{"id":2493,"date":"2021-10-15T11:42:55","date_gmt":"2021-10-15T11:42:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2493"},"modified":"2023-12-05T09:48:03","modified_gmt":"2023-12-05T09:48:03","slug":"summary-testing-the-significance-of-the-correlation-coefficient","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-testing-the-significance-of-the-correlation-coefficient\/","title":{"raw":"Summary: Testing the Significance of the Correlation Coefficient","rendered":"Summary: Testing the Significance of the Correlation Coefficient"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The null hypothesis is the population correlation coefficient is not significantly different from zero. This means there is not a significant linear relationship (correlation) between <em>x<\/em> and <em>y<\/em>. The line should not be used for making predictions.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The alternate hypothesis is the population correlation coefficient is significantly different from zero. This means there is a significant linear relationship (correlation) between <em>x<\/em> and <em>y<\/em> in the population. The line should be used for making predictions.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The <em>p<\/em>-value is calculated using [latex]n - 2[\/latex] degrees of freedom and the test statistic is [latex]t = \\frac{r \\sqrt{n-2}}{\\sqrt{1-r^2}}[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">A critical value approach is an alternative method to doing a test of significance for a correlation coefficient.\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">There are assumptions that need to be verified before doing the test of significance for a correlation coefficient. They are as follows:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The underlying relationship is a linear relationship.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The <em>y<\/em> values for any particular <em>x<\/em> value are normally distributed about the line.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The standard deviations of the population y values about the line are equal for each value of <em>x<\/em>. There is no pattern in a plot of the residuals.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The data are produced from a well-designed, random sample or randomized experiment.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>Coefficient of Correlation:\u00a0<\/strong>a measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable; the formula is\r\n<p style=\"text-align: center;\">[latex]\\LARGE r = \\frac{n \\sum{(xy)} - (\\sum{x})(\\sum{y})}{\\sqrt{[n \\sum{x^2} - (\\sum{x})^2 ] [ n \\sum{y^2} - (\\sum{y})^2 ]}} [\/latex]<\/p>\r\nwhere [latex]n[\/latex] is the number of data points. The coefficient cannot be more than 1 or less than \u20131. The closer the coefficient is to \u00b11, the stronger the evidence of a significant linear relationship between <em>x<\/em> and <em>y<\/em>.","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The null hypothesis is the population correlation coefficient is not significantly different from zero. This means there is not a significant linear relationship (correlation) between <em>x<\/em> and <em>y<\/em>. The line should not be used for making predictions.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The alternate hypothesis is the population correlation coefficient is significantly different from zero. This means there is a significant linear relationship (correlation) between <em>x<\/em> and <em>y<\/em> in the population. The line should be used for making predictions.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The <em>p<\/em>-value is calculated using [latex]n - 2[\/latex] degrees of freedom and the test statistic is [latex]t = \\frac{r \\sqrt{n-2}}{\\sqrt{1-r^2}}[\/latex].<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">A critical value approach is an alternative method to doing a test of significance for a correlation coefficient.\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">There are assumptions that need to be verified before doing the test of significance for a correlation coefficient. They are as follows:\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The underlying relationship is a linear relationship.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The <em>y<\/em> values for any particular <em>x<\/em> value are normally distributed about the line.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The standard deviations of the population y values about the line are equal for each value of <em>x<\/em>. There is no pattern in a plot of the residuals.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The data are produced from a well-designed, random sample or randomized experiment.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>Coefficient of Correlation:\u00a0<\/strong>a measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable; the formula is<\/p>\n<p style=\"text-align: center;\">[latex]\\LARGE r = \\frac{n \\sum{(xy)} - (\\sum{x})(\\sum{y})}{\\sqrt{[n \\sum{x^2} - (\\sum{x})^2 ] [ n \\sum{y^2} - (\\sum{y})^2 ]}}[\/latex]<\/p>\n<p>where [latex]n[\/latex] is the number of data points. The coefficient cannot be more than 1 or less than \u20131. The closer the coefficient is to \u00b11, the stronger the evidence of a significant linear relationship between <em>x<\/em> and <em>y<\/em>.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2493\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-key-terms<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-key-terms\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2493","chapter","type-chapter","status-publish","hentry"],"part":303,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2493","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2493\/revisions"}],"predecessor-version":[{"id":2494,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2493\/revisions\/2494"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/303"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2493\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2493"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2493"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2493"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}