{"id":1249,"date":"2023-06-22T02:09:36","date_gmt":"2023-06-22T02:09:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/errors-in-hypothesis-testing-dig-deeper\/"},"modified":"2025-05-16T03:41:58","modified_gmt":"2025-05-16T03:41:58","slug":"errors-in-hypothesis-testing-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/errors-in-hypothesis-testing-dig-deeper\/","title":{"raw":"Errors in Hypothesis Testing: Fresh Take","rendered":"Errors in Hypothesis Testing: Fresh Take"},"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;Recognize Type I and Type II errors and their consequences&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;}\">Recognize Type I and Type II errors and their consequences.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>If the null hypothesis is true, the sample should lead us to that conclusion. If the null hypothesis is false, the sample should lead us to that conclusion, too. However, sometimes the data might not lead to the correct conclusion.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>errors in hypothesis testing<\/h3>\r\n<p><em>[latex]\u03b1[\/latex]<\/em>\u00a0= probability of a Type I error =\u00a0<strong>P(Type I error)<\/strong>\u00a0= probability of rejecting the null hypothesis when the null hypothesis is true.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><em>[latex]\u03b2[\/latex]<\/em>\u00a0= probability of a Type II error =\u00a0<strong>P(Type II error)<\/strong> = probability of not rejecting the null hypothesis when the null hypothesis is false.<\/p>\r\n<\/section>\r\n<p>The four possible outcomes in hypothesis testing:<\/p>\r\n<ol>\r\n\t<li>The decision is\u00a0not to reject\u00a0[latex]H_0[\/latex] when [latex]H_0[\/latex]\u00a0is true (correct decision).<\/li>\r\n\t<li>The decision is to\u00a0reject [latex]H_0[\/latex] when [latex]H_0[\/latex] is true (incorrect decision known as a Type I error).<\/li>\r\n\t<li>The decision is not to reject [latex]H_0[\/latex]<sub>\u00a0<\/sub>when, in fact, [latex]H_0[\/latex] is false (incorrect decision known as a Type II error).<\/li>\r\n\t<li>The decision is to reject [latex]H_0[\/latex] when [latex]H_0[\/latex] is false (correct decision whose probability is called the Power of the Test).<\/li>\r\n<\/ol>\r\n<section class=\"textbox example\">Suppose the null hypothesis, [latex]H_0[\/latex], is: Frank\u2019s rock climbing equipment is safe.\r\n\r\n<ul>\r\n\t<li><strong>Ty<\/strong><strong>pe I error:<\/strong>\u00a0Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe.<\/li>\r\n\t<li><strong>Type II error:<\/strong>\u00a0Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.<\/li>\r\n<\/ul>\r\n<p>Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<p>[embed]https:\/\/youtu.be\/EowIec7Y8HM[\/embed]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Recognize Type I and Type II errors and their consequences&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;}\">Recognize Type I and Type II errors and their consequences.<\/span><\/li>\n<\/ul>\n<\/section>\n<p>If the null hypothesis is true, the sample should lead us to that conclusion. If the null hypothesis is false, the sample should lead us to that conclusion, too. However, sometimes the data might not lead to the correct conclusion.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>errors in hypothesis testing<\/h3>\n<p><em>[latex]\u03b1[\/latex]<\/em>\u00a0= probability of a Type I error =\u00a0<strong>P(Type I error)<\/strong>\u00a0= probability of rejecting the null hypothesis when the null hypothesis is true.<\/p>\n<p>&nbsp;<\/p>\n<p><em>[latex]\u03b2[\/latex]<\/em>\u00a0= probability of a Type II error =\u00a0<strong>P(Type II error)<\/strong> = probability of not rejecting the null hypothesis when the null hypothesis is false.<\/p>\n<\/section>\n<p>The four possible outcomes in hypothesis testing:<\/p>\n<ol>\n<li>The decision is\u00a0not to reject\u00a0[latex]H_0[\/latex] when [latex]H_0[\/latex]\u00a0is true (correct decision).<\/li>\n<li>The decision is to\u00a0reject [latex]H_0[\/latex] when [latex]H_0[\/latex] is true (incorrect decision known as a Type I error).<\/li>\n<li>The decision is not to reject [latex]H_0[\/latex]<sub>\u00a0<\/sub>when, in fact, [latex]H_0[\/latex] is false (incorrect decision known as a Type II error).<\/li>\n<li>The decision is to reject [latex]H_0[\/latex] when [latex]H_0[\/latex] is false (correct decision whose probability is called the Power of the Test).<\/li>\n<\/ol>\n<section class=\"textbox example\">Suppose the null hypothesis, [latex]H_0[\/latex], is: Frank\u2019s rock climbing equipment is safe.<\/p>\n<ul>\n<li><strong>Ty<\/strong><strong>pe I error:<\/strong>\u00a0Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe.<\/li>\n<li><strong>Type II error:<\/strong>\u00a0Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.<\/li>\n<\/ul>\n<p>Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)<\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Type 1 errors | Inferential statistics | Probability and Statistics | Khan Academy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EowIec7Y8HM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":8,"menu_order":31,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1205,"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\/1249"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1249\/revisions"}],"predecessor-version":[{"id":6768,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1249\/revisions\/6768"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1205"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1249\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1249"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1249"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1249"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}