{"id":241,"date":"2023-08-03T17:38:49","date_gmt":"2023-08-03T17:38:49","guid":{"rendered":"https:\/\/statorials.org\/pt\/intervalo-de-confianca-para-a-media\/"},"modified":"2023-08-03T17:38:49","modified_gmt":"2023-08-03T17:38:49","slug":"intervalo-de-confianca-para-a-media","status":"publish","type":"post","link":"https:\/\/statorials.org\/pt\/intervalo-de-confianca-para-a-media\/","title":{"rendered":"Intervalo de confian\u00e7a para a m\u00e9dia"},"content":{"rendered":"<p>Este artigo explica o que \u00e9 o intervalo de confian\u00e7a para a m\u00e9dia nas estat\u00edsticas e para que ele \u00e9 usado. Da mesma forma, voc\u00ea descobrir\u00e1 como calcular o intervalo de confian\u00e7a da m\u00e9dia, bem como um exerc\u00edcio passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-el-intervalo-de-confianza-para-la-media\"><\/span> Qual \u00e9 o intervalo de confian\u00e7a da m\u00e9dia?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O <strong>intervalo de confian\u00e7a para a m\u00e9dia<\/strong> \u00e9 um intervalo que fornece uma faixa de valores permitidos para a m\u00e9dia de uma popula\u00e7\u00e3o. Por outras palavras, o intervalo de confian\u00e7a para a m\u00e9dia d\u00e1-nos um valor m\u00e1ximo e um valor m\u00ednimo entre os quais liga o valor da m\u00e9dia da popula\u00e7\u00e3o a uma margem de erro.<\/p>\n<p> Por exemplo, se o intervalo de confian\u00e7a de 95% para uma m\u00e9dia populacional for (6,10), isso significa que 95% das vezes a m\u00e9dia populacional estar\u00e1 entre 6 e 10.<\/p>\n<p> Portanto, o intervalo de confian\u00e7a da m\u00e9dia \u00e9 usado para estimar dois valores entre os quais se encontra uma m\u00e9dia populacional. Assim, o intervalo de confian\u00e7a da m\u00e9dia \u00e9 muito \u00fatil para aproximar a m\u00e9dia de uma popula\u00e7\u00e3o quando todos os seus valores s\u00e3o desconhecidos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-intervalo-de-confianza-para-la-media\"><\/span> F\u00f3rmula do intervalo de confian\u00e7a para a m\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Supondo que o processo de <a href=\"https:\/\/statorials.org\/pt\/tipificacao\/\">inser\u00e7\u00e3o de uma vari\u00e1vel<\/a> seja assim:<\/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-f93a5e76c38b2e2556bd62ba66db5947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Z=\\cfrac{X-\\mu}{\\displaystyle\\frac{\\sigma}{\\sqrt{n}}} \\sim N(0,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"165\" style=\"vertical-align: -34px;\"><\/p>\n<\/p>\n<p> O intervalo de confian\u00e7a da m\u00e9dia \u00e9 calculado somando e subtraindo da m\u00e9dia amostral o valor de Z <sub>\u03b1\/2<\/sub> multiplicado pelo desvio padr\u00e3o (\u03c3) e dividido pela raiz quadrada do tamanho da amostra (n). Portanto, a f\u00f3rmula para calcular o intervalo de confian\u00e7a da m\u00e9dia \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-985a374c69b468deb3d6d159fcbe6de5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-z_{\\alpha\/2}\\cdot \\frac{\\sigma}{\\sqrt{n}} \\ , \\ \\overline{x}+z_{\\alpha\/2}\\cdot \\frac{\\sigma}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"251\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Para amostras grandes e n\u00edvel de confian\u00e7a de 95%, o valor cr\u00edtico \u00e9 Z <sub>\u03b1\/2<\/sub> = 1,96 e para n\u00edvel de confian\u00e7a de 99%, o valor cr\u00edtico \u00e9 Z <sub>\u03b1\/2<\/sub> = 2,576.<\/p>\n<p> A f\u00f3rmula acima \u00e9 usada quando a vari\u00e2ncia populacional \u00e9 conhecida. Contudo, se a vari\u00e2ncia populacional for desconhecida, o que \u00e9 o caso mais frequente, <strong>o intervalo de confian\u00e7a da m\u00e9dia \u00e9 calculado utilizando a seguinte f\u00f3rmula:<\/strong><\/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-e40e2c40fbfe4f5aa96c21ff979aae5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\ , \\ \\overline{x}+t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/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-33bd5d525ddf37600aef97b0c8c08f94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t_{\\alpha\/2}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"29\" style=\"vertical-align: -8px;\"><\/p>\n<p> \u00e9 o valor da distribui\u00e7\u00e3o t de Student de n-1 graus de liberdade com probabilidade de \u03b1\/2.<\/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 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-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<\/ul>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/intervalle-de-confiance.png\" alt=\"intervalo de confian\u00e7a\" class=\"wp-image-5249\" width=\"351\" height=\"339\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-del-calculo-de-un-intervalo-de-confianza-para-la-media\"><\/span> Exemplo de c\u00e1lculo de um intervalo de confian\u00e7a para a m\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para que voc\u00ea possa ver como \u00e9 calculado o intervalo de confian\u00e7a para a m\u00e9dia de uma popula\u00e7\u00e3o, deixamos abaixo um exemplo resolvido passo a passo.<\/p>\n<ul>\n<li> Temos uma amostra de 8 observa\u00e7\u00f5es com os valores mostrados abaixo. Qual \u00e9 o intervalo de confian\u00e7a da m\u00e9dia da popula\u00e7\u00e3o no n\u00edvel de confian\u00e7a de 95%?<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"> 206 203 201 212<br \/> 194 176 208 201<\/p>\n<p> Como vimos na se\u00e7\u00e3o anterior, a f\u00f3rmula para obter o intervalo de confian\u00e7a de uma m\u00e9dia populacional quando n\u00e3o conhecemos o desvio padr\u00e3o populacional \u00e9 a seguinte:<\/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-e40e2c40fbfe4f5aa96c21ff979aae5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\ , \\ \\overline{x}+t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Portanto, para determinar o intervalo de confian\u00e7a da m\u00e9dia, devemos primeiro calcular a m\u00e9dia amostral e o desvio padr\u00e3o. <\/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-e4db3aa9e4a141ef4ac81e1816a243ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\mu =200,13 \\\\[4ex]s=11,13\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"87\" style=\"vertical-align: 0px;\"><\/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\/media-aritmetica\/\">Calculadora de m\u00e9dia aritm\u00e9tica<\/a><br \/> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <a href=\"https:\/\/statorials.org\/pt\/desvio-padrao-ou-desvio-padrao\/\">Calculadora de desvio padr\u00e3o<\/a><\/div>\n<p> Como queremos encontrar o intervalo de confian\u00e7a com n\u00edvel de confian\u00e7a de 1-\u03b1=95% e o tamanho da amostra \u00e9 8, precisamos acessar a tabela de distribui\u00e7\u00e3o t de Student e ver qual valor corresponde a t <sub>0,025|7<\/sub> . <\/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-776a555edd1d064cf29a3a354caa5325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1-\\alpha=0,95 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\alpha=0,05 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black}\\ \\alpha\/2=0,025\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"534\" style=\"vertical-align: -5px;\"><\/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-2caa8c4576ebc261acfdb1341dc4567d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}t_{\\alpha\/2| n-1}= \\ \\color{orange}\\bm{?}\\\\[4ex]t_{0,025| 7}=2,365\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"72\" width=\"152\" style=\"vertical-align: 0px;\"><\/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-t-do-aluno\/\">Valores da tabela de distribui\u00e7\u00e3o t de Student<\/a><\/div>\n<p> Ent\u00e3o aplicamos a f\u00f3rmula do intervalo de confian\u00e7a para a m\u00e9dia e realizamos os c\u00e1lculos para encontrar os limites do intervalo: <\/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-e40e2c40fbfe4f5aa96c21ff979aae5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\overline{x}-t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\ , \\ \\overline{x}+t_{\\alpha\/2}\\cdot \\frac{s}{\\sqrt{n}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/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-7c9bc920ed22b19f5b15528eef268878_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(200,13-2,365\\cdot \\frac{11,13}{\\sqrt{8}} \\ , \\ 200,13+2,365\\cdot \\frac{11,13}{\\sqrt{8}} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"394\" style=\"vertical-align: -17px;\"><\/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-322fdcebe92ea2e3868440918d39d431_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(190,82 \\ , \\ 209,43 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Concluindo, o intervalo de confian\u00e7a calculado diz-nos que com um n\u00edvel de confian\u00e7a de 95%, a m\u00e9dia da popula\u00e7\u00e3o estar\u00e1 entre 190,82 e 209,43.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Este artigo explica o que \u00e9 o intervalo de confian\u00e7a para a m\u00e9dia nas estat\u00edsticas e para que ele \u00e9 usado. Da mesma forma, voc\u00ea descobrir\u00e1 como calcular o intervalo de confian\u00e7a da m\u00e9dia, bem como um exerc\u00edcio passo a passo. Qual \u00e9 o intervalo de confian\u00e7a da m\u00e9dia? O intervalo de confian\u00e7a para 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-241","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 Intervalo de confian\u00e7a para a m\u00e9dia - Probabilidade e estat\u00edstica<\/title>\n<meta name=\"description\" content=\"Este artigo explica o que \u00e9 o intervalo de confian\u00e7a para a m\u00e9dia nas estat\u00edsticas e para que ele \u00e9 usado. 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