{"id":2076,"date":"2021-09-27T14:46:00","date_gmt":"2021-09-27T14:46:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2076"},"modified":"2023-12-05T09:34:22","modified_gmt":"2023-12-05T09:34:22","slug":"rare-events-the-sample-decision-and-conclusion-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/rare-events-the-sample-decision-and-conclusion-2\/","title":{"raw":"Decision and Conclusion","rendered":"Decision and Conclusion"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul id=\"list67\">\r\n \t<li>Explain what a <em>p<\/em>-value means about a given test statistic<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2 data-type=\"title\">Decision and Conclusion<\/h2>\r\n<p id=\"fs-idp169707568\">A systematic way to make a decision of whether to reject or to not reject the <strong>null hypothesis<\/strong> is to compare the <em data-effect=\"italics\">p<\/em>-value and a <strong>preset or preconceived <span data-type=\"term\">\u03b1<\/span> (also called a \"significance level\")<\/strong>. A preset <em data-effect=\"italics\">\u03b1<\/em> is the probability of a <span data-type=\"term\">Type I error<\/span> (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.<\/p>\r\nWhen you make a <strong>decision<\/strong> to reject or to not reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>, do as follows:\r\n<div data-type=\"list\" data-list-type=\"bulleted\">\r\n<ul>\r\n \t<li data-type=\"item\">If <em data-effect=\"italics\">\u03b1<\/em> &gt; <em data-effect=\"italics\">p<\/em>-value, reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>. The results of the sample data are significant. There is sufficient evidence to conclude that <em data-effect=\"italics\">H<sub>0<\/sub><\/em> is an incorrect belief and that the\u00a0<strong>alternative hypothesis<\/strong>, <em data-effect=\"italics\">H<sub>a<\/sub><\/em>, may be correct.<\/li>\r\n \t<li data-type=\"item\">If <em data-effect=\"italics\">\u03b1<\/em> \u2264 <em data-effect=\"italics\">p<\/em>-value, do not reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>. The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis,\u00a0<em data-effect=\"italics\">H<sub>a<\/sub>,\u00a0<\/em>may be correct.<\/li>\r\n \t<li data-type=\"item\">When you \"do not reject <em style=\"font-size: 1em;\" data-effect=\"italics\">H<sub>0<\/sub><\/em><span style=\"font-size: 1em;\">,\" it does not mean that you should believe that <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">H<sub>0<\/sub><\/em><span style=\"font-size: 1em;\"> is true. It simply means that the sample data have\u00a0<\/span><strong style=\"font-size: 1em;\">failed<\/strong><span style=\"font-size: 1em;\"> to provide sufficient evidence to cast serious doubt about the truthfulness of <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">H<sub>o<\/sub><\/em><span style=\"font-size: 1em;\">.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"element-734\"><strong>Conclusion:<\/strong> After you make your decision, write a thoughtful <strong>conclusion<\/strong> about the hypotheses in terms of the given problem.<\/p>\r\n\r\n<div class=\"example\" data-type=\"example\"><section>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 2<\/h3>\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div id=\"eip-890\" class=\"problem\" data-type=\"problem\">\r\n\r\nWhen using the p-value to evaluate a hypothesis test, it is sometimes useful to use the following memory device:\r\n\r\nIf the p-value is low, the null must go.\r\n\r\nIf the p-value is high, the null must fly.\r\n\r\nThis memory aid relates a p-value less than the established alpha (the p is low) as rejecting the null hypothesis and, likewise, relates a p-value higher than the established alpha (the p is high) as not rejecting the null hypothesis.\r\n\r\nFill in the blanks.\r\n\r\nReject the null hypothesis when ______________________________________.\r\n\r\nThe results of the sample data _____________________________________.\r\n\r\nDo not reject the null when hypothesis when __________________________________________.\r\n\r\nThe results of the sample data ____________________________________________.\r\n\r\n[reveal-answer q=\"704931\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"704931\"]\r\n\r\nReject the null hypothesis when the <strong>p-value is less than the established alpha value<\/strong>.\r\n\r\nThe results of the sample data\u00a0<strong>support the alternative hypothesis<\/strong>.\r\n\r\nDo not reject the null hypothesis when <strong>the p-value is greater than the established alpha value<\/strong>.\r\n\r\nThe results of the sample data <strong>do not support the alternative hypothesis<\/strong>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 2<\/h3>\r\n<div id=\"fs-idm142970080\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<div id=\"eip-437\" class=\"exercise\" data-type=\"exercise\"><section>It\u2019s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows: H<sub>0<\/sub>: p = 0.50, H<sub>a<\/sub>: p &gt; 0.50\r\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">\u03b1 = 0.0<\/p>\r\n<p class=\"solution ui-solution-visible\" data-type=\"solution\"><em>p<\/em>-value = 0.025<\/p>\r\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">Interpret the results and state a conclusion in simple, non-technical terms.<\/p>\r\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">[reveal-answer q=\"602623\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"602623\"]<\/p>\r\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">Since the p-value is greater than the established alpha value (the p-value is high), we do not reject the null hypothesis. There is not enough evidence to support It\u2019s a\u00a0Boy Genetics Labs' stated claim that their procedures improve the chances of a boy being born.<\/p>\r\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul id=\"list67\">\n<li>Explain what a <em>p<\/em>-value means about a given test statistic<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2 data-type=\"title\">Decision and Conclusion<\/h2>\n<p id=\"fs-idp169707568\">A systematic way to make a decision of whether to reject or to not reject the <strong>null hypothesis<\/strong> is to compare the <em data-effect=\"italics\">p<\/em>-value and a <strong>preset or preconceived <span data-type=\"term\">\u03b1<\/span> (also called a &#8220;significance level&#8221;)<\/strong>. A preset <em data-effect=\"italics\">\u03b1<\/em> is the probability of a <span data-type=\"term\">Type I error<\/span> (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.<\/p>\n<p>When you make a <strong>decision<\/strong> to reject or to not reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>, do as follows:<\/p>\n<div data-type=\"list\" data-list-type=\"bulleted\">\n<ul>\n<li data-type=\"item\">If <em data-effect=\"italics\">\u03b1<\/em> &gt; <em data-effect=\"italics\">p<\/em>-value, reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>. The results of the sample data are significant. There is sufficient evidence to conclude that <em data-effect=\"italics\">H<sub>0<\/sub><\/em> is an incorrect belief and that the\u00a0<strong>alternative hypothesis<\/strong>, <em data-effect=\"italics\">H<sub>a<\/sub><\/em>, may be correct.<\/li>\n<li data-type=\"item\">If <em data-effect=\"italics\">\u03b1<\/em> \u2264 <em data-effect=\"italics\">p<\/em>-value, do not reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>. The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis,\u00a0<em data-effect=\"italics\">H<sub>a<\/sub>,\u00a0<\/em>may be correct.<\/li>\n<li data-type=\"item\">When you &#8220;do not reject <em style=\"font-size: 1em;\" data-effect=\"italics\">H<sub>0<\/sub><\/em><span style=\"font-size: 1em;\">,&#8221; it does not mean that you should believe that <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">H<sub>0<\/sub><\/em><span style=\"font-size: 1em;\"> is true. It simply means that the sample data have\u00a0<\/span><strong style=\"font-size: 1em;\">failed<\/strong><span style=\"font-size: 1em;\"> to provide sufficient evidence to cast serious doubt about the truthfulness of <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">H<sub>o<\/sub><\/em><span style=\"font-size: 1em;\">.<\/span><\/li>\n<\/ul>\n<\/div>\n<p id=\"element-734\"><strong>Conclusion:<\/strong> After you make your decision, write a thoughtful <strong>conclusion<\/strong> about the hypotheses in terms of the given problem.<\/p>\n<div class=\"example\" data-type=\"example\">\n<section>\n<div class=\"textbox exercises\">\n<h3>Example 2<\/h3>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div id=\"eip-890\" class=\"problem\" data-type=\"problem\">\n<p>When using the p-value to evaluate a hypothesis test, it is sometimes useful to use the following memory device:<\/p>\n<p>If the p-value is low, the null must go.<\/p>\n<p>If the p-value is high, the null must fly.<\/p>\n<p>This memory aid relates a p-value less than the established alpha (the p is low) as rejecting the null hypothesis and, likewise, relates a p-value higher than the established alpha (the p is high) as not rejecting the null hypothesis.<\/p>\n<p>Fill in the blanks.<\/p>\n<p>Reject the null hypothesis when ______________________________________.<\/p>\n<p>The results of the sample data _____________________________________.<\/p>\n<p>Do not reject the null when hypothesis when __________________________________________.<\/p>\n<p>The results of the sample data ____________________________________________.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q704931\">Show Answer<\/span><\/p>\n<div id=\"q704931\" class=\"hidden-answer\" style=\"display: none\">\n<p>Reject the null hypothesis when the <strong>p-value is less than the established alpha value<\/strong>.<\/p>\n<p>The results of the sample data\u00a0<strong>support the alternative hypothesis<\/strong>.<\/p>\n<p>Do not reject the null hypothesis when <strong>the p-value is greater than the established alpha value<\/strong>.<\/p>\n<p>The results of the sample data <strong>do not support the alternative hypothesis<\/strong>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it 2<\/h3>\n<div id=\"fs-idm142970080\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<div id=\"eip-437\" class=\"exercise\" data-type=\"exercise\">\n<section>It\u2019s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows: H<sub>0<\/sub>: p = 0.50, H<sub>a<\/sub>: p &gt; 0.50<\/p>\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">\u03b1 = 0.0<\/p>\n<p class=\"solution ui-solution-visible\" data-type=\"solution\"><em>p<\/em>-value = 0.025<\/p>\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">Interpret the results and state a conclusion in simple, non-technical terms.<\/p>\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q602623\">Show Answer<\/span><\/p>\n<div id=\"q602623\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"solution ui-solution-visible\" data-type=\"solution\">Since the p-value is greater than the established alpha value (the p-value is high), we do not reject the null hypothesis. There is not enough evidence to support It\u2019s a\u00a0Boy Genetics Labs&#8217; stated claim that their procedures improve the chances of a boy being born.<\/p>\n<p class=\"solution ui-solution-visible\" data-type=\"solution\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\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-2076\">\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>Rare Events, the Sample, Decision and Conclusion. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/statistics\/pages\/9-4-rare-events-the-sample-and-the-decision-and-conclusion\">https:\/\/openstax.org\/books\/statistics\/pages\/9-4-rare-events-the-sample-and-the-decision-and-conclusion<\/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":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Rare Events, the Sample, Decision and Conclusion\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/statistics\/pages\/9-4-rare-events-the-sample-and-the-decision-and-conclusion\",\"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-2076","chapter","type-chapter","status-publish","hentry"],"part":276,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2076","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":5,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2076\/revisions"}],"predecessor-version":[{"id":3848,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2076\/revisions\/3848"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/276"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2076\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2076"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2076"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2076"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2076"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}