{"id":813,"date":"2025-07-15T19:37:43","date_gmt":"2025-07-15T19:37:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=813"},"modified":"2026-04-01T08:43:34","modified_gmt":"2026-04-01T08:43:34","slug":"linear-models-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-models-learn-it-3\/","title":{"raw":"Linear Models: Learn It 3","rendered":"Linear Models: Learn It 3"},"content":{"raw":"<h2>Drawing and Interpreting Scatterplots<\/h2>\r\nA professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">When expressing pairs of inputs and outputs on a graph, they take the form of (<em>input<\/em>, <em>output<\/em>). In scatter plots, the two variables relate to create each data point,\u00a0(<em>variable 1<\/em>, <em>variable 2<\/em>), but it is often not necessary to declare that one is dependent on the other. In the example below, each\u00a0<em>Age<\/em>\u00a0coordinate corresponds to a\u00a0<em>Final Exam Score <\/em>in the form (<em>age<\/em>,\u00a0<em>score<\/em>). Each corresponding pair is plotted on the graph.<\/section>A <strong>scatter plot<\/strong> is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a <strong>linear model<\/strong>, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions. Below is\u00a0a sample scatter plot.\r\n\r\n[caption id=\"attachment_4938\" align=\"aligncenter\" width=\"449\"]<img class=\"wp-image-4938 \" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214631\/3.3.L.3.Graph1_-300x214.png\" alt=\"Scatter plot, titled 'Final Exam Score VS Age'. The x-axis is the age, and the y-axis is the final exam score. The range of ages are between 20s - 50s, and the range for scores are between upper 50s and 90s.\" width=\"449\" height=\"320\" \/> A scatter plot of age and final exam score variables.[\/caption]\r\n\r\nNotice this scatter plot does <em>not<\/em> indicate a linear relationship. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.\r\n\r\n<section class=\"textbox example\">The table below shows the number of cricket chirps in [latex]15[\/latex] seconds, for several different air temperatures, in degrees Fahrenheit.[footnote]Selected data from http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/. Retrieved Aug 3, 2010.[\/footnote]\r\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>Chirps<\/strong><\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<td>[latex]35[\/latex]<\/td>\r\n<td>[latex]20.4[\/latex]<\/td>\r\n<td>[latex]33[\/latex]<\/td>\r\n<td>[latex]31[\/latex]<\/td>\r\n<td>[latex]35[\/latex]<\/td>\r\n<td>[latex]18.5[\/latex]<\/td>\r\n<td>[latex]37[\/latex]<\/td>\r\n<td>[latex]26[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>[latex]80.5[\/latex]<\/td>\r\n<td>[latex]70.5[\/latex]<\/td>\r\n<td>[latex]57[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]68[\/latex]<\/td>\r\n<td>[latex]72[\/latex]<\/td>\r\n<td>[latex]52[\/latex]<\/td>\r\n<td>[latex]73.5[\/latex]<\/td>\r\n<td>[latex]53[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet's plot the data to determine whether the data appears to have a linear relationship.\r\n\r\n<img class=\"wp-image-4941 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214737\/3.3.L.3.Graph2_-300x261.png\" alt=\"A scatterplot showing the relationship between chirps per minute (x-axis) and temperature in degrees Fahrenheit (y-axis). As the number of chirps increases, the temperature also increases. Points range from about 20\u201345 chirps and 53\u201380 degrees, forming an upward trend.\" width=\"444\" height=\"386\" \/>\r\n\r\nWhat do you think? Does it have a linear relationship?\r\n\r\n[reveal-answer q=\"513473\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"513473\"]Looking at the scatterplot, we can determine if there is a linear relationship between the two variables, chirps and temperature.\r\n\r\n<strong>What to look for:<\/strong>\r\n<ul>\r\n \t<li>If the points form a pattern that looks like a straight line (either rising or falling), it suggests a linear relationship.<\/li>\r\n \t<li>If the points are scattered randomly with no clear pattern, it suggests there is no linear relationship.<\/li>\r\n<\/ul>\r\n<strong>Observation:<\/strong>\r\n<ul>\r\n \t<li>In the scatterplot, as the number of chirps increases, the temperature also seems to increase.<\/li>\r\n \t<li>The points appear to form a rising pattern, indicating that there is a positive linear relationship.<\/li>\r\n \t<li>Although the points do not form a perfectly straight line, they still show a general upward trend.<\/li>\r\n<\/ul>\r\n<strong>Conclusion:<\/strong>\r\n<ul>\r\n \t<li>Yes, the scatterplot shows a linear relationship. Although it is not perfectly linear, as the number of chirps increases, the temperature tends to increase as well.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318764[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318765[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318766[\/ohm_question]<\/section>","rendered":"<h2>Drawing and Interpreting Scatterplots<\/h2>\n<p>A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">When expressing pairs of inputs and outputs on a graph, they take the form of (<em>input<\/em>, <em>output<\/em>). In scatter plots, the two variables relate to create each data point,\u00a0(<em>variable 1<\/em>, <em>variable 2<\/em>), but it is often not necessary to declare that one is dependent on the other. In the example below, each\u00a0<em>Age<\/em>\u00a0coordinate corresponds to a\u00a0<em>Final Exam Score <\/em>in the form (<em>age<\/em>,\u00a0<em>score<\/em>). Each corresponding pair is plotted on the graph.<\/section>\n<p>A <strong>scatter plot<\/strong> is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a <strong>linear model<\/strong>, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions. Below is\u00a0a sample scatter plot.<\/p>\n<figure id=\"attachment_4938\" aria-describedby=\"caption-attachment-4938\" style=\"width: 449px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4938\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214631\/3.3.L.3.Graph1_-300x214.png\" alt=\"Scatter plot, titled 'Final Exam Score VS Age'. The x-axis is the age, and the y-axis is the final exam score. The range of ages are between 20s - 50s, and the range for scores are between upper 50s and 90s.\" width=\"449\" height=\"320\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214631\/3.3.L.3.Graph1_-300x214.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214631\/3.3.L.3.Graph1_-65x46.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214631\/3.3.L.3.Graph1_-225x161.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214631\/3.3.L.3.Graph1_-350x250.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214631\/3.3.L.3.Graph1_.png 490w\" sizes=\"(max-width: 449px) 100vw, 449px\" \/><figcaption id=\"caption-attachment-4938\" class=\"wp-caption-text\">A scatter plot of age and final exam score variables.<\/figcaption><\/figure>\n<p>Notice this scatter plot does <em>not<\/em> indicate a linear relationship. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.<\/p>\n<section class=\"textbox example\">The table below shows the number of cricket chirps in [latex]15[\/latex] seconds, for several different air temperatures, in degrees Fahrenheit.<a class=\"footnote\" title=\"Selected data from http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/. Retrieved Aug 3, 2010.\" id=\"return-footnote-813-1\" href=\"#footnote-813-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>[latex]44[\/latex]<\/td>\n<td>[latex]35[\/latex]<\/td>\n<td>[latex]20.4[\/latex]<\/td>\n<td>[latex]33[\/latex]<\/td>\n<td>[latex]31[\/latex]<\/td>\n<td>[latex]35[\/latex]<\/td>\n<td>[latex]18.5[\/latex]<\/td>\n<td>[latex]37[\/latex]<\/td>\n<td>[latex]26[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>[latex]80.5[\/latex]<\/td>\n<td>[latex]70.5[\/latex]<\/td>\n<td>[latex]57[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]68[\/latex]<\/td>\n<td>[latex]72[\/latex]<\/td>\n<td>[latex]52[\/latex]<\/td>\n<td>[latex]73.5[\/latex]<\/td>\n<td>[latex]53[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let&#8217;s plot the data to determine whether the data appears to have a linear relationship.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4941 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214737\/3.3.L.3.Graph2_-300x261.png\" alt=\"A scatterplot showing the relationship between chirps per minute (x-axis) and temperature in degrees Fahrenheit (y-axis). As the number of chirps increases, the temperature also increases. Points range from about 20\u201345 chirps and 53\u201380 degrees, forming an upward trend.\" width=\"444\" height=\"386\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214737\/3.3.L.3.Graph2_-300x261.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214737\/3.3.L.3.Graph2_-65x56.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214737\/3.3.L.3.Graph2_-225x195.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214737\/3.3.L.3.Graph2_-350x304.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02214737\/3.3.L.3.Graph2_.png 480w\" sizes=\"(max-width: 444px) 100vw, 444px\" \/><\/p>\n<p>What do you think? Does it have a linear relationship?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q513473\">Show Answer<\/button><\/p>\n<div id=\"q513473\" class=\"hidden-answer\" style=\"display: none\">Looking at the scatterplot, we can determine if there is a linear relationship between the two variables, chirps and temperature.<\/p>\n<p><strong>What to look for:<\/strong><\/p>\n<ul>\n<li>If the points form a pattern that looks like a straight line (either rising or falling), it suggests a linear relationship.<\/li>\n<li>If the points are scattered randomly with no clear pattern, it suggests there is no linear relationship.<\/li>\n<\/ul>\n<p><strong>Observation:<\/strong><\/p>\n<ul>\n<li>In the scatterplot, as the number of chirps increases, the temperature also seems to increase.<\/li>\n<li>The points appear to form a rising pattern, indicating that there is a positive linear relationship.<\/li>\n<li>Although the points do not form a perfectly straight line, they still show a general upward trend.<\/li>\n<\/ul>\n<p><strong>Conclusion:<\/strong><\/p>\n<ul>\n<li>Yes, the scatterplot shows a linear relationship. Although it is not perfectly linear, as the number of chirps increases, the temperature tends to increase as well.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318764\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318764&theme=lumen&iframe_resize_id=ohm318764&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318765\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318765&theme=lumen&iframe_resize_id=ohm318765&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318766\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318766&theme=lumen&iframe_resize_id=ohm318766&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-813-1\">Selected data from http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/. Retrieved Aug 3, 2010. <a href=\"#return-footnote-813-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":13,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/813"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/813\/revisions"}],"predecessor-version":[{"id":6090,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/813\/revisions\/6090"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/813\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=813"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=813"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=813"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=813"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}