{"id":278,"date":"2023-08-03T05:03:01","date_gmt":"2023-08-03T05:03:01","guid":{"rendered":"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-diferenca-de-medias\/"},"modified":"2023-08-03T05:03:01","modified_gmt":"2023-08-03T05:03:01","slug":"distribuicao-amostral-da-diferenca-de-medias","status":"publish","type":"post","link":"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-diferenca-de-medias\/","title":{"rendered":"Distribui\u00e7\u00e3o amostral de diferen\u00e7a de m\u00e9dias"},"content":{"rendered":"<p>Este artigo explica o que \u00e9 a distribui\u00e7\u00e3o amostral da diferen\u00e7a entre duas m\u00e9dias nas estat\u00edsticas. Voc\u00ea tamb\u00e9m encontrar\u00e1 a f\u00f3rmula de distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias e, al\u00e9m disso, 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-diferencia-de-medias\"><\/span> Qual \u00e9 a distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias<\/strong> \u00e9 a distribui\u00e7\u00e3o que resulta do c\u00e1lculo das diferen\u00e7as entre as m\u00e9dias de todas as amostras poss\u00edveis de duas popula\u00e7\u00f5es diferentes.<\/p>\n<p> Ou seja, para obter a distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias, devem ser selecionadas todas as amostras poss\u00edveis de duas popula\u00e7\u00f5es de estudo, em seguida \u00e9 calculada a m\u00e9dia de cada amostra selecionada e, por fim, calculada a diferen\u00e7a entre todas as m\u00e9dias das duas popula\u00e7\u00f5es. Assim, o conjunto de valores obtidos ap\u00f3s a aplica\u00e7\u00e3o de todas essas opera\u00e7\u00f5es forma a distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/distribution-de-lechantillon-de-la-difference-de-moyennes.png\" alt=\"distribui\u00e7\u00e3o amostral de diferen\u00e7a de m\u00e9dias\" class=\"wp-image-5927\" width=\"561\" height=\"237\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> A diferen\u00e7a de distribui\u00e7\u00e3o amostral de m\u00e9dias \u00e9 usada para calcular a probabilidade de que a diferen\u00e7a entre duas <a href=\"https:\/\/statorials.org\/pt\/media-amostral\/\">m\u00e9dias de amostras selecionadas aleatoriamente<\/a> de duas popula\u00e7\u00f5es diferentes esteja pr\u00f3xima da diferen\u00e7a nas m\u00e9dias populacionais. <\/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\/distribuicao-amostral-da-media\/\">Distribui\u00e7\u00e3o amostral da m\u00e9dia<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-distribucion-muestral-de-la-diferencia-de-medias\"><\/span> F\u00f3rmula para a distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Se o tamanho da amostra for grande o suficiente (n <sub>1<\/sub> \u226530 e n <sub>2<\/sub> \u226530), a distribui\u00e7\u00e3o amostral da diferen\u00e7a nas m\u00e9dias segue uma distribui\u00e7\u00e3o normal. Mais precisamente, os par\u00e2metros desta distribui\u00e7\u00e3o s\u00e3o calculados da seguinte forma:<\/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-90c67b74b4e9326b7869d641a59725d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\displaystyle \\mu_{\\overline{x_1}-\\overline{x_2}}=\\mu_1-\\mu_2 \\qquad \\sigma_{\\overline{x_1}-\\overline{x_2}}=\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}\\\\[6ex]\\displaystyle N_{\\overline{x_1}-\\overline{x_2}}\\left(\\mu_1-\\mu_2, \\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}\\right) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"151\" width=\"328\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong>Nota:<\/strong> Se ambas as popula\u00e7\u00f5es forem <a href=\"https:\/\/statorials.org\/pt\/distribuicao-normal\/\">distribui\u00e7\u00f5es normais<\/a> , ent\u00e3o a distribui\u00e7\u00e3o amostral da diferen\u00e7a nas m\u00e9dias segue uma distribui\u00e7\u00e3o normal, independentemente do tamanho da amostra.<\/p>\n<p> Portanto, como a distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias \u00e9 definida por uma distribui\u00e7\u00e3o normal, a <strong>f\u00f3rmula para calcular a estat\u00edstica da distribui\u00e7\u00e3o amostral da diferen\u00e7a<\/strong> de m\u00e9dias \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-767964a3f07b303178ee08ec191eef43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{(\\overline{x_1}-\\overline{x_2})-(\\mu_1-\\mu_2)}{\\displaystyle\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"203\" style=\"vertical-align: -52px;\"><\/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-a4071a38558726a684ed069430c89fe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_i}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 a m\u00e9dia da amostra i.<\/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-78e04dacbf6a47efcbdcc0417020dcbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu_i\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\"><\/p>\n<p> \u00e9 a m\u00e9dia da popula\u00e7\u00e3o i.<\/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-dc8c8f782c0ed8b7925012b60e174fa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o desvio padr\u00e3o da popula\u00e7\u00e3o i.<\/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-5f087375b50e0b49186779714206626b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o tamanho da amostra i.<\/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<p> Observe que amostras de popula\u00e7\u00f5es diferentes podem ter tamanhos de amostra diferentes. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-resuelto-de-la-distribucion-muestral-de-la-diferencia-de-medias\"><\/span> Exemplo concreto de distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos a defini\u00e7\u00e3o de distribui\u00e7\u00e3o amostral por diferen\u00e7a de m\u00e9dias e qual \u00e9 sua f\u00f3rmula, veremos um exemplo passo a passo para finalizar a compreens\u00e3o do conceito de distribui\u00e7\u00e3o amostral por diferen\u00e7a de m\u00e9dias.<\/p>\n<ul>\n<li> Num estudo estat\u00edstico, queremos analisar a diferen\u00e7a entre a altura de meninos e meninas de uma determinada idade. Sabemos que a distribui\u00e7\u00e3o que define a popula\u00e7\u00e3o de meninos desta idade tem m\u00e9dia de 157 cm e desvio padr\u00e3o de 9 cm e, por outro lado, a distribui\u00e7\u00e3o que define a popula\u00e7\u00e3o de meninas desta idade tem m\u00e9dia de 148 cm e um desvio padr\u00e3o de 7 cm. Se forem selecionadas uma amostra de 30 meninos desta idade e uma amostra de 35 meninas desta idade, qual \u00e9 a probabilidade de a altura m\u00e9dia da amostra de meninos ser 12 cm maior que a altura m\u00e9dia da amostra de meninas?<\/li>\n<\/ul>\n<p> Para resolver este problema, a primeira coisa a fazer \u00e9 calcular a estat\u00edstica da diferen\u00e7a de m\u00e9dias da distribui\u00e7\u00e3o amostral. Portanto, aplicamos a f\u00f3rmula vista 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-3897bf456851ccd8de2c501bd779dd02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{(\\overline{x_1}-\\overline{x_2})-(\\mu_1-\\mu_2)}{\\displaystyle\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}\\right)}=\\cfrac{12-(157-148)}{\\displaystyle\\sqrt{\\frac{9^2}{30}+\\frac{7^2}{35}}}=1,48\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"415\" style=\"vertical-align: -52px;\"><\/p>\n<\/p>\n<p> Portanto, a probabilidade de a altura m\u00e9dia amostral dos meninos ser 12 cm maior que a altura m\u00e9dia das meninas equivale \u00e0 probabilidade de a vari\u00e1vel Z ser maior que 1,48.<\/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-683cd2a72980115d8d18c906d619370a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P[(\\overline{x_1}-\\overline{x_2})>12]=P[Z>1,48]&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;253&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<p> Procuramos, portanto, a probabilidade de Z&gt;1,48 na <a href=\"https:\/\/statorials.org\/pt\/tabela-de-distribuicao-normal\/\">tabela de Z<\/a> :<\/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-5b83fdc11bd2320249679c4752660451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P[(\\overline{x_1}-\\overline{x_2})>12]=P[Z>1,48]=0,0694&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;331&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<p> Em suma, a probabilidade de a altura m\u00e9dia amostral dos meninos ser 12 cm maior que a altura m\u00e9dia das meninas \u00e9 de 6,94%.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Este artigo explica o que \u00e9 a distribui\u00e7\u00e3o amostral da diferen\u00e7a entre duas m\u00e9dias nas estat\u00edsticas. Voc\u00ea tamb\u00e9m encontrar\u00e1 a f\u00f3rmula de distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias e, al\u00e9m disso, um exerc\u00edcio resolvido passo a passo. Qual \u00e9 a distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias? A distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias \u00e9 a [&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-278","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 diferen\u00e7a de m\u00e9dias<\/title>\n<meta name=\"description\" content=\"Aqui voc\u00ea encontrar\u00e1 qual \u00e9 a diferen\u00e7a de m\u00e9dias na distribui\u00e7\u00e3o amostral, sua f\u00f3rmula e um exemplo passo a passo.\" \/>\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-diferenca-de-medias\/\" \/>\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 diferen\u00e7a de m\u00e9dias\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea encontrar\u00e1 qual \u00e9 a diferen\u00e7a de m\u00e9dias na distribui\u00e7\u00e3o amostral, sua f\u00f3rmula e um exemplo passo a passo.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-diferenca-de-medias\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-03T05:03:01+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/distribution-de-lechantillon-de-la-difference-de-moyennes.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-diferenca-de-medias\/\",\"url\":\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-diferenca-de-medias\/\",\"name\":\"\u25b7 Distribui\u00e7\u00e3o amostral da diferen\u00e7a de m\u00e9dias\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pt\/#website\"},\"datePublished\":\"2023-08-03T05:03:01+00:00\",\"dateModified\":\"2023-08-03T05:03:01+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pt\/#\/schema\/person\/e08f98e8db95e0aa9c310e1b27c9c666\"},\"description\":\"Aqui voc\u00ea encontrar\u00e1 qual \u00e9 a diferen\u00e7a de m\u00e9dias na distribui\u00e7\u00e3o amostral, sua f\u00f3rmula e um exemplo passo a passo.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-diferenca-de-medias\/#breadcrumb\"},\"inLanguage\":\"pt-PT\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-diferenca-de-medias\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pt\/distribuicao-amostral-da-diferenca-de-medias\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Lar\",\"item\":\"https:\/\/statorials.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Distribui\u00e7\u00e3o amostral de diferen\u00e7a de m\u00e9dias\"}]},{\"@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. Com vasta experi\u00eancia e conhecimento na \u00e1rea de estat\u00edstica, estou empenhado em compartilhar meu conhecimento para capacitar os alunos por meio de Statorials. 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