{"id":1735,"date":"2021-09-07T16:36:38","date_gmt":"2021-09-07T16:36:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1735"},"modified":"2023-12-05T09:19:15","modified_gmt":"2023-12-05T09:19:15","slug":"the-standard-normal-distribution-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-standard-normal-distribution-2\/","title":{"raw":"The Empirical Rule","rendered":"The Empirical Rule"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list4253\">\r\n \t<li>Use the empirical rule to determine values at 1, 2, and 3 standard deviations from the mean<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Empirical Rule<\/h2>\r\nIf <em data-redactor-tag=\"em\">X<\/em> is a random variable and has a normal distribution with mean <em data-redactor-tag=\"em\">\u00b5<\/em> and standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>, then the <strong>e<\/strong><strong data-redactor-tag=\"strong\">mpirical rule<\/strong> says the following:\r\n<ul>\r\n \t<li>About 68% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20131<em data-redactor-tag=\"em\">\u03c3<\/em> and +1<em data-redactor-tag=\"em\">\u03c3<\/em> of the mean <em data-redactor-tag=\"em\">\u00b5<\/em> (within one standard deviation of the mean).<\/li>\r\n \t<li>About 95% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20132<em data-redactor-tag=\"em\">\u03c3<\/em> and +2<em data-redactor-tag=\"em\">\u03c3<\/em> of the mean <em data-redactor-tag=\"em\">\u00b5<\/em> (within two standard deviations of the mean).<\/li>\r\n \t<li>About 99.7% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20133<em data-redactor-tag=\"em\">\u03c3<\/em> and +3<em data-redactor-tag=\"em\">\u03c3<\/em> of the mean <em data-redactor-tag=\"em\">\u00b5<\/em>(within three standard deviations of the mean). Notice that almost all the<em data-redactor-tag=\"em\">x<\/em> values lie within three standard deviations of the mean.<\/li>\r\n \t<li>The <em data-redactor-tag=\"em\">z<\/em>-scores for +1<em data-redactor-tag=\"em\">\u03c3<\/em> and \u20131<em data-redactor-tag=\"em\">\u03c3<\/em> are +1 and \u20131, respectively.<\/li>\r\n \t<li>The <em data-redactor-tag=\"em\">z<\/em>-scores for +2<em data-redactor-tag=\"em\">\u03c3<\/em> and \u20132<em data-redactor-tag=\"em\">\u03c3<\/em> are +2 and \u20132, respectively.<\/li>\r\n \t<li>The <em data-redactor-tag=\"em\">z<\/em>-scores for +3<em data-redactor-tag=\"em\">\u03c3<\/em> and \u20133<em data-redactor-tag=\"em\">\u03c3<\/em> are +3 and \u20133 respectively.<\/li>\r\n<\/ul>\r\nThe empirical rule is also known as the 68-95-99.7 rule.\r\n\r\n<a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-07-at-7.34.36-PM.png\"><img class=\"aligncenter wp-image-509 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214545\/Screen-Shot-2015-06-07-at-7.34.36-PM.png\" alt=\"Graph of the empirical Rule. The graph is a bell curve showing 3 standard deviations from mu in both directions with mu being at the center. The standard deviations are labeled -3 sigma, -2 sigma, -1 sigma, 1 sigma, 2 sigma, and 3 sigma. These are centered around mu.\" width=\"682\" height=\"392\" \/><\/a>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe mean height of 15 to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let <em data-redactor-tag=\"em\">X<\/em> = the height of a 15 to 18-year-old male from Chile in 2009 to 2010. Then <em data-redactor-tag=\"em\">X<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(170, 6.28).\r\n\r\na. Suppose a 15 to 18-year-old male from Chile was 168 cm tall from 2009 to 2010. The <em data-redactor-tag=\"em\">z<\/em>-score when <em data-redactor-tag=\"em\">x<\/em> = 168 cm is <em data-redactor-tag=\"em\">z<\/em> = _______. This <em data-redactor-tag=\"em\">z<\/em>-score tells you that <em data-redactor-tag=\"em\">x<\/em> = 168 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).\r\n\r\nb. Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a <em data-redactor-tag=\"em\">z<\/em>-score of <em data-redactor-tag=\"em\">z<\/em> = 1.27. What is the male's height? The <em data-redactor-tag=\"em\">z<\/em>-score (<em data-redactor-tag=\"em\">z<\/em> = 1.27) tells you that the male's height is ________ standard deviations to the __________ (right or left) of the mean.\r\n\r\n[reveal-answer q=\"832268\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"832268\"]\r\n\r\na. \u20130.32, 0.32, left, 170\r\n\r\nb. 177.98, 1.27, right\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h4><\/h4>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nUse the information in the previous Example to answer the following questions.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Suppose a 15 to 18-year-old male from Chile was 176 cm tall from 2009 to 2010. The <em data-redactor-tag=\"em\">z<\/em>-score when <em data-redactor-tag=\"em\">x<\/em> = 176 cm is <em data-redactor-tag=\"em\">z<\/em> = _______. This <em data-redactor-tag=\"em\">z<\/em>-score tells you that <em data-redactor-tag=\"em\">x<\/em> = 176 cm is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).<\/li>\r\n \t<li>Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a <em data-redactor-tag=\"em\">z<\/em>-score of <em data-redactor-tag=\"em\">z<\/em> = \u20132. What is the male's height? The <em data-redactor-tag=\"em\">z<\/em>-score (<em data-redactor-tag=\"em\">z<\/em> = \u20132) tells you that the male's height is ________ standard deviations to the __________ (right or left) of the mean.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFrom 1984 to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let <em data-redactor-tag=\"em\">Y<\/em> = the height of 15 to 18-year-old males from 1984 to 1985. Then <em data-redactor-tag=\"em\">Y<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(172.36, 6.34).\r\n\r\nThe mean height of 15 to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let <em data-redactor-tag=\"em\">X<\/em> = the height of a 15 to 18-year-old male from Chile in 2009 to 2010. Then <em data-redactor-tag=\"em\">X<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(170, 6.28).\r\n\r\nFind the <em data-redactor-tag=\"em\">z<\/em>-scores for <em data-redactor-tag=\"em\">x<\/em> = 160.58 cm and <em data-redactor-tag=\"em\">y<\/em> = 162.85 cm. Interpret each <em data-redactor-tag=\"em\">z<\/em>-score. What can you say about <em data-redactor-tag=\"em\">x<\/em> = 160.58 cm and <em data-redactor-tag=\"em\">y<\/em> = 162.85 cm?\r\n\r\n[reveal-answer q=\"835211\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"835211\"]\r\n\r\nThe <em data-redactor-tag=\"em\">z<\/em>-score for <em data-redactor-tag=\"em\">x<\/em> = 160.58 is <em data-redactor-tag=\"em\">z<\/em> = \u20131.5.\r\n\r\nThe <em data-redactor-tag=\"em\">z<\/em>-score for <em data-redactor-tag=\"em\">y<\/em> = 162.85 is <em data-redactor-tag=\"em\">z<\/em> = \u20131.5.\r\n\r\nBoth <em data-redactor-tag=\"em\">x<\/em> = 160.58 and <em data-redactor-tag=\"em\">y<\/em> = 162.85 deviate the same number of standard deviations from their respective means and in the same direction.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nIn 2012, 1,664,479 students took the SAT exam. The distribution of scores in the verbal section of the SAT had a mean\u00a0<em data-redactor-tag=\"em\">\u00b5<\/em> = 496 and a standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em> = 114. Let <em data-redactor-tag=\"em\">X<\/em> = a SAT exam verbal section score in 2012. Then <em data-redactor-tag=\"em\">X<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(496, 114).\r\n\r\nFind the <em data-redactor-tag=\"em\">z<\/em>-scores for [latex]x_1= 325[\/latex] and [latex]x_2 = 366.21[\/latex]. Interpret each <em data-redactor-tag=\"em\">z<\/em>-score. What can you say about [latex]x_1 = 325[\/latex] and [latex]x_2 = 366.21[\/latex]?\r\n\r\n[reveal-answer q=\"489579\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"489579\"]\r\n\r\nThe <em data-redactor-tag=\"em\">z<\/em>-score for [latex]x_1 = 325[\/latex] is [latex]z_1 = \u20131.14[\/latex].\r\n\r\nThe <em data-redactor-tag=\"em\">z<\/em>-score for [latex]x_2 = 366.21[\/latex] is [latex]z_2 = \u20131.14[\/latex].\r\n\r\nStudent 2 scored closer to the mean than Student 1 and, since they both had negative <em data-redactor-tag=\"em\">z<\/em>-scores, Student 2 had the better score.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 style=\"text-align: start;\"><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose <em data-redactor-tag=\"em\">x<\/em> has a normal distribution with mean 50 and standard deviation 6.\r\n<ul>\r\n \t<li>About 68% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20131<em data-redactor-tag=\"em\">\u03c3<\/em> = (\u20131)(6) = \u20136 and 1<em data-redactor-tag=\"em\">\u03c3<\/em> = (1)(6) = 6 of the mean 50. The values 50 \u2013 6 = 44 and 50 + 6 = 56 are within one standard deviation of the mean 50. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20131 and +1 for 44 and 56, respectively.<\/li>\r\n \t<li>About 95% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20132<em data-redactor-tag=\"em\">\u03c3<\/em> = (\u20132)(6) = \u201312 and 2<em data-redactor-tag=\"em\">\u03c3<\/em> = (2)(6) = 12. The values 50 \u2013 12 = 38 and 50 + 12 = 62 are within two standard deviations of the mean 50. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20132 and +2 for 38 and 62, respectively.<\/li>\r\n \t<li>About 99.7% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20133<em data-redactor-tag=\"em\">\u03c3<\/em> = (\u20133)(6) = \u201318 and 3<em data-redactor-tag=\"em\">\u03c3<\/em>= (3)(6) = 18 of the mean 50. The values 50 \u2013 18 = 32 and 50 + 18 = 68 are within three standard deviations of the mean 50. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20133 and +3 for 32 and 68, respectively<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSuppose <em data-redactor-tag=\"em\">X<\/em> has a normal distribution with mean 25 and standard deviation five. Between what values of <em data-redactor-tag=\"em\">x<\/em> do 68% of the values lie?\r\n\r\n[reveal-answer q=\"725225\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"725225\"]\r\n\r\nBetween 20 and 30.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFrom 1984 to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let <em data-redactor-tag=\"em\">Y<\/em> = the height of 15 to 18-year-old males in 1984 to 1985. Then <em data-redactor-tag=\"em\">Y<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(172.36, 6.34).\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>About 68% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The <em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\r\n \t<li>About 95% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The <em data-redactor-tag=\"em\">z<\/em>-scores are ________________ respectively.<\/li>\r\n \t<li>About 99.7% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two .7% of the values lie between 153.34 and 191.38. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20133 and 3.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"891522\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"891522\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>About 68% of the values lie between 166.02 cm and 178.7 cm. The\u00a0<em data-effect=\"italics\">z<\/em>-scores are \u20131 and 1.<\/li>\r\n \t<li>About 95% of the values lie between 159.68 cm and 185.04 cm. The\u00a0<em data-effect=\"italics\">z<\/em>-scores are \u20132 and 2.<\/li>\r\n \t<li>About 99.7% of the values lie between 153.34 cm and 191.38 cm. The\u00a0<em data-effect=\"italics\">z<\/em>-scores are \u20133 and 3.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nThe scores on a college entrance exam have an approximate normal distribution with mean, <em data-redactor-tag=\"em\">\u00b5<\/em> = 52 points and a standard deviation, <em data-redactor-tag=\"em\">\u03c3<\/em> = 11 points.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>About 68% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\r\n \t<li>About 95% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\r\n \t<li>About 99.7% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"949886\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"949886\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>About 68% of the values lie between the values 41 and 63. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are \u20131 and 1, respectively.<\/li>\r\n \t<li>About 95% of the values lie between the values 30 and 74. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are \u20132 and 2, respectively.<\/li>\r\n \t<li>About 99.7% of the values lie between the values 19 and 85. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are \u20133 and 3, respectively.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list4253\">\n<li>Use the empirical rule to determine values at 1, 2, and 3 standard deviations from the mean<\/li>\n<\/ul>\n<\/div>\n<h2>The Empirical Rule<\/h2>\n<p>If <em data-redactor-tag=\"em\">X<\/em> is a random variable and has a normal distribution with mean <em data-redactor-tag=\"em\">\u00b5<\/em> and standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>, then the <strong>e<\/strong><strong data-redactor-tag=\"strong\">mpirical rule<\/strong> says the following:<\/p>\n<ul>\n<li>About 68% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20131<em data-redactor-tag=\"em\">\u03c3<\/em> and +1<em data-redactor-tag=\"em\">\u03c3<\/em> of the mean <em data-redactor-tag=\"em\">\u00b5<\/em> (within one standard deviation of the mean).<\/li>\n<li>About 95% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20132<em data-redactor-tag=\"em\">\u03c3<\/em> and +2<em data-redactor-tag=\"em\">\u03c3<\/em> of the mean <em data-redactor-tag=\"em\">\u00b5<\/em> (within two standard deviations of the mean).<\/li>\n<li>About 99.7% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20133<em data-redactor-tag=\"em\">\u03c3<\/em> and +3<em data-redactor-tag=\"em\">\u03c3<\/em> of the mean <em data-redactor-tag=\"em\">\u00b5<\/em>(within three standard deviations of the mean). Notice that almost all the<em data-redactor-tag=\"em\">x<\/em> values lie within three standard deviations of the mean.<\/li>\n<li>The <em data-redactor-tag=\"em\">z<\/em>-scores for +1<em data-redactor-tag=\"em\">\u03c3<\/em> and \u20131<em data-redactor-tag=\"em\">\u03c3<\/em> are +1 and \u20131, respectively.<\/li>\n<li>The <em data-redactor-tag=\"em\">z<\/em>-scores for +2<em data-redactor-tag=\"em\">\u03c3<\/em> and \u20132<em data-redactor-tag=\"em\">\u03c3<\/em> are +2 and \u20132, respectively.<\/li>\n<li>The <em data-redactor-tag=\"em\">z<\/em>-scores for +3<em data-redactor-tag=\"em\">\u03c3<\/em> and \u20133<em data-redactor-tag=\"em\">\u03c3<\/em> are +3 and \u20133 respectively.<\/li>\n<\/ul>\n<p>The empirical rule is also known as the 68-95-99.7 rule.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-07-at-7.34.36-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-509 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214545\/Screen-Shot-2015-06-07-at-7.34.36-PM.png\" alt=\"Graph of the empirical Rule. The graph is a bell curve showing 3 standard deviations from mu in both directions with mu being at the center. The standard deviations are labeled -3 sigma, -2 sigma, -1 sigma, 1 sigma, 2 sigma, and 3 sigma. These are centered around mu.\" width=\"682\" height=\"392\" \/><\/a><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The mean height of 15 to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let <em data-redactor-tag=\"em\">X<\/em> = the height of a 15 to 18-year-old male from Chile in 2009 to 2010. Then <em data-redactor-tag=\"em\">X<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(170, 6.28).<\/p>\n<p>a. Suppose a 15 to 18-year-old male from Chile was 168 cm tall from 2009 to 2010. The <em data-redactor-tag=\"em\">z<\/em>-score when <em data-redactor-tag=\"em\">x<\/em> = 168 cm is <em data-redactor-tag=\"em\">z<\/em> = _______. This <em data-redactor-tag=\"em\">z<\/em>-score tells you that <em data-redactor-tag=\"em\">x<\/em> = 168 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).<\/p>\n<p>b. Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a <em data-redactor-tag=\"em\">z<\/em>-score of <em data-redactor-tag=\"em\">z<\/em> = 1.27. What is the male&#8217;s height? The <em data-redactor-tag=\"em\">z<\/em>-score (<em data-redactor-tag=\"em\">z<\/em> = 1.27) tells you that the male&#8217;s height is ________ standard deviations to the __________ (right or left) of the mean.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q832268\">Show Answer<\/span><\/p>\n<div id=\"q832268\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. \u20130.32, 0.32, left, 170<\/p>\n<p>b. 177.98, 1.27, right<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h4><\/h4>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Use the information in the previous Example to answer the following questions.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Suppose a 15 to 18-year-old male from Chile was 176 cm tall from 2009 to 2010. The <em data-redactor-tag=\"em\">z<\/em>-score when <em data-redactor-tag=\"em\">x<\/em> = 176 cm is <em data-redactor-tag=\"em\">z<\/em> = _______. This <em data-redactor-tag=\"em\">z<\/em>-score tells you that <em data-redactor-tag=\"em\">x<\/em> = 176 cm is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).<\/li>\n<li>Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a <em data-redactor-tag=\"em\">z<\/em>-score of <em data-redactor-tag=\"em\">z<\/em> = \u20132. What is the male&#8217;s height? The <em data-redactor-tag=\"em\">z<\/em>-score (<em data-redactor-tag=\"em\">z<\/em> = \u20132) tells you that the male&#8217;s height is ________ standard deviations to the __________ (right or left) of the mean.<\/li>\n<\/ol>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>From 1984 to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let <em data-redactor-tag=\"em\">Y<\/em> = the height of 15 to 18-year-old males from 1984 to 1985. Then <em data-redactor-tag=\"em\">Y<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(172.36, 6.34).<\/p>\n<p>The mean height of 15 to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let <em data-redactor-tag=\"em\">X<\/em> = the height of a 15 to 18-year-old male from Chile in 2009 to 2010. Then <em data-redactor-tag=\"em\">X<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(170, 6.28).<\/p>\n<p>Find the <em data-redactor-tag=\"em\">z<\/em>-scores for <em data-redactor-tag=\"em\">x<\/em> = 160.58 cm and <em data-redactor-tag=\"em\">y<\/em> = 162.85 cm. Interpret each <em data-redactor-tag=\"em\">z<\/em>-score. What can you say about <em data-redactor-tag=\"em\">x<\/em> = 160.58 cm and <em data-redactor-tag=\"em\">y<\/em> = 162.85 cm?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q835211\">Show Answer<\/span><\/p>\n<div id=\"q835211\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em data-redactor-tag=\"em\">z<\/em>-score for <em data-redactor-tag=\"em\">x<\/em> = 160.58 is <em data-redactor-tag=\"em\">z<\/em> = \u20131.5.<\/p>\n<p>The <em data-redactor-tag=\"em\">z<\/em>-score for <em data-redactor-tag=\"em\">y<\/em> = 162.85 is <em data-redactor-tag=\"em\">z<\/em> = \u20131.5.<\/p>\n<p>Both <em data-redactor-tag=\"em\">x<\/em> = 160.58 and <em data-redactor-tag=\"em\">y<\/em> = 162.85 deviate the same number of standard deviations from their respective means and in the same direction.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>In 2012, 1,664,479 students took the SAT exam. The distribution of scores in the verbal section of the SAT had a mean\u00a0<em data-redactor-tag=\"em\">\u00b5<\/em> = 496 and a standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em> = 114. Let <em data-redactor-tag=\"em\">X<\/em> = a SAT exam verbal section score in 2012. Then <em data-redactor-tag=\"em\">X<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(496, 114).<\/p>\n<p>Find the <em data-redactor-tag=\"em\">z<\/em>-scores for [latex]x_1= 325[\/latex] and [latex]x_2 = 366.21[\/latex]. Interpret each <em data-redactor-tag=\"em\">z<\/em>-score. What can you say about [latex]x_1 = 325[\/latex] and [latex]x_2 = 366.21[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q489579\">Show Answer<\/span><\/p>\n<div id=\"q489579\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em data-redactor-tag=\"em\">z<\/em>-score for [latex]x_1 = 325[\/latex] is [latex]z_1 = \u20131.14[\/latex].<\/p>\n<p>The <em data-redactor-tag=\"em\">z<\/em>-score for [latex]x_2 = 366.21[\/latex] is [latex]z_2 = \u20131.14[\/latex].<\/p>\n<p>Student 2 scored closer to the mean than Student 1 and, since they both had negative <em data-redactor-tag=\"em\">z<\/em>-scores, Student 2 had the better score.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3 style=\"text-align: start;\"><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose <em data-redactor-tag=\"em\">x<\/em> has a normal distribution with mean 50 and standard deviation 6.<\/p>\n<ul>\n<li>About 68% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20131<em data-redactor-tag=\"em\">\u03c3<\/em> = (\u20131)(6) = \u20136 and 1<em data-redactor-tag=\"em\">\u03c3<\/em> = (1)(6) = 6 of the mean 50. The values 50 \u2013 6 = 44 and 50 + 6 = 56 are within one standard deviation of the mean 50. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20131 and +1 for 44 and 56, respectively.<\/li>\n<li>About 95% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20132<em data-redactor-tag=\"em\">\u03c3<\/em> = (\u20132)(6) = \u201312 and 2<em data-redactor-tag=\"em\">\u03c3<\/em> = (2)(6) = 12. The values 50 \u2013 12 = 38 and 50 + 12 = 62 are within two standard deviations of the mean 50. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20132 and +2 for 38 and 62, respectively.<\/li>\n<li>About 99.7% of the <em data-redactor-tag=\"em\">x<\/em> values lie between \u20133<em data-redactor-tag=\"em\">\u03c3<\/em> = (\u20133)(6) = \u201318 and 3<em data-redactor-tag=\"em\">\u03c3<\/em>= (3)(6) = 18 of the mean 50. The values 50 \u2013 18 = 32 and 50 + 18 = 68 are within three standard deviations of the mean 50. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20133 and +3 for 32 and 68, respectively<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Suppose <em data-redactor-tag=\"em\">X<\/em> has a normal distribution with mean 25 and standard deviation five. Between what values of <em data-redactor-tag=\"em\">x<\/em> do 68% of the values lie?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725225\">Show Answer<\/span><\/p>\n<div id=\"q725225\" class=\"hidden-answer\" style=\"display: none\">\n<p>Between 20 and 30.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>From 1984 to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let <em data-redactor-tag=\"em\">Y<\/em> = the height of 15 to 18-year-old males in 1984 to 1985. Then <em data-redactor-tag=\"em\">Y<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(172.36, 6.34).<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>About 68% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The <em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\n<li>About 95% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The <em data-redactor-tag=\"em\">z<\/em>-scores are ________________ respectively.<\/li>\n<li>About 99.7% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two .7% of the values lie between 153.34 and 191.38. The <em data-redactor-tag=\"em\">z<\/em>-scores are \u20133 and 3.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q891522\">Show Answer<\/span><\/p>\n<div id=\"q891522\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>About 68% of the values lie between 166.02 cm and 178.7 cm. The\u00a0<em data-effect=\"italics\">z<\/em>-scores are \u20131 and 1.<\/li>\n<li>About 95% of the values lie between 159.68 cm and 185.04 cm. The\u00a0<em data-effect=\"italics\">z<\/em>-scores are \u20132 and 2.<\/li>\n<li>About 99.7% of the values lie between 153.34 cm and 191.38 cm. The\u00a0<em data-effect=\"italics\">z<\/em>-scores are \u20133 and 3.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>The scores on a college entrance exam have an approximate normal distribution with mean, <em data-redactor-tag=\"em\">\u00b5<\/em> = 52 points and a standard deviation, <em data-redactor-tag=\"em\">\u03c3<\/em> = 11 points.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>About 68% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\n<li>About 95% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\n<li>About 99.7% of the <em data-redactor-tag=\"em\">y<\/em> values lie between what two values? These values are ________________. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are ________________, respectively.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q949886\">Show Answer<\/span><\/p>\n<div id=\"q949886\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>About 68% of the values lie between the values 41 and 63. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are \u20131 and 1, respectively.<\/li>\n<li>About 95% of the values lie between the values 30 and 74. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are \u20132 and 2, respectively.<\/li>\n<li>About 99.7% of the values lie between the values 19 and 85. The\u00a0<em data-redactor-tag=\"em\">z<\/em>-scores are \u20133 and 3, respectively.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\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-1735\">\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>Statistics, The Standard Normal Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/statistics\/pages\/6-1-the-standard-normal-distribution\">https:\/\/openstax.org\/books\/statistics\/pages\/6-1-the-standard-normal-distribution<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":169134,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Statistics, The Standard Normal Distribution\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/statistics\/pages\/6-1-the-standard-normal-distribution\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1735","chapter","type-chapter","status-publish","hentry"],"part":256,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1735","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1735\/revisions"}],"predecessor-version":[{"id":3665,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1735\/revisions\/3665"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/256"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1735\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1735"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1735"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1735"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1735"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}