{"id":315,"date":"2017-04-15T03:21:28","date_gmt":"2017-04-15T03:21:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/introduction-to-normal-random-variables-3-of-6\/"},"modified":"2017-05-31T00:13:12","modified_gmt":"2017-05-31T00:13:12","slug":"introduction-to-normal-random-variables-3-of-6","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/chapter\/introduction-to-normal-random-variables-3-of-6\/","title":{"raw":"Normal Random Variables (3 of 6)","rendered":"Normal Random Variables (3 of 6)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use a normal probability distribution to estimate probabilities and identify unusual events.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h2>The Empirical Rule in a Context<\/h2>\r\nSuppose that foot length of a randomly chosen adult male is a normal random variable with mean [latex]\\mathrm{\u03bc}=11[\/latex] and standard deviation [latex]\\mathrm{\u03c3}=1.5[\/latex] . Then the empirical rule lets us sketch the probability distribution of <em>X<\/em> as follows:\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032125\/m6_probability_topic_6_2_m6_normal_random_variables_2_image34.png\" alt=\"Probability distribution of X (foot length), where the curve is black, and the data is in blue\" width=\"360\" height=\"301\" \/>\r\n<ul style=\"list-style-type: none\">\r\n \t<li><strong>(a)<\/strong> What is the probability that a randomly chosen adult male will have a foot length between 8 and 14 inches?<\/li>\r\n \t<li><strong>Answer:<\/strong> 0.95, or 95%<\/li>\r\n \t<li><strong>(b)<\/strong> An adult male is almost guaranteed (0.997 probability) to have a foot length between what two values?<\/li>\r\n \t<li><strong>Answer:<\/strong> 6.5 and 15.5 inches<\/li>\r\n \t<li><strong>(c)<\/strong> The probability is only 2.5% that an adult male will have a foot length greater than how many inches?<\/li>\r\n \t<li><strong>Answer:<\/strong> 14 inches<\/li>\r\n<\/ul>\r\nNinety-five percent of the area is within 2 standard deviations of the mean, so 2.5% of the area is in the tail above 2 standard deviations. The <em>x<\/em>-value 2 standard deviations above the mean is 14 inches.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032127\/m6_probability_topic_6_2_m6_normal_random_variables_2_image35.png\" alt=\"Normal curve showing 95% of area is within 2 SD of the mean\" width=\"357\" height=\"270\" \/>\r\n\r\n<\/div>\r\nNow you should try a few: questions (d), (e), and (f) are presented in the Learn By Doing activity. Use the figure preceding question (a) to help you.\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3567\r\n\r\n<\/div>\r\n<strong>Comment<\/strong>\r\n\r\nNotice that there are two types of problems we may want to solve: those like (a) and, from the Learn By Doing activity, (d) and (e), in which a particular interval of values of a normal random variable is given and we are asked to find a probability; and those like (b), (c), and, from the Learn By Doing, (f), in which a probability is given and we are asked to identify values of the normal random variable.\r\n\r\n&nbsp;","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use a normal probability distribution to estimate probabilities and identify unusual events.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<h2>The Empirical Rule in a Context<\/h2>\n<p>Suppose that foot length of a randomly chosen adult male is a normal random variable with mean [latex]\\mathrm{\u03bc}=11[\/latex] and standard deviation [latex]\\mathrm{\u03c3}=1.5[\/latex] . Then the empirical rule lets us sketch the probability distribution of <em>X<\/em> as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032125\/m6_probability_topic_6_2_m6_normal_random_variables_2_image34.png\" alt=\"Probability distribution of X (foot length), where the curve is black, and the data is in blue\" width=\"360\" height=\"301\" \/><\/p>\n<ul style=\"list-style-type: none\">\n<li><strong>(a)<\/strong> What is the probability that a randomly chosen adult male will have a foot length between 8 and 14 inches?<\/li>\n<li><strong>Answer:<\/strong> 0.95, or 95%<\/li>\n<li><strong>(b)<\/strong> An adult male is almost guaranteed (0.997 probability) to have a foot length between what two values?<\/li>\n<li><strong>Answer:<\/strong> 6.5 and 15.5 inches<\/li>\n<li><strong>(c)<\/strong> The probability is only 2.5% that an adult male will have a foot length greater than how many inches?<\/li>\n<li><strong>Answer:<\/strong> 14 inches<\/li>\n<\/ul>\n<p>Ninety-five percent of the area is within 2 standard deviations of the mean, so 2.5% of the area is in the tail above 2 standard deviations. The <em>x<\/em>-value 2 standard deviations above the mean is 14 inches.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032127\/m6_probability_topic_6_2_m6_normal_random_variables_2_image35.png\" alt=\"Normal curve showing 95% of area is within 2 SD of the mean\" width=\"357\" height=\"270\" \/><\/p>\n<\/div>\n<p>Now you should try a few: questions (d), (e), and (f) are presented in the Learn By Doing activity. Use the figure preceding question (a) to help you.<\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3567\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3567&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3567\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p><strong>Comment<\/strong><\/p>\n<p>Notice that there are two types of problems we may want to solve: those like (a) and, from the Learn By Doing activity, (d) and (e), in which a particular interval of values of a normal random variable is given and we are asked to find a probability; and those like (b), (c), and, from the Learn By Doing, (f), in which a probability is given and we are asked to identify values of the normal random variable.<\/p>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-315\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":163,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"2ce02974-da6b-4303-b00f-95b8660cb774","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-315","chapter","type-chapter","status-publish","hentry"],"part":258,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/315","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/315\/revisions"}],"predecessor-version":[{"id":1415,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/315\/revisions\/1415"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/parts\/258"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/315\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/media?parent=315"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapter-type?post=315"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/contributor?post=315"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/license?post=315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}