{"id":1067,"date":"2023-06-22T01:45:31","date_gmt":"2023-06-22T01:45:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/connection-between-binomial-and-normal-distributions-apply-it-1\/"},"modified":"2024-02-10T00:09:28","modified_gmt":"2024-02-10T00:09:28","slug":"connection-between-binomial-and-normal-distributions-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/connection-between-binomial-and-normal-distributions-apply-it-1\/","title":{"raw":"Connection Between Binomial and Normal Distributions: Apply It 1","rendered":"Connection Between Binomial and Normal Distributions: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use normal probability distribution to calculate binomial probabilities&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Use normal probability distribution to calculate binomial probabilities<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the conditions for applying a normal distribution to approximate a binomial distribution&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the conditions for applying a normal distribution to approximate a binomial distribution<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Connection between Binomial and Normal Distributions<\/h2>\r\n<p><img class=\"aligncenter wp-image-1694\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/13184719\/Picture17.jpg\" alt=\"A basketball player about to take a shot.\" width=\"317\" height=\"205\" \/><\/p>\r\n<h3>Is it large enough?<\/h3>\r\n<p>Previously, we\u00a0learned about the role that [latex] p [\/latex] plays in the shape of the binomial distribution. The binomial probability distribution is skewed right if [latex] p &lt; 0.5 [\/latex], symmetric and approximately bell shaped if [latex] p= 0.5 [\/latex], and skewed left if [latex] p &gt; 0.5 [\/latex].<\/p>\r\n<p>Let\u2019s discuss the role that [latex] n [\/latex] plays in its shape. For a fixed [latex] p [\/latex], as the number of trials, [latex] n [\/latex], in a binomial experiment increases, the probability distribution of the random variable [latex] X [\/latex] becomes nearly symmetric and bell shaped.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>condition for a normal distribution<\/h3>\r\n<p>The binomial distribution can be approximated well by the normal distribution when n is large enough so that the expected number of successes, [latex] np [\/latex], and the expected number of failures, [latex] n(1-p) [\/latex], are both at least [latex]10[\/latex].<\/p>\r\n<p>That is: The probability distribution will be approximately symmetric and bell shaped if<\/p>\r\n<ul>\r\n\t<li>[latex] np \\geq 10[\/latex] AND<\/li>\r\n\t<li>[latex] n(1-p) \\geq 10 [\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1539[\/ohm2_question]<\/section>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]905[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use normal probability distribution to calculate binomial probabilities&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Use normal probability distribution to calculate binomial probabilities<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the conditions for applying a normal distribution to approximate a binomial distribution&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the conditions for applying a normal distribution to approximate a binomial distribution<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>The Connection between Binomial and Normal Distributions<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1694\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/13184719\/Picture17.jpg\" alt=\"A basketball player about to take a shot.\" width=\"317\" height=\"205\" \/><\/p>\n<h3>Is it large enough?<\/h3>\n<p>Previously, we\u00a0learned about the role that [latex]p[\/latex] plays in the shape of the binomial distribution. The binomial probability distribution is skewed right if [latex]p < 0.5[\/latex], symmetric and approximately bell shaped if [latex]p= 0.5[\/latex], and skewed left if [latex]p > 0.5[\/latex].<\/p>\n<p>Let\u2019s discuss the role that [latex]n[\/latex] plays in its shape. For a fixed [latex]p[\/latex], as the number of trials, [latex]n[\/latex], in a binomial experiment increases, the probability distribution of the random variable [latex]X[\/latex] becomes nearly symmetric and bell shaped.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>condition for a normal distribution<\/h3>\n<p>The binomial distribution can be approximated well by the normal distribution when n is large enough so that the expected number of successes, [latex]np[\/latex], and the expected number of failures, [latex]n(1-p)[\/latex], are both at least [latex]10[\/latex].<\/p>\n<p>That is: The probability distribution will be approximately symmetric and bell shaped if<\/p>\n<ul>\n<li>[latex]np \\geq 10[\/latex] AND<\/li>\n<li>[latex]n(1-p) \\geq 10[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1539\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1539&theme=lumen&iframe_resize_id=ohm1539&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm905\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=905&theme=lumen&iframe_resize_id=ohm905&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2912,"module-header":"apply_it","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\/1067"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1067\/revisions"}],"predecessor-version":[{"id":5534,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1067\/revisions\/5534"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/2912"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1067\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1067"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1067"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1067"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1067"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}