{"id":259,"date":"2023-08-03T11:26:47","date_gmt":"2023-08-03T11:26:47","guid":{"rendered":"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/"},"modified":"2023-08-03T11:26:47","modified_gmt":"2023-08-03T11:26:47","slug":"teste-de-hipotese-para-a-media","status":"publish","type":"post","link":"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/","title":{"rendered":"Teste de hip\u00f3tese para a m\u00e9dia"},"content":{"rendered":"<p>Este artigo explica o que \u00e9 o teste de hip\u00f3tese para a m\u00e9dia nas estat\u00edsticas. Assim, voc\u00ea encontrar\u00e1 a f\u00f3rmula do teste de hip\u00f3tese da m\u00e9dia 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-prueba-de-hipotesis-para-la-media\"><\/span> O que \u00e9 teste de hip\u00f3tese para a m\u00e9dia?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>O teste de hip\u00f3teses para a m\u00e9dia<\/strong> \u00e9 um m\u00e9todo estat\u00edstico usado para rejeitar ou rejeitar a hip\u00f3tese nula de uma m\u00e9dia populacional.<\/p>\n<p> Mais especificamente, o teste de hip\u00f3tese para a m\u00e9dia envolve calcular a estat\u00edstica de teste e compar\u00e1-la com o valor cr\u00edtico para rejeitar ou n\u00e3o a hip\u00f3tese nula.<\/p>\n<p> Deve-se notar que os testes de hip\u00f3teses t\u00eam nomes diferentes; em estat\u00edstica, eles tamb\u00e9m s\u00e3o chamados de contrastes de hip\u00f3teses, testes de hip\u00f3teses ou testes de signific\u00e2ncia. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-prueba-de-hipotesis-para-la-media\"><\/span> F\u00f3rmula de teste de hip\u00f3teses para m\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir veremos como a estat\u00edstica do teste de hip\u00f3tese para a m\u00e9dia \u00e9 calculada. No entanto, a f\u00f3rmula varia ligeiramente dependendo se a vari\u00e2ncia \u00e9 conhecida ou n\u00e3o, ent\u00e3o veremos primeiro como isso \u00e9 feito quando a vari\u00e2ncia \u00e9 conhecida e depois quando a vari\u00e2ncia \u00e9 desconhecida.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"con-varianza-conocida\"><\/span> Com desvio conhecido<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>f\u00f3rmula da hip\u00f3tese de teste para a m\u00e9dia com vari\u00e2ncia conhecida<\/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-fdf1817a07bfb8c1eda9a8d8ccd8c828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle Z=\\frac{\\overline{x}-\\mu}{\\displaystyle \\frac{\\sigma}{\\sqrt{n}}} \" title=\"Rendered by QuickLaTeX.com\" height=\"57\" 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>\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 a estat\u00edstica do teste de hip\u00f3tese para a m\u00e9dia.<\/li>\n<li>\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>\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> \u00e9 o valor m\u00e9dio proposto.<\/li>\n<li>\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> \u00e9 o desvio padr\u00e3o da popula\u00e7\u00e3o.<\/li>\n<li>\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<p> Uma vez calculada a estat\u00edstica do teste de hip\u00f3tese para a m\u00e9dia, o resultado deve ser interpretado para rejeitar ou rejeitar a hip\u00f3tese nula:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Se o teste de hip\u00f3tese para a m\u00e9dia for bilateral, a hip\u00f3tese nula \u00e9 rejeitada se o valor absoluto da estat\u00edstica for maior que o valor cr\u00edtico Z <sub>\u03b1\/2<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Se o teste de hip\u00f3tese para a m\u00e9dia corresponder \u00e0 cauda direita, a hip\u00f3tese nula ser\u00e1 rejeitada se a estat\u00edstica for maior que o valor cr\u00edtico Z <sub>\u03b1<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Se o teste de hip\u00f3tese para a m\u00e9dia corresponder \u00e0 cauda esquerda, a hip\u00f3tese nula ser\u00e1 rejeitada se a estat\u00edstica for menor que o valor cr\u00edtico -Z <sub>\u03b1<\/sub> .<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0e2ccadfc369eb7543b8f86dfccc528e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}H_1: \\mu\\neq \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } |Z|>Z_{\\alpha\/2} \\text{ se rechaza } H_0\\\\[3ex]H_1: \\mu> \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } Z>Z_{\\alpha} \\text{ se rechaza } H_0\\\\[3ex]H_1: \\mu< \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } Z<-Z_{\\alpha} \\text{ se rechaza } H_0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"440\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Neste caso, os valores cr\u00edticos s\u00e3o obtidos da <a href=\"https:\/\/statorials.org\/pt\/tabela-de-distribuicao-normal\/\">tabela de distribui\u00e7\u00e3o normal padronizada<\/a> .<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"con-varianza-desconocida\"><\/span> Com varia\u00e7\u00e3o desconhecida<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>f\u00f3rmula da hip\u00f3tese de teste para a m\u00e9dia com vari\u00e2ncia desconhecida<\/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-f8277d83b2d28325f25f5d118486200f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\overline{x}-\\mu}{\\displaystyle \\frac{s}{\\sqrt{n}}} \" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"75\" 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>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 a estat\u00edstica de teste de hip\u00f3tese para a m\u00e9dia, que \u00e9 definida por uma <a href=\"https:\/\/statorials.org\/pt\/distribuicao-de-alunos\/\">distribui\u00e7\u00e3o t de Student<\/a> .<\/li>\n<li>\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>\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> \u00e9 o valor m\u00e9dio proposto.<\/li>\n<li>\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>\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<p> Como antes, o resultado calculado da estat\u00edstica de teste deve ser interpretado com o valor cr\u00edtico para rejeitar ou n\u00e3o a hip\u00f3tese nula:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Se o teste de hip\u00f3tese para a m\u00e9dia for bilateral, a hip\u00f3tese nula \u00e9 rejeitada se o valor absoluto da estat\u00edstica for maior que o valor cr\u00edtico t <sub>\u03b1\/2|n-1<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Se o teste de hip\u00f3tese para a m\u00e9dia corresponder \u00e0 cauda direita, a hip\u00f3tese nula ser\u00e1 rejeitada se a estat\u00edstica for maior que o valor cr\u00edtico t <sub>\u03b1|n-1<\/sub> .<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Se o teste de hip\u00f3tese para a m\u00e9dia corresponder \u00e0 cauda esquerda, a hip\u00f3tese nula ser\u00e1 rejeitada se a estat\u00edstica for menor que o valor cr\u00edtico -t <sub>\u03b1|n-1<\/sub> .<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-31fb206b75a47181c7c673f54ba28ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}H_1: \\mu\\neq \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } |t|>t_{\\alpha\/2|n-1} \\text{ se rechaza } H_0\\\\[3ex]H_1: \\mu> \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } t>t_{\\alpha|n-1} \\text{ se rechaza } H_0\\\\[3ex]H_1: \\mu< \\mu_0 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Si } t<-t_{\\alpha|n-1} \\text{ se rechaza } H_0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"112\" width=\"457\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Quando a vari\u00e2ncia \u00e9 desconhecida, os valores cr\u00edticos do teste s\u00e3o obtidos na tabela de distribui\u00e7\u00e3o de Student. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-resuelto-de-la-prueba-de-hipotesis-para-la-media\"><\/span> Exemplo do mundo real de teste de hip\u00f3teses para a m\u00e9dia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para entender completamente o conceito de teste de hip\u00f3tese para a m\u00e9dia populacional, voc\u00ea pode ver abaixo um exemplo real desse tipo de teste de hip\u00f3tese.<\/p>\n<ul>\n<li> Uma empresa de tecnologia afirma que a bateria do laptop que vende dura 6 horas. Verificamos se esta hip\u00f3tese \u00e9 falsa realizando um teste de hip\u00f3tese com n\u00edvel de signific\u00e2ncia \u03b1 = 0,05. Para isso, decide-se adquirir 20 unidades e observar a dura\u00e7\u00e3o da bateria de cada computador (os valores s\u00e3o expressos em horas):<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"> 5,2 5,9 7,1 4,2 6,5<br \/> 8,5 4,6 6,8 6,9 5,8<br \/> 5,1 6,5 7,0 5,3 6,2<br \/> 5,7 6,6 7,5 5,1 6,1<\/p>\n<p> Neste caso, as <a href=\"https:\/\/statorials.org\/pt\/hipotese-nula-e-alternativa\/\">hip\u00f3teses nula e alternativa<\/a> do teste de hip\u00f3tese sobre a m\u00e9dia s\u00e3o as seguintes:<\/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-329ffe392783b8bee1eef642d1a45f53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases}H_0: \\mu=6\\\\[2ex] H_1:\\mu\\neq 6 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"93\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para determinar a estat\u00edstica de teste, primeiro precisamos calcular a m\u00e9dia amostral e o desvio padr\u00e3o amostral:<\/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-ef07dbee8f95fc767bc069bd738bb493_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x}=6,13 \\qquad s=1,05\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"169\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Como n\u00e3o conhecemos a vari\u00e2ncia populacional, para obter a estat\u00edstica do teste precisamos aplicar a f\u00f3rmula de teste de hip\u00f3tese para a m\u00e9dia com vari\u00e2ncia desconhecida: <\/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-f8277d83b2d28325f25f5d118486200f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\overline{x}-\\mu}{\\displaystyle \\frac{s}{\\sqrt{n}}} \" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"75\" style=\"vertical-align: -34px;\"><\/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-447879c9f526afed74796d347fd69c97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{6,13-6}{\\displaystyle \\frac{1,05}{\\sqrt{20}}} \" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"98\" style=\"vertical-align: -38px;\"><\/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-e0e63621349c12eec107c43152b5213c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=0,68\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Agora precisamos encontrar o valor cr\u00edtico do teste de hip\u00f3tese, ent\u00e3o procuramos na <a href=\"https:\/\/statorials.org\/pt\/tabela-de-distribuicao-t-do-aluno\/\">tabela de distribui\u00e7\u00e3o t de Student<\/a> o valor correspondente. Os graus de liberdade do t de Student s\u00e3o um a menos que o tamanho da amostra (20-1=19) e, por outro lado, a probabilidade correspondente \u00e9 metade do n\u00edvel de signific\u00e2ncia (0,05\/2= 0,025), uma vez que se trata de uma probabilidade bilateral. testando hip\u00f3teses.<\/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-8154fa6e5bfee35698c6dc26928cd98c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha=0,05 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black}\\ \\alpha\/2=0,025\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"300\" 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-e85692dfb2fb2522025566dc205b8117_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| 19}=2,093\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"72\" width=\"152\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Concluindo, por se tratar de um teste de hip\u00f3tese bilateral e o valor absoluto da estat\u00edstica de teste ser menor que o valor cr\u00edtico, a hip\u00f3tese nula n\u00e3o \u00e9 rejeitada, mas a hip\u00f3tese alternativa \u00e9 rejeitada. <\/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-0e46ceecc8ebcda6dd88ec4d3771c285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0,68<2,093 \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ \\text{Se rechaza } H_1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"345\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"prueba-de-hipotesis-para-la-diferencia-de-medias\"><\/span> Teste de hip\u00f3teses para diferen\u00e7a de m\u00e9dias<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O <strong>teste de hip\u00f3tese de diferen\u00e7a de m\u00e9dias<\/strong> \u00e9 usado para rejeitar ou aceitar a hip\u00f3tese nula de que as m\u00e9dias de duas popula\u00e7\u00f5es s\u00e3o iguais.<\/p>\n<p> Assim, a hip\u00f3tese nula de um teste de hip\u00f3tese para a diferen\u00e7a de duas m\u00e9dias \u00e9 sempre 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-2945a64bf97e1c02ac7fb56bbf1f215b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"H_0: \\mu_1=\\mu_2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Embora a hip\u00f3tese alternativa possa ser uma das tr\u00eas seguintes:<\/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-c612ac7651faad9faa195f37fdf6edef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}H_1:\\mu_1\\neq \\mu_2\\\\[2ex]H_1:\\mu_1>\\mu_2\\\\[2ex]H_1:\\mu_1<\\mu_2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"92\" width=\"97\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o, a <strong>f\u00f3rmula para calcular a estat\u00edstica do teste de hip\u00f3tese para a diferen\u00e7a de m\u00e9dias quando a vari\u00e2ncia \u00e9 conhecida<\/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-83c9d3ddf8c2ff8b9cf2b0fa4ac8082e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle Z=\\frac{\\displaystyle \\overline{x_1}-\\overline{x_2}}{\\displaystyle\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"75\" width=\"123\" 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>\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 a estat\u00edstica de teste de hip\u00f3tese para a diferen\u00e7a de duas m\u00e9dias com vari\u00e2ncia conhecida, que segue uma distribui\u00e7\u00e3o normal padr\u00e3o.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-1436bb55d331d1efe559fd1e55245851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 a m\u00e9dia da amostra 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f06b28a75570ecb4df1371dea2d9f2b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 a m\u00e9dia da amostra 2.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-c2e8ce841b1fc1e199b133e6f24f6f51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_1^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"18\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 a vari\u00e2ncia da popula\u00e7\u00e3o 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-d5fd430c9ff19057933c215e683ace41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sigma_2^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"18\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 a vari\u00e2ncia da popula\u00e7\u00e3o 2.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-cebc0a013985f2695aeb53ded9e7afb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o tamanho da amostra 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bf1e42c248eee22a0911c24c95fe28f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o tamanho da amostra 2.<\/li>\n<\/ul>\n<p> Por outro lado, a <strong>f\u00f3rmula para calcular a estat\u00edstica do teste de hip\u00f3tese para a diferen\u00e7a de m\u00e9dias quando a vari\u00e2ncia \u00e9 desconhecida<\/strong> \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-e9fbc7bf6d5bbaeb18a3992a3f373a55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle t=\\frac{\\displaystyle \\overline{x_1}-\\overline{x_2}}{\\displaystyle\\sqrt{\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"75\" width=\"317\" 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>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 a estat\u00edstica de teste de hip\u00f3tese para a diferen\u00e7a de duas m\u00e9dias com vari\u00e2ncia desconhecida, que segue a distribui\u00e7\u00e3o t de Student.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-1436bb55d331d1efe559fd1e55245851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 a m\u00e9dia da amostra 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f06b28a75570ecb4df1371dea2d9f2b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{x_2}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 a m\u00e9dia da amostra 2.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-59ae3c411ec80f65d56e397f87d29753_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s_1^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"15\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 a vari\u00e2ncia da amostra 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-c785df8c080b6984c6205f881426fb10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s_2^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"15\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 a vari\u00e2ncia da amostra 2.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-cebc0a013985f2695aeb53ded9e7afb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o tamanho da amostra 1.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bf1e42c248eee22a0911c24c95fe28f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o tamanho da amostra 2.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Este artigo explica o que \u00e9 o teste de hip\u00f3tese para a m\u00e9dia nas estat\u00edsticas. Assim, voc\u00ea encontrar\u00e1 a f\u00f3rmula do teste de hip\u00f3tese da m\u00e9dia e, al\u00e9m disso, um exerc\u00edcio resolvido passo a passo. O que \u00e9 teste de hip\u00f3tese para a m\u00e9dia? O teste de hip\u00f3teses para a m\u00e9dia \u00e9 um m\u00e9todo estat\u00edstico [&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-259","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 Teste de hip\u00f3tese para a m\u00e9dia<\/title>\n<meta name=\"description\" content=\"Aqui voc\u00ea encontrar\u00e1 o que \u00e9 o teste de hip\u00f3tese para m\u00e9dia, sua f\u00f3rmula e um exemplo concreto de teste de hip\u00f3tese para 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\/teste-de-hipotese-para-a-media\/\" \/>\n<meta property=\"og:locale\" content=\"pt_PT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Teste de hip\u00f3tese para a m\u00e9dia\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea encontrar\u00e1 o que \u00e9 o teste de hip\u00f3tese para m\u00e9dia, sua f\u00f3rmula e um exemplo concreto de teste de hip\u00f3tese para m\u00e9dia.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-03T11:26:47+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-fdf1817a07bfb8c1eda9a8d8ccd8c828_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=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/\",\"url\":\"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/\",\"name\":\"\u25b7 Teste de hip\u00f3tese para a m\u00e9dia\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pt\/#website\"},\"datePublished\":\"2023-08-03T11:26:47+00:00\",\"dateModified\":\"2023-08-03T11:26:47+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pt\/#\/schema\/person\/e08f98e8db95e0aa9c310e1b27c9c666\"},\"description\":\"Aqui voc\u00ea encontrar\u00e1 o que \u00e9 o teste de hip\u00f3tese para m\u00e9dia, sua f\u00f3rmula e um exemplo concreto de teste de hip\u00f3tese para m\u00e9dia.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/#breadcrumb\"},\"inLanguage\":\"pt-PT\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pt\/teste-de-hipotese-para-a-media\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Lar\",\"item\":\"https:\/\/statorials.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Teste de hip\u00f3tese para a 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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