{"id":2163,"date":"2023-07-27T23:11:06","date_gmt":"2023-07-27T23:11:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=2163"},"modified":"2025-05-16T03:49:09","modified_gmt":"2025-05-16T03:49:09","slug":"connecting-tests-and-intervals-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/connecting-tests-and-intervals-learn-it-4\/","title":{"raw":"Connecting Tests and Intervals: Learn It 4","rendered":"Connecting Tests and Intervals: Learn It 4"},"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;Explain how a confidence interval is related to a two-sided hypothesis test&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;}\">Explain how a confidence interval is related to a two-sided hypothesis test.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Confidence Interval - Hypothesis Testing<\/h2>\r\n<p>A confidence interval provides a range of population values with which a sample statistic is consistent at a given confidence level. In many cases, a confidence interval can also be used to either reject or not reject the null hypothesis, and therefore perform the same function as the typical hypothesis test.<\/p>\r\n<p>The Lego experiment considered the population parameter of the difference of proportions, [latex] p_{1} -p_{2} [\/latex]. The confidence intervals constructed were <strong>two-tailed<\/strong>, since [latex] -z^{*} [\/latex] is the point on the standard normal distribution such that the proportion of area under the curve between [latex] -z^{*} [\/latex] and [latex] +z^{*}[\/latex] is [latex] C [\/latex], the confidence level.<\/p>\r\n<section class=\"textbox proTip\">The two-tailed confidence intervals with a confidence level of\u00a0[latex]C [\/latex] correspond to two-tailed hypothesis tests with a significance level of [latex] 1-C[\/latex].<\/section>\r\n<p>For example, a [latex]95\\%[\/latex] confidence interval corresponds to a hypothesis test with a significance level of [latex]5\\%[\/latex], or [latex] \\alpha = 0.05[\/latex]. Similarly, a [latex]99\\%[\/latex] confidence interval corresponds to a hypothesis test with a significance level of [latex]1\\%[\/latex], or [latex]\\alpha =0.01 [\/latex].<\/p>\r\n<p>Let\u2019s connect the result of the confidence interval to the conclusion of the hypothesis test.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]11083[\/ohm2_question]<\/section>\r\n<p>Typically, the conclusion drawn from a two-tailed confidence interval is usually the same as the conclusion drawn from a two-tailed hypothesis test.<\/p>\r\n<section>\r\n<section class=\"textbox proTip\">\r\n<ul>\r\n\t<li>If a confidence interval contains the hypothesized parameter, a hypothesis test at the [latex]0.05[\/latex] level will <em>almost always<\/em> fail to reject the null hypothesis.<\/li>\r\n\t<li>If the [latex]95\\%[\/latex] confidence interval does not contain the hypothesized parameter, a hypothesis test at the [latex]0.05[\/latex] level will <em>almost always<\/em> reject the null hypothesis.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<\/section>\r\n<p>While this does not always hold for tests of proportions, a confidence interval typically provides more information about reasonable values of the parameter.<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Explain how a confidence interval is related to a two-sided hypothesis test&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;}\">Explain how a confidence interval is related to a two-sided hypothesis test.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Confidence Interval &#8211; Hypothesis Testing<\/h2>\n<p>A confidence interval provides a range of population values with which a sample statistic is consistent at a given confidence level. In many cases, a confidence interval can also be used to either reject or not reject the null hypothesis, and therefore perform the same function as the typical hypothesis test.<\/p>\n<p>The Lego experiment considered the population parameter of the difference of proportions, [latex]p_{1} -p_{2}[\/latex]. The confidence intervals constructed were <strong>two-tailed<\/strong>, since [latex]-z^{*}[\/latex] is the point on the standard normal distribution such that the proportion of area under the curve between [latex]-z^{*}[\/latex] and [latex]+z^{*}[\/latex] is [latex]C[\/latex], the confidence level.<\/p>\n<section class=\"textbox proTip\">The two-tailed confidence intervals with a confidence level of\u00a0[latex]C[\/latex] correspond to two-tailed hypothesis tests with a significance level of [latex]1-C[\/latex].<\/section>\n<p>For example, a [latex]95\\%[\/latex] confidence interval corresponds to a hypothesis test with a significance level of [latex]5\\%[\/latex], or [latex]\\alpha = 0.05[\/latex]. Similarly, a [latex]99\\%[\/latex] confidence interval corresponds to a hypothesis test with a significance level of [latex]1\\%[\/latex], or [latex]\\alpha =0.01[\/latex].<\/p>\n<p>Let\u2019s connect the result of the confidence interval to the conclusion of the hypothesis test.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm11083\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=11083&theme=lumen&iframe_resize_id=ohm11083&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Typically, the conclusion drawn from a two-tailed confidence interval is usually the same as the conclusion drawn from a two-tailed hypothesis test.<\/p>\n<section>\n<section class=\"textbox proTip\">\n<ul>\n<li>If a confidence interval contains the hypothesized parameter, a hypothesis test at the [latex]0.05[\/latex] level will <em>almost always<\/em> fail to reject the null hypothesis.<\/li>\n<li>If the [latex]95\\%[\/latex] confidence interval does not contain the hypothesized parameter, a hypothesis test at the [latex]0.05[\/latex] level will <em>almost always<\/em> reject the null hypothesis.<\/li>\n<\/ul>\n<\/section>\n<\/section>\n<p>While this does not always hold for tests of proportions, a confidence interval typically provides more information about reasonable values of the parameter.<\/p>\n","protected":false},"author":12,"menu_order":43,"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":"learn_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\/2163"}],"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\/12"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2163\/revisions"}],"predecessor-version":[{"id":6777,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2163\/revisions\/6777"}],"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\/2163\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=2163"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=2163"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=2163"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=2163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}