{"id":276,"date":"2023-08-03T05:34:09","date_gmt":"2023-08-03T05:34:09","guid":{"rendered":"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/"},"modified":"2023-08-03T05:34:09","modified_gmt":"2023-08-03T05:34:09","slug":"distribuicao-amostral-da-media","status":"publish","type":"post","link":"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/","title":{"rendered":"Distribui\u00e7\u00e3o amostral da m\u00e9dia"},"content":{"rendered":"<p>Este artigo explica o que \u00e9 a distribui\u00e7\u00e3o amostral da m\u00e9dia nas estat\u00edsticas. Voc\u00ea tamb\u00e9m encontrar\u00e1 a f\u00f3rmula de distribui\u00e7\u00e3o amostral m\u00e9dia e um exerc\u00edcio resolvido passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-distribucion-muestral-de-la-media\"><\/span> Qual \u00e9 a distribui\u00e7\u00e3o amostral da m\u00e9dia?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>distribui\u00e7\u00e3o amostral da m\u00e9dia<\/strong> (ou <strong>distribui\u00e7\u00e3o amostral das m\u00e9dias<\/strong> ) \u00e9 a distribui\u00e7\u00e3o que resulta do c\u00e1lculo da m\u00e9dia amostral de cada amostra poss\u00edvel de uma popula\u00e7\u00e3o. Ou seja, o conjunto de <a href=\"https:\/\/statorials.org\/pt\/media-amostral\/\">m\u00e9dias amostrais<\/a> de todas as amostras poss\u00edveis de uma popula\u00e7\u00e3o forma a distribui\u00e7\u00e3o amostral da m\u00e9dia.<\/p>\n<p> Ou em outras palavras, se estudarmos todas as amostras que podem ser retiradas de uma popula\u00e7\u00e3o e calcularmos a m\u00e9dia de cada uma das amostras, o conjunto de valores calculados forma uma distribui\u00e7\u00e3o amostral da m\u00e9dia amostral.<\/p>\n<p> Nas estat\u00edsticas, a distribui\u00e7\u00e3o amostral da m\u00e9dia \u00e9 usada para calcular a probabilidade de aproxima\u00e7\u00e3o do valor da m\u00e9dia da popula\u00e7\u00e3o ao analisar uma \u00fanica amostra. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-distribucion-muestral-de-la-media\"><\/span> F\u00f3rmula para a distribui\u00e7\u00e3o amostral da m\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dada uma popula\u00e7\u00e3o que segue uma <a href=\"https:\/\/statorials.org\/pt\/distribuicao-normal\/\">distribui\u00e7\u00e3o de probabilidade normal<\/a> com m\u00e9dia<\/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-05d9eae892416bd34247a25207f8b718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"><\/p>\n<p> e desvio padr\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-eaaf379fee5e67946f3fedf5631047b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> e amostras de tamanho s\u00e3o extra\u00eddas<\/p>\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> , a distribui\u00e7\u00e3o amostral da m\u00e9dia tamb\u00e9m ser\u00e1 definida por uma distribui\u00e7\u00e3o normal com as seguintes caracter\u00edsticas:<\/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-44571aa7337b095ab9c9fa1f746e93a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\mu_{\\overline{x}}=\\mu \\qquad \\sigma_{\\overline{x}}=\\cfrac{\\sigma}{\\sqrt{n}}\\\\[4ex]\\displaystyle N_{\\overline{x}}\\left(\\mu, \\frac{\\sigma}{\\sqrt{n}}\\right) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ouro<\/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-8ed084decbdfb365889aae767cf63e81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_{\\overline{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"20\" style=\"vertical-align: -4px;\"><\/p>\n<p> \u00e9 a m\u00e9dia da distribui\u00e7\u00e3o amostral da m\u00e9dia e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-9067e28d896c7e5278763081c6cc40d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_{\\overline{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o seu desvio padr\u00e3o. Al\u00e9m disso,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-7d78e2a2f2fae99a53eb087263cbb478_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\sigma}{\\sqrt{n}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"26\" style=\"vertical-align: -16px;\"><\/p>\n<p> \u00e9 o erro padr\u00e3o da distribui\u00e7\u00e3o amostral.<\/p>\n<p> <strong>Nota:<\/strong> Se a popula\u00e7\u00e3o n\u00e3o seguir uma distribui\u00e7\u00e3o normal, mas o tamanho da amostra for grande (n&gt;30), a distribui\u00e7\u00e3o amostral da m\u00e9dia tamb\u00e9m pode ser aproximada da distribui\u00e7\u00e3o normal anterior pelo teorema do limite central.<\/p>\n<p> Portanto, como a distribui\u00e7\u00e3o amostral da m\u00e9dia segue uma distribui\u00e7\u00e3o normal, a <strong>f\u00f3rmula para calcular qualquer probabilidade relacionada \u00e0 m\u00e9dia amostral<\/strong> \u00e9:<\/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-ecd8bcb78b739c50d01b8bad563e5cb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{\\overline{x}-\\mu}{\\displaystyle\\frac{\\sigma}{\\sqrt{n}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"81\" style=\"vertical-align: -34px;\"><\/p>\n<\/p>\n<p style=\"margin-bottom:5px\"> Ouro:<\/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> \u00e9 a m\u00e9dia amostral.<\/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-05d9eae892416bd34247a25207f8b718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"><\/p>\n<p> Esta \u00e9 a m\u00e9dia da popula\u00e7\u00e3o.<\/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> \u00e9 o desvio padr\u00e3o da popula\u00e7\u00e3o.<\/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> \u00e9 o tamanho da amostra.<\/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> \u00e9 uma vari\u00e1vel definida pela distribui\u00e7\u00e3o normal padr\u00e3o N(0,1). <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-resuelto-de-la-distribucion-muestral-de-la-media\"><\/span> Exemplo do mundo real da distribui\u00e7\u00e3o amostral da m\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de ver a defini\u00e7\u00e3o de distribui\u00e7\u00e3o amostral de m\u00e9dia e quais s\u00e3o suas f\u00f3rmulas associadas, vamos resolver um exemplo para entender melhor o conceito.<\/p>\n<ul>\n<li> O peso dos universit\u00e1rios segue distribui\u00e7\u00e3o normal com m\u00e9dia de 68 kg e desvio padr\u00e3o de 9 kg. Determinado:\n<ol>\n<li> Qual \u00e9 a probabilidade de que a m\u00e9dia de uma amostra aleat\u00f3ria de 25 alunos seja inferior a 66 kg?<\/li>\n<li> Se forem colhidas 300 amostras com tamanho de 25 alunos cada, quantas m\u00e9dias amostrais ter\u00e3o valor inferior a 66 kg?<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<p> Em primeiro lugar devemos calcular o valor da estat\u00edstica correspondente, para isso aplicamos a f\u00f3rmula que vimos acima:<\/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-b132f95081cac0e4028d5e8148514671_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle Z=\\frac{\\overline{x}-\\mu}{\\displaystyle\\frac{\\sigma}{\\sqrt{n}}}=\\frac{66-68}{\\displaystyle\\frac{9}{\\sqrt{25}}}=-1,11\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"238\" style=\"vertical-align: -38px;\"><\/p>\n<\/p>\n<p> A probabilidade que procuramos \u00e9, portanto, aquela correspondente ao valor Z=-1,11 da cauda esquerda da distribui\u00e7\u00e3o normal padr\u00e3o, que pode ser facilmente obtida na tabela de probabilidade Z. Portanto, usamos a tabela Z para determinar a probabilidade que o problema nos pede: <\/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-0db3a516f2be44a78c4f69958a9f0d51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P[\\overline{x}\\leq 66]=P[Z\\leq -1,11] =0,1335\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"283\" style=\"vertical-align: -5px;\"><\/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>Veja:<\/strong> <a href=\"https:\/\/statorials.org\/pt\/tabela-de-distribuicao-normal\/\">Tabela Z<\/a><\/div>\n<p> Agora que sabemos a probabilidade de que a m\u00e9dia de uma amostra aleat\u00f3ria seja inferior a 66 kg, para saber a m\u00e9dia do n\u00famero de amostras inferior a 66 kg tomando 300 amostras iguais, precisamos multiplicar a probabilidade calculada pelo n\u00famero total de amostras colhidas:<\/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-7d2519cbeabf4b272e6f5a1463472768_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0,1335\\cdot 300 =40,05 \\approx 40\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"200\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Portanto, aproximadamente 40 das amostras extra\u00eddas ter\u00e3o em m\u00e9dia menos de 66 kg.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Este artigo explica o que \u00e9 a distribui\u00e7\u00e3o amostral da m\u00e9dia nas estat\u00edsticas. Voc\u00ea tamb\u00e9m encontrar\u00e1 a f\u00f3rmula de distribui\u00e7\u00e3o amostral m\u00e9dia e um exerc\u00edcio resolvido passo a passo. Qual \u00e9 a distribui\u00e7\u00e3o amostral da m\u00e9dia? A distribui\u00e7\u00e3o amostral da m\u00e9dia (ou distribui\u00e7\u00e3o amostral das m\u00e9dias ) \u00e9 a distribui\u00e7\u00e3o que resulta do c\u00e1lculo da [&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-276","post","type-post","status-publish","format-standard","hentry","category-estatisticas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Distribui\u00e7\u00e3o amostral da m\u00e9dia<\/title>\n<meta name=\"description\" content=\"Aqui voc\u00ea encontrar\u00e1 o que \u00e9 a distribui\u00e7\u00e3o amostral da m\u00e9dia, sua f\u00f3rmula e um exemplo concreto da distribui\u00e7\u00e3o amostral da m\u00e9dia.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/\" \/>\n<meta property=\"og:locale\" content=\"pt_PT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Distribui\u00e7\u00e3o amostral da m\u00e9dia\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea encontrar\u00e1 o que \u00e9 a distribui\u00e7\u00e3o amostral da m\u00e9dia, sua f\u00f3rmula e um exemplo concreto da distribui\u00e7\u00e3o amostral da m\u00e9dia.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-03T05:34:09+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-05d9eae892416bd34247a25207f8b718_l3.png\" \/>\n<meta name=\"author\" content=\"Dr. benjamim anderson\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Dr. benjamim anderson\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tempo estimado de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/\",\"url\":\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/\",\"name\":\"\u25b7 Distribui\u00e7\u00e3o amostral da m\u00e9dia\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pt\/#website\"},\"datePublished\":\"2023-08-03T05:34:09+00:00\",\"dateModified\":\"2023-08-03T05:34:09+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pt\/#\/schema\/person\/e08f98e8db95e0aa9c310e1b27c9c666\"},\"description\":\"Aqui voc\u00ea encontrar\u00e1 o que \u00e9 a distribui\u00e7\u00e3o amostral da m\u00e9dia, sua f\u00f3rmula e um exemplo concreto da distribui\u00e7\u00e3o amostral da m\u00e9dia.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/#breadcrumb\"},\"inLanguage\":\"pt-PT\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-media\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Lar\",\"item\":\"https:\/\/statorials.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Distribui\u00e7\u00e3o amostral da m\u00e9dia\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/statorials.org\/pt\/#website\",\"url\":\"https:\/\/statorials.org\/pt\/\",\"name\":\"Statorials\",\"description\":\"O seu guia para a literacia estat\u00edstica!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/statorials.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-PT\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/statorials.org\/pt\/#\/schema\/person\/e08f98e8db95e0aa9c310e1b27c9c666\",\"name\":\"Dr. benjamim anderson\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-PT\",\"@id\":\"https:\/\/statorials.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/statorials.org\/pt\/wp-content\/uploads\/2023\/10\/Dr.-Benjamin-Anderson-96x96.jpg\",\"contentUrl\":\"https:\/\/statorials.org\/pt\/wp-content\/uploads\/2023\/10\/Dr.-Benjamin-Anderson-96x96.jpg\",\"caption\":\"Dr. benjamim anderson\"},\"description\":\"Ol\u00e1, sou Benjamin, um professor aposentado de estat\u00edstica que se tornou professor dedicado na Statorials. 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