{"id":1054,"date":"2023-06-22T01:45:26","date_gmt":"2023-06-22T01:45:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-dig-deeper\/"},"modified":"2024-03-14T19:10:37","modified_gmt":"2024-03-14T19:10:37","slug":"normal-distribution-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/normal-distribution-dig-deeper\/","title":{"raw":"Normal Distribution: Fresh Take","rendered":"Normal Distribution: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li class=\"li1\">Understand the properties, characteristics, and importance of a normal distribution in statistical analysis.<\/li>\r\n\t<li class=\"li1\">Explain how changing the mean and standard deviation will change the characteristics of a normal curve.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>Recall that a normal distribution is a mathematical model with a smooth bell-shaped curve.<\/p>\r\n<section class=\"textbox example\">The final exam scores in a statistics class were normally distributed with a mean of [latex]63[\/latex]and a standard deviation of five.<img class=\"aligncenter wp-image-5872 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/04012153\/STAT_8D_Preview_Q10.png\" alt=\"A normal distribution curve with boxes marked A, B, C, D, E, F, G corresponding to ticks on the x-axis.\" width=\"936\" height=\"496\" \/><br \/>\r\n<strong>(a) <\/strong>What is the value of each tick mark labeled above? [reveal-answer q=\"322532\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"322532\"]<br \/>\r\n<img class=\"aligncenter wp-image-5873 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/04012418\/Screenshot-2023-01-03-at-6.24.10-PM.png\" alt=\"A normal distribution curve with the ticks having numbers corresponding to the boxes from the above graph. A=48, B=53, C=58, D=63, E=68, F=73, G=78.\" width=\"884\" height=\"486\" \/>[\/hidden-answer]<strong>(b)<\/strong> The score of ______ is [latex]1[\/latex] standard deviation above the mean.<br \/>\r\n[reveal-answer q=\"913853\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"913853\"]The score of <strong>68<\/strong> is [latex]1[\/latex] standard deviation above the mean.[\/hidden-answer]\r\n\r\n<p><strong>(c)<\/strong> The score of ______ has a [latex]z[\/latex]-score of [latex]-3[\/latex].<br \/>\r\n[reveal-answer q=\"78010\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"78010\"][latex]z[\/latex]-score [latex]=-3[\/latex] means that it is 3 standard deviations below the mean. So, the answer is [latex]48[\/latex].[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Find the mean and standard deviation of the following curves.<img class=\"aligncenter wp-image-5874 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/04013002\/m6_probability_topic_6_2_normalrandomvariables_1_image2.png\" alt=\"A normal curve in black with a mean of 10, a curve in red with a mean of 10, and a curve in green with a mean of 14.\" width=\"530\" height=\"334\" \/><strong>(a)\u00a0<\/strong>What is the mean and standard deviation of the black color normal distribution curve?<br \/>\r\n[reveal-answer q=\"533130\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"533130\"][latex]\\mu = 10[\/latex] and [latex]\\sigma=1[\/latex][\/hidden-answer]\r\n\r\n<p><strong>(b) <\/strong>What is the mean and standard deviation of the red color normal distribution curve?<br \/>\r\n[reveal-answer q=\"533131\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"533131\"][latex]\\mu = 10[\/latex] and [latex]\\sigma=2[\/latex][\/hidden-answer]<\/p>\r\n<p><strong>(c) <\/strong>What is the mean and standard deviation of the green color normal distribution curve?<br \/>\r\n[reveal-answer q=\"533132\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"533132\"][latex]\\mu = 14[\/latex] and [latex]\\sigma=1[\/latex][\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li class=\"li1\">Understand the properties, characteristics, and importance of a normal distribution in statistical analysis.<\/li>\n<li class=\"li1\">Explain how changing the mean and standard deviation will change the characteristics of a normal curve.<\/li>\n<\/ul>\n<\/section>\n<p>Recall that a normal distribution is a mathematical model with a smooth bell-shaped curve.<\/p>\n<section class=\"textbox example\">The final exam scores in a statistics class were normally distributed with a mean of [latex]63[\/latex]and a standard deviation of five.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5872 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/04012153\/STAT_8D_Preview_Q10.png\" alt=\"A normal distribution curve with boxes marked A, B, C, D, E, F, G corresponding to ticks on the x-axis.\" width=\"936\" height=\"496\" \/><br \/>\n<strong>(a) <\/strong>What is the value of each tick mark labeled above? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q322532\">Show answer<\/button><\/p>\n<div id=\"q322532\" class=\"hidden-answer\" style=\"display: none\">\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5873 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/04012418\/Screenshot-2023-01-03-at-6.24.10-PM.png\" alt=\"A normal distribution curve with the ticks having numbers corresponding to the boxes from the above graph. A=48, B=53, C=58, D=63, E=68, F=73, G=78.\" width=\"884\" height=\"486\" \/><\/div>\n<\/div>\n<p><strong>(b)<\/strong> The score of ______ is [latex]1[\/latex] standard deviation above the mean.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q913853\">Show answer<\/button><\/p>\n<div id=\"q913853\" class=\"hidden-answer\" style=\"display: none\">The score of <strong>68<\/strong> is [latex]1[\/latex] standard deviation above the mean.<\/div>\n<\/div>\n<p><strong>(c)<\/strong> The score of ______ has a [latex]z[\/latex]-score of [latex]-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q78010\">Show answer<\/button><\/p>\n<div id=\"q78010\" class=\"hidden-answer\" style=\"display: none\">[latex]z[\/latex]-score [latex]=-3[\/latex] means that it is 3 standard deviations below the mean. So, the answer is [latex]48[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Find the mean and standard deviation of the following curves.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5874 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/04013002\/m6_probability_topic_6_2_normalrandomvariables_1_image2.png\" alt=\"A normal curve in black with a mean of 10, a curve in red with a mean of 10, and a curve in green with a mean of 14.\" width=\"530\" height=\"334\" \/><strong>(a)\u00a0<\/strong>What is the mean and standard deviation of the black color normal distribution curve?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q533130\">Show answer<\/button><\/p>\n<div id=\"q533130\" class=\"hidden-answer\" style=\"display: none\">[latex]\\mu = 10[\/latex] and [latex]\\sigma=1[\/latex]<\/div>\n<\/div>\n<p><strong>(b) <\/strong>What is the mean and standard deviation of the red color normal distribution curve?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q533131\">Show answer<\/button><\/p>\n<div id=\"q533131\" class=\"hidden-answer\" style=\"display: none\">[latex]\\mu = 10[\/latex] and [latex]\\sigma=2[\/latex]<\/div>\n<\/div>\n<p><strong>(c) <\/strong>What is the mean and standard deviation of the green color normal distribution curve?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q533132\">Show answer<\/button><\/p>\n<div id=\"q533132\" class=\"hidden-answer\" style=\"display: none\">[latex]\\mu = 14[\/latex] and [latex]\\sigma=1[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":8,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":3053,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1054"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1054\/revisions"}],"predecessor-version":[{"id":5957,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1054\/revisions\/5957"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/3053"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1054\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1054"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1054"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1054"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}