{"id":7138,"date":"2011-11-08T23:24:41","date_gmt":"2011-11-08T23:24:41","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7138"},"modified":"2022-01-25T19:34:09","modified_gmt":"2022-01-25T19:34:09","slug":"numa-fabrica-de-material-eletrico","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7138","title":{"rendered":"Numa f\u00e1brica de material el\u00e9trico"},"content":{"rendered":"<p><ul id='GTTabs_ul_7138' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7138' class='GTTabs_curr'><a  id=\"7138_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7138' ><a  id=\"7138_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7138'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Numa f\u00e1brica de material el\u00e9trico, recolheu-se uma amostra de 400 l\u00e2mpadas, para estudar o seu tempo de dura\u00e7\u00e3o.<\/p>\n<table class=\" aligncenter\" style=\"width: 50%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">Horas de vida<\/td>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">200 a 399<\/td>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">400 a 599<\/td>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">600 a 799<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">N.\u00ba de l\u00e2mpadas<\/td>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">48<\/td>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">203<\/td>\n<td style=\"text-align: center; border: #6b8e23 1px solid;\">149<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/lampada.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7141\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7141\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/lampada.jpg\" data-orig-size=\"194,259\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"L\u00e2mpada\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/lampada.jpg\" class=\"alignright size-full wp-image-7141\" title=\"L\u00e2mpada\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/lampada.jpg\" alt=\"\" width=\"116\" height=\"155\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/lampada.jpg 194w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/lampada-112x150.jpg 112w\" sizes=\"auto, (max-width: 116px) 100vw, 116px\" \/><\/a>Construa um histograma e calcule a esperan\u00e7a matem\u00e1tica e o desvio padr\u00e3o da distribui\u00e7\u00e3o.<\/li>\n<li>Qual a percentagem de l\u00e2mpadas que duram menos do que $\\mu -2\\sigma $?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7138' onClick='GTTabs_show(1,7138)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7138'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Apresenta-se um histograma do tempo de dura\u00e7\u00e3o de uma amostra de 400 l\u00e2mpadas:<br \/>\n<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/histograma.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7140\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7140\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/histograma.jpg\" data-orig-size=\"376,212\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Histograma\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/histograma.jpg\" class=\"aligncenter wp-image-7140 size-full\" title=\"Histograma\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/histograma.jpg\" alt=\"\" width=\"376\" height=\"212\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/histograma.jpg 376w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/histograma-300x169.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/histograma-150x84.jpg 150w\" sizes=\"auto, (max-width: 376px) 100vw, 376px\" \/><\/a><br \/>\nA esperan\u00e7a matem\u00e1tica e o desvio padr\u00e3o s\u00e3o:<br \/>\n\\[\\mu =\\frac{48\\times 300+203\\times 500+149\\times 700}{48+203+149}=\\frac{220200}{400}=550,5\\]<br \/>\n\\[\\sigma =\\sqrt{\\frac{48\\times {{(300-550,5)}^{2}}+203\\times {{(500-550,5)}^{2}}+149\\times {{(700-550,5)}^{2}}}{400}}=\\frac{\\sqrt{68599}}{2}\\approx 131\\]<br \/>\n\u00ad<\/li>\n<li>Ora, $\\mu -2\\sigma \\approx 550,5-2\\times 131=288,5$ horas.<br \/>\nAssim, temos \\[\\frac{48}{200}=\\frac{x}{88,5}\\] donde $x\\approx 21$.<br \/>\n\u00c9 de esperar que cerca de 21 l\u00e2mpadas tenham dura\u00e7\u00e3o inferior a 288,5 horas, a que corresponde\u00a0$\\frac{21}{400}=5,25$% do total da amostra.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7138' onClick='GTTabs_show(0,7138)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Numa f\u00e1brica de material el\u00e9trico, recolheu-se uma amostra de 400 l\u00e2mpadas, para estudar o seu tempo de dura\u00e7\u00e3o. Horas de vida 200 a 399 400 a 599 600 a 799 N.\u00ba&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21025,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,247,184],"series":[],"class_list":["post-7138","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-desvio-padrao","tag-histograma"],"views":2149,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/12V1Pag172-34_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7138"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7138\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21025"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7138"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}