{"id":26826,"date":"2021-11-05T13:00:41","date_gmt":"2021-11-05T13:00:41","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26826"},"modified":"2024-05-22T07:37:57","modified_gmt":"2024-05-22T07:37:57","slug":"como-calcular-a-distribuicao-normal-usando-a-tabela","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/como-calcular-a-distribuicao-normal-usando-a-tabela\/","title":{"rendered":"Como calcular a distribui\u00e7\u00e3o normal usando a tabela?"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n  <center><\/p>\n<h1>Como calcular a distribui\u00e7\u00e3o normal usando a tabela?<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\">\n      <strong>Resumo<\/strong><br \/>\n      <em>Nesta aula, abordaremos o tema da distribui\u00e7\u00e3o normal, uma das distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais comuns. Analisaremos como uma vari\u00e1vel aleat\u00f3ria X com par\u00e2metros \u03bc e \u03c3 pode seguir uma distribui\u00e7\u00e3o normal e como essa distribui\u00e7\u00e3o pode ser padronizada por meio de uma substitui\u00e7\u00e3o na integral. No entanto, notaremos que, apesar da simplifica\u00e7\u00e3o atrav\u00e9s da padroniza\u00e7\u00e3o, o c\u00e1lculo anal\u00edtico da integral continua sendo um desafio, pois n\u00e3o existem express\u00f5es alg\u00e9bricas que permitam expressar seu resultado. Para superar esse problema, apresentaremos uma tabela da distribui\u00e7\u00e3o normal padr\u00e3o constru\u00edda no Excel, que oferece aproxima\u00e7\u00f5es num\u00e9ricas aos valores da fun\u00e7\u00e3o. Explicaremos detalhadamente como usar a tabela, incluindo a interpreta\u00e7\u00e3o das probabilidades acumuladas como \u00e1reas sob a curva. Finalmente, a aula conclui com uma s\u00e9rie de exerc\u00edcios pr\u00e1ticos focados no c\u00e1lculo das probabilidades e estimativas usando a tabela da distribui\u00e7\u00e3o normal e o conceito de distribui\u00e7\u00e3o normal padr\u00e3o.<\/em>\n    <\/p>\n<p>  <\/center><br \/>\n  <\/p>\n<p style=\"text-align:center;\">\n    <strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><br \/>\n    Ao final desta aula, o aluno ser\u00e1 capaz de:\n  <\/p>\n<ol>\n<li><strong>Demonstrar<\/strong> a rela\u00e7\u00e3o entre a distribui\u00e7\u00e3o normal padr\u00e3o e a distribui\u00e7\u00e3o normal com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span><\/span> estabelecidos, por meio de uma substitui\u00e7\u00e3o integral.<\/li>\n<li><strong>Resolver<\/strong> problemas pr\u00e1ticos usando a tabela de distribui\u00e7\u00e3o normal padr\u00e3o.<\/li>\n<li><strong>Construir<\/strong> e usar uma tabela de distribui\u00e7\u00e3o normal padr\u00e3o usando Excel.<\/li>\n<\/ol>\n<p>  <center><br \/>\n    <strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n    <a href=\"#1\">O problema e a solu\u00e7\u00e3o da distribui\u00e7\u00e3o normal<\/a><br \/>\n    <a href=\"#2\">Como uso esta tabela?<\/a><br \/>\n    <a href=\"#3\">Exerc\u00edcios<\/a><br \/>\n  <\/center><br \/>\n  <center><br \/>\n    <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/GdoFK696k3s\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n  <\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>O problema e a solu\u00e7\u00e3o da distribui\u00e7\u00e3o normal<\/h2>\n<p style=\"text-align:justify;\">Uma das <a href=\"http:\/\/toposuranos.com\/material\/es\/distribuciones-continuas-de-probabilidad\/\" rel=\"noopener\" target=\"_blank\">distribui\u00e7\u00f5es cont\u00ednuas<\/a> de probabilidade mais utilizadas \u00e9 a distribui\u00e7\u00e3o normal. Uma vari\u00e1vel aleat\u00f3ria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> tem distribui\u00e7\u00e3o normal com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu,\\sigma<\/span><\/span> se satisfaz que<\/p>\n<p class=\"eq\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X\\leq x) =\\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^x \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{-\\frac{1}{2} \\left(\\frac{t-\\mu}{\\sigma}\\right)^2}dt <\/span><\/span><\/p>\n<p>e tamb\u00e9m <a href=\"https:\/\/www.youtube.com\/watch?v=kdxgrB1h98g&amp;t=1188s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">vimos<\/span><\/strong><\/a> que essa distribui\u00e7\u00e3o pode ser \u00abpadronizada\u00bb se fizermos na integral uma substitui\u00e7\u00e3o da forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle z= \\frac{t-\\mu}{\\sigma},<\/span><\/span> obtendo:<\/p>\n<p class=\"eq\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^{\\frac{x-\\mu}{\\sigma}} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{t^2}{2} }dt = \\Phi_{0,1}\\left(\\frac{x-\\mu}{\\sigma} \\right) <\/span><\/span><\/p>\n<p style=\"text-align:justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=GdoFK696k3s&amp;t=270s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Embora este processo de padroniza\u00e7\u00e3o<\/strong><\/span><\/a> simplifique a integral, ainda n\u00e3o \u00e9 suficiente para nos permitir calcular uma \u00fanica probabilidade para esta distribui\u00e7\u00e3o. Acontece que avaliar esta integral de forma anal\u00edtica n\u00e3o \u00e9 algo que possamos realizar; na verdade, n\u00e3o existem express\u00f5es alg\u00e9bricas que permitam expressar o resultado destas integrais e isso nos coloca em dificuldades. No entanto, se tivermos aproxima\u00e7\u00f5es num\u00e9ricas para alguns valores da fun\u00e7\u00e3o de distribui\u00e7\u00e3o normal padr\u00e3o, j\u00e1 podemos come\u00e7ar a fazer algumas estimativas. Esses valores est\u00e3o resumidos nesta <strong>tabela para a distribui\u00e7\u00e3o normal padr\u00e3o<\/strong> que constru\u00ed no Excel para voc\u00ea &gt;:D<\/p>\n<p><center><br \/>\n  <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26819 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26819 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb.png 784w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb-300x296.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/tabla-distribucion-normalb-768x758.png 768w\" sizes=\"(max-width: 784px) 100vw, 784px\" \/><\/noscript><br \/>\n  <em>Voc\u00ea pode baixar esta tabela em formato Excel no seguinte <a href=\"https:\/\/www.dropbox.com\/s\/9yff75b70wd75xn\/tabla%20-%20distribucion%20normal%20estandar.xlsx?dl=0\" rel=\"noopener\" target=\"_blank\">link.<\/a> Recomendo revisar este arquivo porque assim voc\u00ea aprender\u00e1 a montar sua pr\u00f3pria tabela caso n\u00e3o encontre uma \ud83d\ude42<\/em><br \/>\n<\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Como uso esta tabela?<\/h2>\n<p style=\"text-align:justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=GdoFK696k3s&amp;t=622s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>Para usar esta tabela sem dificuldade<\/strong><\/span><\/a> voc\u00ea deve lembrar que, al\u00e9m de indicar uma probabilidade acumulada, esta representa uma \u00e1rea sob a curva (da fun\u00e7\u00e3o de densidade). Isso pode ser visto representado na figura a seguir:<\/p>\n<p><center><br \/>\n  <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26820 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png\" alt=\"\" width=\"784\" height=\"774\" class=\"aligncenter size-full wp-image-26820 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab.png 784w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab-300x296.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/comousarlatablab-768x758.png 768w\" sizes=\"(max-width: 784px) 100vw, 784px\" \/><\/noscript><br \/>\n<\/center><\/p>\n<p style=\"text-align:justify;\">Compreendido isso, o pr\u00f3ximo passo \u00e9 explicar o uso em si da tabela. O que vemos aqui \u00e9 que o valor <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z<\/span><\/span> em que vamos avaliar a fun\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(z)<\/span><\/span> aparece separado em duas partes: na coluna, voc\u00ea encontrar\u00e1 sua expans\u00e3o at\u00e9 o primeiro d\u00edgito decimal, e na linha, voc\u00ea encontrar\u00e1 o segundo d\u00edgito decimal; de modo que, por exemplo: o n\u00famero <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z=1,72<\/span><\/span> pode ser encontrado como se estivesse formado por \u00abcoordenadas\u00bb: a vertical \u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1,7,<\/span><\/span> e a horizontal <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0,02.<\/span><\/span> Finalmente, a aproxima\u00e7\u00e3o num\u00e9rica de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(1,72)<\/span><\/span> aparecer\u00e1 no bloco de coordenadas \u00ab1,7\u00bb e \u00ab0,02\u00bb, que mostra um valor de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0,957284.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Exerc\u00edcios<\/h2>\n<ol>\n<li>Estime o valor de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(2,93)<\/span><\/span><\/li>\n<li>Estime o valor de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu=12,\\sigma=3}(11,5)<\/span><\/span><\/li>\n<li>Uma vari\u00e1vel aleat\u00f3ria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> tem distribui\u00e7\u00e3o normal <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N(\\mu=37,\\sigma=9).<\/span><\/span> Calcule <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(35 \\lt X \\lt 43).<\/span><\/span><\/li>\n<li>A quantidade de tomates que apodrecem diariamente na banca de um feirante tem uma m\u00e9dia de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu=50<\/span><\/span> com um desvio padr\u00e3o de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma=15.<\/span><\/span> Se o feirante leva tomates 3 vezes por semana, estime o n\u00famero de dias no m\u00eas em que mais de 60 tomates ir\u00e3o apodrecer.<\/li>\n<li>Assumindo que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X \\sim N(\\mu,\\sigma)<\/span><\/span> calcule as seguintes probabilidades:\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\mu - \\sigma \\leq X \\leq \\mu + \\sigma)<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X \\leq \\mu - \\sigma \\vee \\mu + \\sigma \\leq X)<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X \\leq \\mu - n\\sigma \\vee \\mu + n\\sigma \\leq X)<\/span><\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><center><br \/>\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6tEAj-_dd_c\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Como calcular a distribui\u00e7\u00e3o normal usando a tabela? Resumo Nesta aula, abordaremos o tema da distribui\u00e7\u00e3o normal, uma das distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais comuns. Analisaremos como uma vari\u00e1vel aleat\u00f3ria X com par\u00e2metros \u03bc e \u03c3 pode seguir uma distribui\u00e7\u00e3o normal e como essa distribui\u00e7\u00e3o pode ser padronizada por meio de uma substitui\u00e7\u00e3o na integral. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26821,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":15,"footnotes":""},"categories":[571,672],"tags":[],"class_list":["post-26826","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica-pt","category-probabilidades-e-estatistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Como calcular a distribui\u00e7\u00e3o normal usando a tabela? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda a usar a tabela padronizada da distribui\u00e7\u00e3o normal para calcular probabilidades e resolver problemas pr\u00e1ticos.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/pt\/como-calcular-a-distribuicao-normal-usando-a-tabela\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Como calcular a distribui\u00e7\u00e3o normal usando a tabela?\" \/>\n<meta property=\"og:description\" content=\"Aprenda a usar a tabela padronizada da distribui\u00e7\u00e3o normal para calcular probabilidades e resolver problemas pr\u00e1ticos.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/pt\/como-calcular-a-distribuicao-normal-usando-a-tabela\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-11-05T13:00:41+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-22T07:37:57+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/distnormal.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Como calcular a distribui\u00e7\u00e3o normal usando a tabela?\" \/>\n<meta name=\"twitter:description\" content=\"Aprenda a usar a tabela padronizada da distribui\u00e7\u00e3o normal para calcular probabilidades e resolver problemas pr\u00e1ticos.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/11\/distnormal.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/como-calcular-a-distribuicao-normal-usando-a-tabela\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/como-calcular-a-distribuicao-normal-usando-a-tabela\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Como calcular a distribui\u00e7\u00e3o normal usando a tabela?\",\"datePublished\":\"2021-11-05T13:00:41+00:00\",\"dateModified\":\"2024-05-22T07:37:57+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/como-calcular-a-distribuicao-normal-usando-a-tabela\\\/\"},\"wordCount\":862,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/como-calcular-a-distribuicao-normal-usando-a-tabela\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/11\\\/distnormal.jpg\",\"articleSection\":[\"Matem\u00e1tica\",\"Probabilidades e Estat\u00edstica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/como-calcular-a-distribuicao-normal-usando-a-tabela\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/como-calcular-a-distribuicao-normal-usando-a-tabela\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/como-calcular-a-distribuicao-normal-usando-a-tabela\\\/\",\"name\":\"Como calcular a distribui\u00e7\u00e3o normal usando a tabela? 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