{"id":1453,"date":"2023-06-22T02:28:46","date_gmt":"2023-06-22T02:28:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/test-for-significance-of-slope-fresh-take\/"},"modified":"2025-05-17T02:34:27","modified_gmt":"2025-05-17T02:34:27","slug":"test-for-significance-of-slope-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/test-for-significance-of-slope-fresh-take\/","title":{"raw":"Test for Significance of Slope \u2013 Fresh Take","rendered":"Test for Significance of Slope \u2013 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;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<section class=\"textbox recall\"><strong>Steps for Hypothesis Testing for Significance of Slope:<\/strong>\r\n<ol>\r\n\t<li>Write out the null and alternative hypotheses.\r\n\r\n<ul>\r\n\t<li>Null Hypothesis: [latex]\\beta_1 = 0[\/latex]<\/li>\r\n\t<li>Alternative Hypothesis: [latex]\\beta_1 \\ne 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Check the conditions for the hypothesis test. For testing a one-sample z-test for proportions, we require:\r\n\r\n<ul>\r\n\t<li>A random sample of data<\/li>\r\n\t<li>A linear trend<\/li>\r\n\t<li>No obvious trends in the residual plot<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Calculate the test statistic: [latex]t=\\dfrac{b-0}{[\\text{std. error of }b]} = \\dfrac{b}{SE_b}[\/latex]<\/li>\r\n\t<li>Calculate a P-value.<\/li>\r\n\t<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<br \/>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Decision<\/strong><\/td>\r\n<td><strong>Conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\r\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\r\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<p>[embed]https:\/\/youtu.be\/Mpd83AuDTrU[\/embed]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>","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<section class=\"textbox recall\"><strong>Steps for Hypothesis Testing for Significance of Slope:<\/strong><\/p>\n<ol>\n<li>Write out the null and alternative hypotheses.\n<ul>\n<li>Null Hypothesis: [latex]\\beta_1 = 0[\/latex]<\/li>\n<li>Alternative Hypothesis: [latex]\\beta_1 \\ne 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Check the conditions for the hypothesis test. For testing a one-sample z-test for proportions, we require:\n<ul>\n<li>A random sample of data<\/li>\n<li>A linear trend<\/li>\n<li>No obvious trends in the residual plot<\/li>\n<\/ul>\n<\/li>\n<li>Calculate the test statistic: [latex]t=\\dfrac{b-0}{[\\text{std. error of }b]} = \\dfrac{b}{SE_b}[\/latex]<\/li>\n<li>Calculate a P-value.<\/li>\n<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<br \/>\n<table>\n<tbody>\n<tr>\n<td><strong>Decision<\/strong><\/td>\n<td><strong>Conclusion<\/strong><\/td>\n<\/tr>\n<tr>\n<td>If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<tr>\n<td>If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Using a P-value to make conclusions in a test about slope | AP Statistics | Khan Academy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Mpd83AuDTrU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":8,"menu_order":13,"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":"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\/1453"}],"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":2,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1453\/revisions"}],"predecessor-version":[{"id":6894,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1453\/revisions\/6894"}],"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\/1453\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1453"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1453"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1453"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}