{"id":233,"date":"2017-04-15T03:19:56","date_gmt":"2017-04-15T03:19:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/exponential-relationships-1-of-6\/"},"modified":"2017-06-06T21:03:29","modified_gmt":"2017-06-06T21:03:29","slug":"exponential-relationships-1-of-6","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/chapter\/exponential-relationships-1-of-6\/","title":{"raw":"Exponential Relationships (1 of 6)","rendered":"Exponential Relationships (1 of 6)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use an exponential model (when appropriate) to describe the relationship between two quantitative variables. Interpret the model in context.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn our first example of exponential relationships, we investigate a nonlinear model for growth in a population over time.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h2>The Return of the Bald Eagle<\/h2>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031948\/m4_non_linear_models_ExponentialRelationships1of2_image1.png\" alt=\"Bald eagle\" width=\"318\" height=\"423\" \/>\r\n\r\nDuring the mid-20th century, the population of bald eagles in the lower 48 states declined substantially. A highly toxic pesticide, DDT, was the main cause of the decline. DDT causes damage to bird egg shells. By 1963, bald eagles were in danger of complete extinction. Only 417 pairs of bald eagles remained. In 1967, the bald eagle became an official endangered species. Then in 1972, the EPA banned the use of DDT in the United States. The impact of the ban was a dramatic turnaround in the fate of the bald eagle.\r\n\r\nHere is the data. Note that in the table, we defined <em>t<\/em>, our explanatory variable, to be <em>Years after 1950<\/em>. The response variable is the number of bald eagle pairs that are mating.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031950\/m4_non_linear_models_ExponentialRelationships1of2_image2.png\" alt=\"Data table showing the number of bald eagle pairs mating after 1950, where t = the number of years after 1950\" width=\"329\" height=\"382\" \/>\r\n\r\nOur goal is to find an equation to model this relationship.\r\n\r\nHere is a scatterplot of the data. We can see that the relationship appears somewhat linear, particularly for years after 1980 (<em>t<\/em> = 30). The correlation coefficient for this data set is high, <em>r<\/em> = 0.914.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031952\/m4_non_linear_models_ExponentialRelationships1of2_image3.png\" alt=\"Scatterplot of eagle pairs\" width=\"381\" height=\"273\" \/>\r\n\r\nThe least squares regression line is Predicted eagle pairs = \u22123,878.11 + 185.4<em>t<\/em>. Below is a scatterplot of the data with the least-squares regression line and the residual plot.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031953\/m4_non_linear_models_ExponentialRelationships1of2_image4.png\" alt=\"Scatterplot with least-squares regression of eagle pairs; corresponding residual plot\" width=\"265\" height=\"533\" \/>\r\n\r\nWe see a clear pattern in the residuals, suggesting that a linear model does not capture patterns in the data.\r\n\r\nNote: This is a reminder that a large <em>r<\/em>-value does not guarantee that a linear model is a good fit.\r\n\r\n<strong>Conclusion:<\/strong> The data set for eagle pairs is clearly nonlinear. We need a better model.\r\n\r\nIn the scatterplot below, we fit an <strong>exponential <\/strong>model to the data. Notice how well this model describes the relationship between the variables. There is very little scatter about the exponential curve. There is a strong, positive exponential relationship between these variables.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031955\/m4_non_linear_models_ExponentialRelationships1of2_image5.png\" alt=\"Scatterplot showing strong, positive exponential relationship\" width=\"367\" height=\"360\" \/>\r\n\r\nThe <strong>equation of the exponential model<\/strong> is Predicted eagle pairs = 121 (1.083)<sup><em>t<\/em><\/sup>.\r\n\r\nNote: In this equation, the <em>t<\/em>-variable is an exponent. Sometimes you will see this written with the caret symbol: ^. So Predicted <em>y<\/em> = 121 (1.083)<sup><em>t<\/em><\/sup> and Predicted <em>y<\/em> = 121(1.083) ^ <em>t<\/em> mean the same thing.\r\n\r\nNow we use the exponential model to make predictions about the number of bald eagle mating pairs. We also compare the predictions from the exponential model to the linear model. Because there is a strong exponential relationship and a weaker linear relationship in the data, we expect the predictions from the exponential model to be better.\r\n\r\nIn 1963 (<em>t<\/em> = 13), there were 417 mating pairs.\r\n<ul>\r\n \t<li>According to the linear model: Predicted eagle pairs = \u20133,878.11 + 185.4 (13) = (\u20131,468). Obviously, a negative value does not make sense for a count of eagle pairs.<\/li>\r\n \t<li>According to the exponential model: <em>Predicted eagle pairs<\/em> = 121 (1.083)<sup>13<\/sup> = 341. So the exponential model underestimates by 417 \u2013 341 = 76 mating pairs. But this is a much better prediction than we got from the linear model.<\/li>\r\n<\/ul>\r\nIn 2000 (<em>t<\/em> = 50), there were 6,471 mating pairs.\r\n<ul>\r\n \t<li>According to the linear model: Predicted eagle pairs = \u20133,878.11 + 185.4 (50) = 5,392. So the linear model underestimates by 6,471 \u2013 5392 = 1,079 mating pairs.<\/li>\r\n \t<li>According to the exponential model: <em>Predicted eagle pairs<\/em> = 121 (1.083)<sup>50<\/sup> = 6,519. So the exponential model overestimates by 6,591 \u2013 6,471 = 48 mating pairs. This is a much better prediction than we got from the linear model.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3518\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3519\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3520\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use an exponential model (when appropriate) to describe the relationship between two quantitative variables. Interpret the model in context.<\/li>\n<\/ul>\n<\/div>\n<p>In our first example of exponential relationships, we investigate a nonlinear model for growth in a population over time.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<h2>The Return of the Bald Eagle<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031948\/m4_non_linear_models_ExponentialRelationships1of2_image1.png\" alt=\"Bald eagle\" width=\"318\" height=\"423\" \/><\/p>\n<p>During the mid-20th century, the population of bald eagles in the lower 48 states declined substantially. A highly toxic pesticide, DDT, was the main cause of the decline. DDT causes damage to bird egg shells. By 1963, bald eagles were in danger of complete extinction. Only 417 pairs of bald eagles remained. In 1967, the bald eagle became an official endangered species. Then in 1972, the EPA banned the use of DDT in the United States. The impact of the ban was a dramatic turnaround in the fate of the bald eagle.<\/p>\n<p>Here is the data. Note that in the table, we defined <em>t<\/em>, our explanatory variable, to be <em>Years after 1950<\/em>. The response variable is the number of bald eagle pairs that are mating.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031950\/m4_non_linear_models_ExponentialRelationships1of2_image2.png\" alt=\"Data table showing the number of bald eagle pairs mating after 1950, where t = the number of years after 1950\" width=\"329\" height=\"382\" \/><\/p>\n<p>Our goal is to find an equation to model this relationship.<\/p>\n<p>Here is a scatterplot of the data. We can see that the relationship appears somewhat linear, particularly for years after 1980 (<em>t<\/em> = 30). The correlation coefficient for this data set is high, <em>r<\/em> = 0.914.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031952\/m4_non_linear_models_ExponentialRelationships1of2_image3.png\" alt=\"Scatterplot of eagle pairs\" width=\"381\" height=\"273\" \/><\/p>\n<p>The least squares regression line is Predicted eagle pairs = \u22123,878.11 + 185.4<em>t<\/em>. Below is a scatterplot of the data with the least-squares regression line and the residual plot.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031953\/m4_non_linear_models_ExponentialRelationships1of2_image4.png\" alt=\"Scatterplot with least-squares regression of eagle pairs; corresponding residual plot\" width=\"265\" height=\"533\" \/><\/p>\n<p>We see a clear pattern in the residuals, suggesting that a linear model does not capture patterns in the data.<\/p>\n<p>Note: This is a reminder that a large <em>r<\/em>-value does not guarantee that a linear model is a good fit.<\/p>\n<p><strong>Conclusion:<\/strong> The data set for eagle pairs is clearly nonlinear. We need a better model.<\/p>\n<p>In the scatterplot below, we fit an <strong>exponential <\/strong>model to the data. Notice how well this model describes the relationship between the variables. There is very little scatter about the exponential curve. There is a strong, positive exponential relationship between these variables.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031955\/m4_non_linear_models_ExponentialRelationships1of2_image5.png\" alt=\"Scatterplot showing strong, positive exponential relationship\" width=\"367\" height=\"360\" \/><\/p>\n<p>The <strong>equation of the exponential model<\/strong> is Predicted eagle pairs = 121 (1.083)<sup><em>t<\/em><\/sup>.<\/p>\n<p>Note: In this equation, the <em>t<\/em>-variable is an exponent. Sometimes you will see this written with the caret symbol: ^. So Predicted <em>y<\/em> = 121 (1.083)<sup><em>t<\/em><\/sup> and Predicted <em>y<\/em> = 121(1.083) ^ <em>t<\/em> mean the same thing.<\/p>\n<p>Now we use the exponential model to make predictions about the number of bald eagle mating pairs. We also compare the predictions from the exponential model to the linear model. Because there is a strong exponential relationship and a weaker linear relationship in the data, we expect the predictions from the exponential model to be better.<\/p>\n<p>In 1963 (<em>t<\/em> = 13), there were 417 mating pairs.<\/p>\n<ul>\n<li>According to the linear model: Predicted eagle pairs = \u20133,878.11 + 185.4 (13) = (\u20131,468). Obviously, a negative value does not make sense for a count of eagle pairs.<\/li>\n<li>According to the exponential model: <em>Predicted eagle pairs<\/em> = 121 (1.083)<sup>13<\/sup> = 341. So the exponential model underestimates by 417 \u2013 341 = 76 mating pairs. But this is a much better prediction than we got from the linear model.<\/li>\n<\/ul>\n<p>In 2000 (<em>t<\/em> = 50), there were 6,471 mating pairs.<\/p>\n<ul>\n<li>According to the linear model: Predicted eagle pairs = \u20133,878.11 + 185.4 (50) = 5,392. So the linear model underestimates by 6,471 \u2013 5392 = 1,079 mating pairs.<\/li>\n<li>According to the exponential model: <em>Predicted eagle pairs<\/em> = 121 (1.083)<sup>50<\/sup> = 6,519. So the exponential model overestimates by 6,591 \u2013 6,471 = 48 mating pairs. This is a much better prediction than we got from the linear model.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3518\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3518&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3518\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3519\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3519&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3519\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3520\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3520&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3520\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/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-233\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":163,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"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-233","chapter","type-chapter","status-publish","hentry"],"part":225,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/233","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/233\/revisions"}],"predecessor-version":[{"id":1397,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/233\/revisions\/1397"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/233\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/media?parent=233"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/pressbooks\/v2\/chapter-type?post=233"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/contributor?post=233"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-wmopen-concepts-statistics\/wp-json\/wp\/v2\/license?post=233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}