{"id":1447,"date":"2023-06-22T02:28:41","date_gmt":"2023-06-22T02:28:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/test-for-significance-of-slope-learn-it-2\/"},"modified":"2025-05-17T02:32:42","modified_gmt":"2025-05-17T02:32:42","slug":"test-for-significance-of-slope-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/test-for-significance-of-slope-learn-it-2\/","title":{"raw":"Test for Significance of Slope - Learn It 2","rendered":"Test for Significance of Slope &#8211; Learn It 2"},"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;Perform a test for significance of slope and interpret the results&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6913,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Perform a test for significance of slope and interpret the results<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the conditions that are necessary to perform a test for significance of slope&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6913,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Check the conditions that are necessary to perform a test for significance of slope<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2><strong>Slope of the Population<\/strong><strong> Regression Line<\/strong><\/h2>\r\n<p>In general, the sign of [latex]r[\/latex]\u00a0(positive, negative, or [latex]0[\/latex]) will be the same as the sign of [latex]b[\/latex]. This tells us that in studying and understanding the correlation coefficient, we simultaneously have information about the slope of the line of best fit.<\/p>\r\n<p>When the line of best fit is estimated, the slope, [latex]b[\/latex], is calculated using sample data. The slope, [latex]b[\/latex], is an estimate of the <strong>slope of the population<\/strong><strong> regression line<\/strong>, [latex]\\beta_1[\/latex]. This is similar to the relationship between the sample mean, [latex]\\bar{x}[\/latex], and the population mean, [latex]\\mu[\/latex], that we previously studied.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3051[\/ohm2_question]<\/section>\r\n<h2>Hypothesis Test for Significance of Slope<\/h2>\r\n<p>We want to conduct a hypothesis test to find out whether or not two quantitative variables have a significant linear relationship. When two variables do NOT have a significant linear relationship, the true value of the slope of the population line is [latex]0: \\beta_1 = 0[\/latex]. That is, the population regression line is a horizontal line, and the value of [latex]y[\/latex] in the simple linear regression model does not depend on\u00a0 [latex]x[\/latex]<em>.<\/em><\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>null and alternative hypotheses<\/h3>\r\n<p>To carry out a <strong>hypothesis<\/strong> <strong>test for significance of slope,<\/strong> often referred to as a <strong>model utility test<\/strong>, we will test the following:<\/p>\r\n<ul>\r\n\t<li>Null hypothesis: [latex]H_0: \\beta_1=0[\/latex]<\/li>\r\n\t<li>Alternative hypothesis:\u00a0[latex]H_{A}: \\beta_1 \\ne 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3052[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Perform a test for significance of slope and interpret the results&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6913,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Perform a test for significance of slope and interpret the results<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the conditions that are necessary to perform a test for significance of slope&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6913,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Check the conditions that are necessary to perform a test for significance of slope<\/span><\/li>\n<\/ul>\n<\/section>\n<h2><strong>Slope of the Population<\/strong><strong> Regression Line<\/strong><\/h2>\n<p>In general, the sign of [latex]r[\/latex]\u00a0(positive, negative, or [latex]0[\/latex]) will be the same as the sign of [latex]b[\/latex]. This tells us that in studying and understanding the correlation coefficient, we simultaneously have information about the slope of the line of best fit.<\/p>\n<p>When the line of best fit is estimated, the slope, [latex]b[\/latex], is calculated using sample data. The slope, [latex]b[\/latex], is an estimate of the <strong>slope of the population<\/strong><strong> regression line<\/strong>, [latex]\\beta_1[\/latex]. This is similar to the relationship between the sample mean, [latex]\\bar{x}[\/latex], and the population mean, [latex]\\mu[\/latex], that we previously studied.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3051\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3051&theme=lumen&iframe_resize_id=ohm3051&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Hypothesis Test for Significance of Slope<\/h2>\n<p>We want to conduct a hypothesis test to find out whether or not two quantitative variables have a significant linear relationship. When two variables do NOT have a significant linear relationship, the true value of the slope of the population line is [latex]0: \\beta_1 = 0[\/latex]. That is, the population regression line is a horizontal line, and the value of [latex]y[\/latex] in the simple linear regression model does not depend on\u00a0 [latex]x[\/latex]<em>.<\/em><\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>null and alternative hypotheses<\/h3>\n<p>To carry out a <strong>hypothesis<\/strong> <strong>test for significance of slope,<\/strong> often referred to as a <strong>model utility test<\/strong>, we will test the following:<\/p>\n<ul>\n<li>Null hypothesis: [latex]H_0: \\beta_1=0[\/latex]<\/li>\n<li>Alternative hypothesis:\u00a0[latex]H_{A}: \\beta_1 \\ne 0[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3052\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3052&theme=lumen&iframe_resize_id=ohm3052&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1438,"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\/1447"}],"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\/1447\/revisions"}],"predecessor-version":[{"id":6891,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1447\/revisions\/6891"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1438"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1447\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1447"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1447"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1447"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1447"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}