{"id":1397,"date":"2023-07-26T13:29:57","date_gmt":"2023-07-26T13:29:57","guid":{"rendered":"https:\/\/statorials.org\/tr\/normal-yaklasim\/"},"modified":"2023-07-26T13:29:57","modified_gmt":"2023-07-26T13:29:57","slug":"normal-yaklasim","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/normal-yaklasim\/","title":{"rendered":"Normal binom yakla\u015f\u0131m\u0131: tan\u0131m ve \u00f6rnek"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\"><em>E\u011fer<\/em> <a href=\"https:\/\/statorials.org\/tr\/rastgele-degiskenler\/\" target=\"_blank\" rel=\"noopener\">_<\/a> <a href=\"https:\/\/statorials.org\/tr\/binom-dagilimi-1\/\" target=\"_blank\" rel=\"noopener\">_<\/a> <em>_<\/em> <em>_<\/em> <em>_<\/em><\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">\u00b5 = np<\/span><\/li>\n<li> <span style=\"color: #000000;\">\u03c3 = \u221a <span style=\"border-top: 1px solid black;\">np(1-p)<\/span><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">E\u011fer <em>n<\/em> yeterince b\u00fcy\u00fckse, binom da\u011f\u0131l\u0131m\u0131yla ilgili olas\u0131l\u0131klara yakla\u015fmak i\u00e7in normal da\u011f\u0131l\u0131m\u0131 kullanabiliriz. Buna <strong>normal binom yakla\u015f\u0131m\u0131<\/strong> denir.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><em>N&#8217;nin<\/em> &#8220;yeterince b\u00fcy\u00fck&#8221; olmas\u0131 i\u00e7in a\u015fa\u011f\u0131daki kriterleri kar\u015f\u0131lamas\u0131 gerekir:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">np \u2265 5<\/span><\/li>\n<li> <span style=\"color: #000000;\">n(1-p) \u2265 5<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Her iki kriter de kar\u015f\u0131land\u0131\u011f\u0131nda, binom da\u011f\u0131l\u0131m\u0131yla ilgili olas\u0131l\u0131k sorular\u0131n\u0131 yan\u0131tlamak i\u00e7in normal da\u011f\u0131l\u0131m\u0131 kullanabiliriz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Ancak normal da\u011f\u0131l\u0131m s\u00fcrekli bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131 iken binom da\u011f\u0131l\u0131m\u0131 ayr\u0131k bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r, dolay\u0131s\u0131yla olas\u0131l\u0131klar\u0131 hesaplarken s\u00fcreklilik d\u00fczeltmesi uygulamam\u0131z gerekir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Basit\u00e7e s\u00f6ylemek gerekirse <strong>s\u00fcreklilik d\u00fczeltmesi<\/strong> , ayr\u0131k bir x de\u011ferine 0,5 ekleme veya \u00e7\u0131karma i\u015flemine verilen add\u0131r.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u00d6rne\u011fin, bir paran\u0131n 100 at\u0131\u015fta 45 defadan az veya e\u015fit tura gelme olas\u0131l\u0131\u011f\u0131n\u0131 bulmak istedi\u011fimizi varsayal\u0131m. Yani P(X \u2264 45)&#8217;i bulmak istiyoruz. Binom da\u011f\u0131l\u0131m\u0131na yakla\u015fmak amac\u0131yla normal da\u011f\u0131l\u0131m\u0131 kullanmak i\u00e7in bunun yerine P(X \u2264 45,5) buluruz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">A\u015fa\u011f\u0131daki tablo, bulmaya \u00e7al\u0131\u015ft\u0131\u011f\u0131n\u0131z olas\u0131l\u0131k t\u00fcr\u00fcne ba\u011fl\u0131 olarak ne zaman 0,5 eklemeniz veya \u00e7\u0131karman\u0131z gerekti\u011fini g\u00f6sterir:<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<th> <span style=\"color: #000000;\"><strong>Binom da\u011f\u0131l\u0131m\u0131n\u0131 kullan\u0131n<\/strong><\/span><\/th>\n<th> <span style=\"color: #000000;\"><strong>S\u00fcreklilik d\u00fczeltmesiyle normal da\u011f\u0131l\u0131m\u0131n kullan\u0131lmas\u0131<\/strong><\/span><\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X = 45<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">44,5 &lt; X &lt; 45,5<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X \u2264 45<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X &lt; 45,5<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X &lt; 45<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X &lt; 44,5<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X \u2265 45<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X &gt; 44,5<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X &gt; 45<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">X &gt; 45,5<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p> <span style=\"color: #000000;\">A\u015fa\u011f\u0131daki ad\u0131m ad\u0131m \u00f6rnek, binom da\u011f\u0131l\u0131m\u0131na yakla\u015fmak i\u00e7in normal da\u011f\u0131l\u0131m\u0131n nas\u0131l kullan\u0131laca\u011f\u0131n\u0131 g\u00f6sterir.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>\u00d6rnek: binomun normal yakla\u015f\u0131m\u0131<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Bir madeni paran\u0131n 100 at\u0131\u015fta 43 veya daha az kez tura gelme olas\u0131l\u0131\u011f\u0131n\u0131 bilmek istedi\u011fimizi varsayal\u0131m.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Bu durumda a\u015fa\u011f\u0131daki de\u011ferlere sahibiz:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>n<\/strong> (deneme say\u0131s\u0131) = 100<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>X<\/strong> (ba\u015far\u0131 say\u0131s\u0131) = 43<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>p<\/strong> (belirli bir denemede ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131) = 0,50<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Paran\u0131n 43 veya daha az tura gelme olas\u0131l\u0131\u011f\u0131n\u0131 hesaplamak i\u00e7in a\u015fa\u011f\u0131daki ad\u0131mlar\u0131 kullanabiliriz:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Ad\u0131m 1: \u00d6rnek boyutunun normal yakla\u015f\u0131m\u0131 kullanacak kadar b\u00fcy\u00fck oldu\u011funu do\u011frulay\u0131n.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">\u00d6ncelikle a\u015fa\u011f\u0131daki kriterlerin kar\u015f\u0131lan\u0131p kar\u015f\u0131lanmad\u0131\u011f\u0131n\u0131 kontrol etmemiz gerekiyor:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">np \u2265 5<\/span><\/li>\n<li> <span style=\"color: #000000;\">n(1-p) \u2265 5<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Bu durumda elimizde:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">np = 100*0,5 = 50<\/span><\/li>\n<li> <span style=\"color: #000000;\">n(1-p) = 100*(1 \u2013 0,5) = 100*0,5 = 50<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Her iki say\u0131 da 5&#8217;ten b\u00fcy\u00fckt\u00fcr, dolay\u0131s\u0131yla normal yakla\u015f\u0131m\u0131 g\u00fcvenle kullanabiliriz.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Ad\u0131m 2: Uygulanacak s\u00fcreklilik d\u00fczeltmesini belirleyin.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Yukar\u0131daki tabloya bak\u0131ld\u0131\u011f\u0131nda X \u2264 43 \u015feklinde olas\u0131l\u0131k ile \u00e7al\u0131\u015f\u0131rken 0,5 eklememiz gerekti\u011fini g\u00f6r\u00fcyoruz. B\u00f6ylece P(X&lt; 43,5) bulaca\u011f\u0131z.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Ad\u0131m 3: Binom da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131n\u0131 (\u03bc) ve standart sapmas\u0131n\u0131 (\u03c3) bulun.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>\u00b5<\/strong> = n*p = 100*0,5 = 50<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>\u03c3<\/strong> = \u221a <span style=\"text-decoration: overline;\">n*p*(1-p)<\/span> = \u221a <span style=\"text-decoration: overline;\">100*.5*(1-.5)<\/span> = \u221a <span style=\"text-decoration: overline;\">25<\/span> = 5<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Ad\u0131m 4: \u00d6nceki ad\u0131mda bulunan ortalama ve standart sapmay\u0131 kullanarak z-puan\u0131n\u0131 bulun.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>z<\/strong> = (x \u2013 \u03bc) \/ \u03c3 = (43,5 \u2013 50) \/ 5 = -6,5 \/ 5 = -1,3.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Ad\u0131m 5: Z puan\u0131yla ili\u015fkili olas\u0131l\u0131\u011f\u0131 bulun.<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">-1,3&#8217;\u00fcn solundaki standart normal e\u011frinin alt\u0131ndaki alan\u0131n <strong>0,0968<\/strong> oldu\u011funu bulmak i\u00e7in <a href=\"https:\/\/statorials.org\/tr\/hizli-normal-cdf-hesaplayici\/\" target=\"_blank\" rel=\"noopener\">normal CDF hesaplay\u0131c\u0131s\u0131n\u0131<\/a> kullanabiliriz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Yani bir paran\u0131n 100 at\u0131\u015fta 43 kereden az veya e\u015fit tura gelme olas\u0131l\u0131\u011f\u0131 <strong>0,0968&#8217;dir<\/strong> .<\/span><\/p>\n<hr>\n<p> <span style=\"color: #000000;\">Bu \u00f6rnek a\u015fa\u011f\u0131dakileri g\u00f6stermektedir:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">Rastgele bir de\u011fi\u015fkenin binom da\u011f\u0131l\u0131m\u0131n\u0131 takip etti\u011fi bir durumla kar\u015f\u0131la\u015ft\u0131k.<\/span><\/li>\n<li> <span style=\"color: #000000;\">Bu rastgele de\u011fi\u015fkenin belirli bir de\u011fer alma olas\u0131l\u0131\u011f\u0131n\u0131 bulmak istedik.<\/span><\/li>\n<li> <span style=\"color: #000000;\">\u00d6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc (n = 100 deneme) yeterince b\u00fcy\u00fck oldu\u011fundan, binom da\u011f\u0131l\u0131m\u0131na yakla\u015fmak i\u00e7in normal da\u011f\u0131l\u0131m\u0131 kullanabildik.<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Bu, binom da\u011f\u0131l\u0131m\u0131yla ilgili olas\u0131l\u0131klar\u0131 bulmak i\u00e7in normal yakla\u015f\u0131m\u0131n nas\u0131l kullan\u0131laca\u011f\u0131na dair tam bir \u00f6rnektir.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>E\u011fer _ _ _ _ _ \u00b5 = np \u03c3 = \u221a np(1-p) E\u011fer n yeterince b\u00fcy\u00fckse, binom da\u011f\u0131l\u0131m\u0131yla ilgili olas\u0131l\u0131klara yakla\u015fmak i\u00e7in normal da\u011f\u0131l\u0131m\u0131 kullanabiliriz. Buna normal binom yakla\u015f\u0131m\u0131 denir. N&#8217;nin &#8220;yeterince b\u00fcy\u00fck&#8221; olmas\u0131 i\u00e7in a\u015fa\u011f\u0131daki kriterleri kar\u015f\u0131lamas\u0131 gerekir: np \u2265 5 n(1-p) \u2265 5 Her iki kriter de kar\u015f\u0131land\u0131\u011f\u0131nda, binom da\u011f\u0131l\u0131m\u0131yla ilgili olas\u0131l\u0131k [&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-1397","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>Normal binom yakla\u015f\u0131m\u0131: tan\u0131m ve \u00f6rnek<\/title>\n<meta name=\"description\" content=\"Bu e\u011fitimde, bir tan\u0131m ve bir \u00f6rnek de dahil olmak \u00fczere, binom da\u011f\u0131l\u0131m\u0131n\u0131n normal 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