcset=\"\" sizes=\"auto, \"><br \/> <span style=\"color: #000000;\">Bu da\u011f\u0131l\u0131m\u0131n daha \u00e7ok normal da\u011f\u0131l\u0131ma benzeyen bir &#8220;\u00e7an&#8221; \u015fekline sahip oldu\u011funa dikkat edin. Bunun nedeni, 5&#8217;lik numuneler ald\u0131\u011f\u0131m\u0131zda, numune ortalamalar\u0131m\u0131z aras\u0131ndaki fark\u0131n \u00e7ok daha d\u00fc\u015f\u00fck olmas\u0131, dolay\u0131s\u0131yla ortalama 2 in\u00e7 veya 6 in\u00e7&#8217;e yak\u0131n numuneler alma olas\u0131l\u0131\u011f\u0131m\u0131z\u0131n daha d\u00fc\u015f\u00fck olmas\u0131 ve ortalama 2 in\u00e7 veya 6 in\u00e7&#8217;e yak\u0131n numuneler elde etme olas\u0131l\u0131\u011f\u0131m\u0131z\u0131n daha y\u00fcksek olmas\u0131d\u0131r. 6 in\u00e7. ortalama, ger\u00e7ek n\u00fcfus ortalamas\u0131na 4 in\u00e7 daha yak\u0131nd\u0131r.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131<\/span> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u03bc = 4&#8217;t\u00fcr<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131: <strong><sup>s2<\/sup> = <sup>\u03c32<\/sup> \/ n = 1,33 \/ 5 = 0,266<\/strong><\/span><\/p>\n<h4> <span style=\"color: #000000;\"><strong>D\u00fczg\u00fcn da\u011f\u0131l\u0131mdan rastgele 30 \u00f6rnek alma<\/strong><\/span><\/h4>\n<p> <span style=\"color: #000000;\">\u015eimdi ayn\u0131 deneyi tekrarlad\u0131\u011f\u0131m\u0131z\u0131 hayal edin, ancak bu sefer 30 kaplumba\u011fadan rastgele \u00f6rnekler al\u0131yoruz ve her seferinde ortalama kabuk geni\u015fli\u011fini buluyoruz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">30 kaplumba\u011faya ait t\u00fcm \u00f6rneklerin ortalama kabuk geni\u015fli\u011fini temsil edecek bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1483 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_uniforme5-2.jpg\" alt=\"30 ki\u015filik bir \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc i\u00e7in merkezi limit teoremi\" width=\"430\" height=\"269\" srcset=\"\" sizes=\"auto, \"><br \/> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n \u00f6nceki iki da\u011f\u0131l\u0131mdan daha da \u00e7an \u015feklinde ve \u00e7ok daha dar oldu\u011funa dikkat edin.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131<\/span> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u03bc = 4&#8217;t\u00fcr<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131 <strong><sup>s2<\/sup> = <sup>\u03c32<\/sup> \/ n = 1,33 \/ 30 = 0,044&#8217;t\u00fcr.<\/strong><\/span><\/p>\n<h3> <strong><span style=\"color: #000000;\">Ki-kare da\u011f\u0131l\u0131m\u0131<\/span><\/strong><\/h3>\n<p> <span style=\"color: #000000;\">Belirli bir \u015fehirdeki aile ba\u015f\u0131na d\u00fc\u015fen evcil hayvan say\u0131s\u0131n\u0131n \u00fc\u00e7 serbestlik derecesine sahip ki-kare da\u011f\u0131l\u0131m\u0131n\u0131 takip etti\u011fini varsayal\u0131m. Hayvanlar\u0131n familyalara g\u00f6re da\u011f\u0131l\u0131m\u0131n\u0131 temsil eden bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1488 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_chi1.jpg\" alt=\"Ki-kare da\u011f\u0131l\u0131m\u0131 i\u00e7in merkezi limit teoremi\" width=\"494\" height=\"272\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Ki-kare da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131 basit\u00e7e serbestlik derecesinin (df) say\u0131s\u0131d\u0131r. Bu durumda <strong>\u03bc<\/strong> = <strong>3<\/strong> .<\/span><\/p>\n<p> <span style=\"color: #000000;\">Ki-kare da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131 2 * df&#8217;dir. Bu durumda <strong><sup>\u03c32<\/sup><\/strong> = 2 * 3 = <strong>6<\/strong> olur.<\/span><\/p>\n<h4> <span style=\"color: #000000;\"><strong>Rastgele 2 \u00f6rnek alma<\/strong><\/span><\/h4>\n<p> <span style=\"color: #000000;\">Bu pop\u00fclasyondan 2 aileden rastgele bir \u00f6rnek ald\u0131\u011f\u0131m\u0131z\u0131 ve her ailedeki evcil hayvan say\u0131s\u0131n\u0131 sayd\u0131\u011f\u0131m\u0131z\u0131 hayal edin. Birinci ailenin 4 evcil hayvan\u0131, ikinci ailenin ise 1 evcil hayvan\u0131 oldu\u011funu varsayal\u0131m. 2 aileden olu\u015fan bu \u00f6rneklem i\u00e7in ortalama evcil hayvan say\u0131s\u0131 2,5&#8217;tir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Daha sonra bu pop\u00fclasyondan 2 aileden olu\u015fan ba\u015fka bir rastgele \u00f6rnek ald\u0131\u011f\u0131m\u0131z\u0131 ve her ailedeki evcil hayvan say\u0131s\u0131n\u0131 tekrar sayd\u0131\u011f\u0131m\u0131z\u0131 hayal edin. Birinci ailenin 6 evcil hayvan\u0131, ikinci ailenin ise 4 evcil hayvan\u0131 oldu\u011funu varsayal\u0131m. Bu 2 aile \u00f6rne\u011finin ortalama evcil hayvan say\u0131s\u0131 5&#8217;tir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Tekrar tekrar 2 aileden rastgele \u00f6rnekler ald\u0131\u011f\u0131m\u0131z\u0131 ve her seferinde ortalama evcil hayvan say\u0131s\u0131n\u0131 bulmaya devam etti\u011fimizi hayal edin.<\/span><\/p>\n<p> <span style=\"color: #000000;\">2 aileden al\u0131nan t\u00fcm bu \u00f6rneklerin ortalama evcil hayvan say\u0131s\u0131n\u0131 temsil edecek bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1489 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_chi2.jpg\" alt=\"Ki-kare da\u011f\u0131l\u0131m \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc 2 olan merkezi limit teoremi\" width=\"442\" height=\"296\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131<\/span> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u03bc = 3&#8217;t\u00fcr<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131: <strong>s <sup>2<\/sup> = \u03c3 <sup>2<\/sup> \/ n = 6 \/ 2 = 3<\/strong><\/span><\/p>\n<h4> <span style=\"color: #000000;\"><strong>10&#8217;dan rastgele numune alma<\/strong><\/span><\/h4>\n<p> <span style=\"color: #000000;\">\u015eimdi ayn\u0131 deneyi tekrarlad\u0131\u011f\u0131m\u0131z\u0131 hayal edin, ancak bu sefer 10 aileden rastgele \u00f6rnekler al\u0131yoruz ve her seferinde aile ba\u015f\u0131na ortalama hayvan say\u0131s\u0131n\u0131 buluyoruz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">10 aileye ait t\u00fcm bu \u00f6rneklerde aile ba\u015f\u0131na d\u00fc\u015fen ortalama hayvan say\u0131s\u0131n\u0131 temsil edecek bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1490 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_chi3.jpg\" alt=\"Ki-kare da\u011f\u0131l\u0131ml\u0131 merkezi limit teoremi\" width=\"442\" height=\"295\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131<\/span> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u03bc = 3&#8217;t\u00fcr<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131: <strong><sup>s2<\/sup> = <sup>\u03c32<\/sup> \/ n = 6\/10 = 0,6<\/strong><\/span><\/p>\n<h4> <span style=\"color: #000000;\"><strong>Rastgele 30 \u00f6rnek alma<\/strong><\/span><\/h4>\n<p> <span style=\"color: #000000;\">\u015eimdi ayn\u0131 deneyi tekrarlad\u0131\u011f\u0131m\u0131z\u0131 hayal edin, ancak bu sefer 30 aileden rastgele \u00f6rnekler al\u0131yoruz ve her seferinde aile ba\u015f\u0131na ortalama hayvan say\u0131s\u0131n\u0131 buluyoruz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">30 familyadan olu\u015fan t\u00fcm bu \u00f6rneklerde aile ba\u015f\u0131na d\u00fc\u015fen ortalama hayvan say\u0131s\u0131n\u0131 temsil edecek bir histogram yapsayd\u0131k \u015f\u00f6yle g\u00f6r\u00fcn\u00fcrd\u00fc:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-1491 size-full\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/clt_chi4.jpg\" alt=\"Ki-kare da\u011f\u0131l\u0131ml\u0131 merkezi limit teoreminin histogram\u0131\" width=\"441\" height=\"294\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131<\/span> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u03bc = 3&#8217;t\u00fcr<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131 <strong><sup>s2<\/sup> = <sup>\u03c32<\/sup> \/ n = 6\/30 = 0,2&#8217;dir.<\/strong><\/span><\/p>\n<h2> <strong><span style=\"color: #000000;\">\u00d6zet<\/span><\/strong><\/h2>\n<p> <span style=\"color: #000000;\">Bu iki \u00f6rnekten ana \u00e7\u0131kar\u0131mlar \u015funlard\u0131r:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">N\u00fcfus da\u011f\u0131l\u0131m\u0131 normal olmasa <em>bile, \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc yeterince b\u00fcy\u00fckse, \u00f6rneklem ortalamas\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 yakla\u015f\u0131k olarak normaldir<\/em> . Yukar\u0131daki iki \u00f6rnekte ne d\u00fczg\u00fcn da\u011f\u0131l\u0131m ne de ki-kare da\u011f\u0131l\u0131m\u0131 normaldi (&#8220;\u00e7an&#8221; \u015feklinde de\u011fildiler), ancak yeterince b\u00fcy\u00fck bir \u00f6rnek ald\u0131\u011f\u0131m\u0131zda \u00f6rnek ortalamas\u0131n\u0131n da\u011f\u0131l\u0131m\u0131 \u015funa d\u00f6n\u00fc\u015ft\u00fc: Normal olmak.<\/span><\/li>\n<li> <span style=\"color: #000000;\">\u00d6rnek boyutu ne kadar b\u00fcy\u00fck olursa, \u00f6rnek ortalamas\u0131n\u0131n varyans\u0131 o kadar d\u00fc\u015f\u00fck olur.<\/span><\/li>\n<\/ul>\n<h2> <span style=\"color: #000000;\"><strong>\u201cYeterince b\u00fcy\u00fck\u201d\u00fc tan\u0131mlay\u0131n<\/strong><\/span><\/h2>\n<p> <span style=\"color: #000000;\">Merkezi limit teoreminin, pop\u00fclasyon da\u011f\u0131l\u0131m\u0131 normal olmasa bile, \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc <strong>&#8220;yeterince b\u00fcy\u00fck&#8221;<\/strong> se, bir \u00f6rneklem ortalamas\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n yakla\u015f\u0131k olarak normal oldu\u011funu belirtti\u011fini hat\u0131rlay\u0131n.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Merkezi limit teoreminin uygulanmas\u0131 i\u00e7in bir numunenin ne kadar b\u00fcy\u00fck olmas\u0131 gerekti\u011fine dair kesin bir tan\u0131m yoktur, ancak genel olarak bu, numunenin geldi\u011fi pop\u00fclasyon da\u011f\u0131l\u0131m\u0131n\u0131n \u00e7arp\u0131kl\u0131\u011f\u0131na ba\u011fl\u0131d\u0131r:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">N\u00fcfus da\u011f\u0131l\u0131m\u0131 simetrikse, 15 kadar k\u00fc\u00e7\u00fck bir \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc bazen yeterli olabilir.<\/span><\/li>\n<li> <span style=\"color: #000000;\">N\u00fcfus da\u011f\u0131l\u0131m\u0131 \u00e7arp\u0131ksa genellikle en az 30 ki\u015fiden olu\u015fan bir \u00f6rneklem gereklidir.<\/span><\/li>\n<li> <span style=\"color: #000000;\">N\u00fcfus da\u011f\u0131l\u0131m\u0131 a\u015f\u0131r\u0131 derecede \u00e7arp\u0131ksa 40 veya daha fazla ki\u015fiden olu\u015fan bir \u00f6rneklem gerekli olabilir.<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Bu konu hakk\u0131nda daha fazla bilgi i\u00e7in <a href=\"https:\/\/statorials.org\/tr\/buyuk-ornek-durumu\/\" target=\"_blank\" rel=\"noopener noreferrer\">B\u00fcy\u00fck Bir Numuneyi Ko\u015fulland\u0131rma<\/a> hakk\u0131ndaki bu e\u011fitime g\u00f6z at\u0131n.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Merkezi limit teoremi,pop\u00fclasyon da\u011f\u0131l\u0131m\u0131 normal olmasa bile, \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc yeterince b\u00fcy\u00fckse, bir \u00f6rneklem ortalamas\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n yakla\u015f\u0131k olarak normal oldu\u011funu belirtir. Merkezi limit teoremi ayr\u0131ca \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n a\u015fa\u011f\u0131daki \u00f6zelliklere sahip olaca\u011f\u0131n\u0131 belirtir: 1. \u00d6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131 n\u00fcfus da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131na e\u015fit olacakt\u0131r: x = \u00b5 2. \u00d6rneklem da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131, ana k\u00fctle da\u011f\u0131l\u0131m\u0131n\u0131n varyans\u0131n\u0131n \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fcne b\u00f6l\u00fcnmesine [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-429","post","type-post","status-publish","format-standard","hentry","category-rehber"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Merkezi limit teoremi: tan\u0131m + \u00f6rnekler - 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