{"id":271,"date":"2023-02-20T17:14:18","date_gmt":"2023-02-20T17:14:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-apply-it-1\/"},"modified":"2025-05-11T23:20:18","modified_gmt":"2025-05-11T23:20:18","slug":"coefficient-of-determination-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/coefficient-of-determination-apply-it-1\/","title":{"raw":"Coefficient of Determination: Apply It 1","rendered":"Coefficient of Determination: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Describe how the slope, shape of the data, and the coefficient of determination are connected.<\/li>\r\n\t<li>Find [latex]R^2[\/latex] and describe how [latex]R^2[\/latex] describes the relationship in a data set.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Thinking About Education<\/h2>\r\n<p>Let's approach this activity from the perspective of the secretary of education in your state. You noticed that many public school students in your state are not showing good results on their high school math exams. So, you would like to introduce a policy change that will lead to better results. Your first step should be to collect data about high school students to see what factors best predict their math performance.<\/p>\r\n\r\n[caption id=\"attachment_627\" align=\"aligncenter\" width=\"284\"]<img class=\"wp-image-627 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-284x300.jpg\" alt=\"Students and a teacher standing and smiling in front of a chalkboard that reads &quot;Back to School&quot;\" width=\"284\" height=\"300\" \/> Figure 1. Before introducing new education policies, it\u2019s important to understand the students they\u2019ll affect\u2014collecting data from students can help identify what influences math success.[\/caption]\r\n\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1297[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1298[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1299[\/ohm2_question]<\/section>\r\n<section>By comparing [latex]R^{2}[\/latex], we can make some decisions regarding the data set because\u00a0[latex]R^{2}[\/latex] represents the percent of variation in the dependent (predicted) variable [latex]y[\/latex]\u00a0that can be explained by variation in the independent (explanatory) variable\u00a0[latex]x[\/latex]\u00a0using the regression (best-fit) line.\r\n\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1301[\/ohm2_question]<\/section>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1300[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1302[\/ohm2_question]<\/section>\r\n<h3><strong>\u201cCorrelation does not imply causation.\u201d<\/strong><\/h3>\r\n<p>A common mistake people make when describing the relationship between two quantitative variables is that they confuse\u00a0association\u00a0and\u00a0causation.\u00a0This confusion often occurs when there is a strong relationship between the two quantitative variables.<\/p>\r\n<p>In the case of a linear relationship, people mistakenly interpret an [latex]r[\/latex]-value that is close to [latex]1[\/latex] or [latex]-1[\/latex] or an [latex]R^{2}[\/latex] that is close to [latex]1[\/latex] as evidence that the explanatory variable causes changes in the response variable. In this case, the correct interpretation is that there is a <strong>statistical relationship<\/strong> between the variables, not a causal link. In other words, the explanatory variable and the response variable vary together in a predictable way. There is an <strong>association<\/strong> between the variables. But this should not be interpreted as a cause-and-effect relationship.<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Describe how the slope, shape of the data, and the coefficient of determination are connected.<\/li>\n<li>Find [latex]R^2[\/latex] and describe how [latex]R^2[\/latex] describes the relationship in a data set.<\/li>\n<\/ul>\n<\/section>\n<h2>Thinking About Education<\/h2>\n<p>Let&#8217;s approach this activity from the perspective of the secretary of education in your state. You noticed that many public school students in your state are not showing good results on their high school math exams. So, you would like to introduce a policy change that will lead to better results. Your first step should be to collect data about high school students to see what factors best predict their math performance.<\/p>\n<figure id=\"attachment_627\" aria-describedby=\"caption-attachment-627\" style=\"width: 284px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-627 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-284x300.jpg\" alt=\"Students and a teacher standing and smiling in front of a chalkboard that reads &quot;Back to School&quot;\" width=\"284\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-284x300.jpg 284w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-969x1024.jpg 969w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-768x812.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-1453x1536.jpg 1453w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-1200x1268.jpg 1200w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-65x69.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-225x238.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172-350x370.jpg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/02\/28184411\/pexels-rodnae-productions-8500302-scaled-e1671830152172.jpg 1689w\" sizes=\"(max-width: 284px) 100vw, 284px\" \/><figcaption id=\"caption-attachment-627\" class=\"wp-caption-text\">Figure 1. Before introducing new education policies, it\u2019s important to understand the students they\u2019ll affect\u2014collecting data from students can help identify what influences math success.<\/figcaption><\/figure>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1297\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1297&theme=lumen&iframe_resize_id=ohm1297&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1298\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1298&theme=lumen&iframe_resize_id=ohm1298&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1299\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1299&theme=lumen&iframe_resize_id=ohm1299&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>By comparing [latex]R^{2}[\/latex], we can make some decisions regarding the data set because\u00a0[latex]R^{2}[\/latex] represents the percent of variation in the dependent (predicted) variable [latex]y[\/latex]\u00a0that can be explained by variation in the independent (explanatory) variable\u00a0[latex]x[\/latex]\u00a0using the regression (best-fit) line.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1301\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1301&theme=lumen&iframe_resize_id=ohm1301&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1300\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1300&theme=lumen&iframe_resize_id=ohm1300&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1302\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1302&theme=lumen&iframe_resize_id=ohm1302&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3><strong>\u201cCorrelation does not imply causation.\u201d<\/strong><\/h3>\n<p>A common mistake people make when describing the relationship between two quantitative variables is that they confuse\u00a0association\u00a0and\u00a0causation.\u00a0This confusion often occurs when there is a strong relationship between the two quantitative variables.<\/p>\n<p>In the case of a linear relationship, people mistakenly interpret an [latex]r[\/latex]-value that is close to [latex]1[\/latex] or [latex]-1[\/latex] or an [latex]R^{2}[\/latex] that is close to [latex]1[\/latex] as evidence that the explanatory variable causes changes in the response variable. In this case, the correct interpretation is that there is a <strong>statistical relationship<\/strong> between the variables, not a causal link. In other words, the explanatory variable and the response variable vary together in a predictable way. There is an <strong>association<\/strong> between the variables. But this should not be interpreted as a cause-and-effect relationship.<\/p>\n<\/section>\n","protected":false},"author":12,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Students posing with teacher image\",\"author\":\"RODNAE Productions\",\"organization\":\"Pexels\",\"url\":\"https:\/\/www.pexels.com\/photo\/group-of-people-standing-beside-chalkboard-8500302\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":225,"module-header":"apply_it","content_attributions":[{"type":"cc","description":"Students posing with teacher image","author":"RODNAE Productions","organization":"Pexels","url":"https:\/\/www.pexels.com\/photo\/group-of-people-standing-beside-chalkboard-8500302\/","project":"","license":"cc0","license_terms":""}],"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\/271"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/271\/revisions"}],"predecessor-version":[{"id":6660,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/271\/revisions\/6660"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/271\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=271"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=271"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=271"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=271"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}