{"id":3839,"date":"2022-03-15T23:15:53","date_gmt":"2022-03-15T23:15:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3839"},"modified":"2022-05-23T19:45:13","modified_gmt":"2022-05-23T19:45:13","slug":"corequisite-support-activity-for-6-a","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/corequisite-support-activity-for-6-a\/","title":{"raw":"Corequisite Support Activity for 6.A:  Exploring Lines of Best Fit","rendered":"Corequisite Support Activity for 6.A:  Exploring Lines of Best Fit"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>what you'll need to know:<\/h3>\r\nIn this support activity you\u2019ll become familiar with the following:\r\n<ul>\r\n \t<li>Use a given trend to create an appropriately labeled graph from a data table.<\/li>\r\n \t<li>Interpret an ordered pair (x, y) that represents a data point on a graph.<\/li>\r\n \t<li>Use a given graph to make a linear prediction.<\/li>\r\n \t<li>Write a linear equation to describe a given scenario.<\/li>\r\n \t<li>Identify the explanatory and response variables of a scenario.<\/li>\r\n \t<li>Interpret the slope and y-intercept in the context of statistics given the equation of linearly related variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the next preview assignment and in the next class, you will need to know how to write a linear equation that describes a situation where there is a constant rate of change. In this activity, you'll prepare for that by following steps to model data from two variables using a linear equation. You will create a table of variable values then use that data to plot points on a graph. Then you'll use the graph and data to write a linear equation that describes the scenario. You may have seen linear equations before. See the recall box below for a refresher on the slope-intercept form of a linear equation and how to represent linearly related variable values as ordered pairs in order to create a graph that models the data.\r\n<div class=\"textbox examples\">\r\n<h3>Recall<\/h3>\r\nWhen modeling data from bivariate data (two variables) with a linear relationship, we need to understand how to write and graph a linear equation.\r\n<ul>\r\n \t<li>Core skill:\u00a0[reveal-answer q=\"10624\"]Identify the parts of a linear equation in slope-intercept form[\/reveal-answer][hidden-answer a=\"10624\"]\r\n<ul>\r\n \t<li>Given two variables, let's name them [latex]x[\/latex] and [latex]y[\/latex], that have a relationship in which the value of [latex]y[\/latex] is dependent upon the value of [latex]x[\/latex], we can describe the relationship using a\u00a0<strong>linear equation<\/strong>.<\/li>\r\n \t<li>Ex. If a certain tree grows two inches in a year, we can say that a 7 inch tall sapling will grow 2 inches each year thereafter by letting the number of years be represented by [latex]x[\/latex] and the height of the tree in inches be represented by [latex]y[\/latex].\r\n<ul>\r\n \t<li>\"The height [latex]y[\/latex] is equivalent to the starting height plus 2 inches per year.\"<\/li>\r\n \t<li>Mathematically, we can write [latex]y = 7 + 2x[\/latex].<\/li>\r\n \t<li>Equivalently, we can write\u00a0[latex]y = 2x + 7[\/latex] which is in a form we call \"slope-intercept form.\"<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The familiar slope-intercept form of a linear equation you probably learned about in algebra is [latex]y=mx+b[\/latex], where\r\n<ul>\r\n \t<li>[latex]x[\/latex] and [latex]y[\/latex] are, respectively, the explanatory and response variables,<\/li>\r\n \t<li>[latex]m[\/latex] represents the constant rate of change (<strong><span style=\"background-color: #ffff99;\">the slope<\/span><\/strong>), and<\/li>\r\n \t<li>[latex]b[\/latex] represents the initial value in a given scenario (which can be zero or negative).\u00a0[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Core skill:\u00a0[reveal-answer q=\"143779\"]Graph a linear equation[\/reveal-answer][hidden-answer a=\"143779\"]\r\n<ul>\r\n \t<li>To graph a linear equation in the <strong><span style=\"background-color: #ffff99;\">coordinate plane<\/span><\/strong>, identify a paired set of values,\u00a0[latex]x[\/latex] and [latex]y[\/latex] and write them as an ordered pair [latex]\\left(x, y\\right)[\/latex].<\/li>\r\n \t<li>For example, using the sapling growth rate from the core skill above, a 7 inch sapling is predicted to grow 2 inches per year thereafter.\r\n<ul>\r\n \t<li>At the start (zero years thereafter), it is 7 inches tall. That information can be represented by the ordered pair [latex]\\left(0, 7\\right)[\/latex].<\/li>\r\n \t<li>One year later, it will have grown 2 inches, giving the ordered pair\u00a0[latex]\\left(1, 9\\right)[\/latex].<\/li>\r\n \t<li>The predicted height of the sapling five years thereafter would be represented by the ordered pair\u00a0[latex]\\left(5, 17\\right)[\/latex]. That is, at a steady growth rate of 2 inches per year, the sapling would have picked up [latex]5\\ast2=10[\/latex] inches of growth, making it 17 inches tall. <span style=\"background-color: #ffff99;\">See the student resource on <strong>slope<\/strong> for more information about constant rates of change<\/span>.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>To graph an ordered pair such as\u00a0[latex]\\left(5, 17\\right)[\/latex] on a plane, locate the point on the plane where 5 on the horizontal axis intersects with 17 on the vertical axis, and place a dot there. Once you have places several such points on the plane for two linearly related variables, you will be able to draw a straight line between them.\u00a0\u00a0<span style=\"background-color: #ffff99;\">See the student resource on the coordinate plane for more information about graphing ordered pairs<\/span>.\u00a0[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n[<span style=\"background-color: #ffff99;\">Coordinate Plane<\/span>], [<span style=\"background-color: #ffff99;\">Slope<\/span>] [&lt;-- link to these from DC Student Resources please, in the places indicated above]\r\n\r\n<\/div>\r\nIn this activity, we'll follow several steps to see how to model bivariate data that is linearly related. Let's begin by watching the story of a special dog's journey to a healthier life. The video below follows the dog Kai on a one year journey to lose 100 pounds. As you watch the video, note how Kai's small, constant changes eventually add up to a big change.\r\n<h2>Life Lessons from Kai<\/h2>\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-HFFSYcEMCI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nA small and constant change can create a straight path to a happier life!\r\n\r\nThe first two questions below may help you identify and process any emotions related to watching the video. Watching Kail's journey may stir thoughts about something in your life that you would like to change. It could be anything at all, not just a physical characteristic like Kai's. Maybe you would like to work on punctuality or study skills, getting more sleep, or hitting the gym more regularly.\u00a0 Whatever it is, reframe your thoughts and answer Question 1\u00a0 below individually. You don't have to write down or share your answer with anyone.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\n1) Reflect on how the video made you feel. Is there something in your life that you would like to change? [You are not required to write this answer down if you would prefer to keep it to yourself.]\r\n\r\n[reveal-answer q=\"295305\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"295305\"]Examples include increased GPA, faster run time, greater personal best weight lifted, etc. You don't need to answer this if you would prefer to keep it to yourself![\/hidden-answer]\r\n\r\n<\/div>\r\nNow, answer Question 2 on your own then compare your answers with a partner.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\n2) List some phrases or ideas from this video that might help you meet a goal or life challenge. [This is required.]\r\n\r\n[reveal-answer q=\"400432\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"400432\"]These don't need to be word-for-word from the video. Just list some ideas that stood out to you as helpful for any person to achieve a goal.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Modeling and Interpreting a Linear Relationship<\/h3>\r\nWhen Kai met Pam, he was almost 100 pounds overweight, and for this reason alone, his owner wanted him euthanized. Kai\u2019s life was in danger and he was miserable, but Pam rescued him. Here are the details of Kai\u2019s journey:\r\n\r\n<strong>Initial Weight:<\/strong> Kai weighed 173 pounds before Pam agreed to foster him. She modified his diet and began walking him three times daily.\r\n\r\n<strong>Constant Rate of Change:<\/strong> A reasonable goal was for Kai to lose about two pounds each week.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n3)\u00a0 Assuming Kai meets his goal of losing two pounds per week, let\u2019s track his weight from week to week for the first six weeks. Complete the following chart.\r\n<div align=\"center\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Week<\/td>\r\n<td>Kai\u2019s weight<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>173<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>171<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"877164\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"877164\"]Complete the chart by filling in the weight that results from each decrease of 2 pounds per week.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 4<\/h3>\r\n4) Create a graph to keep track of Kai\u2019s weight loss for the first six weeks, assuming he manages to consistently lose two pounds each week. Label your axes accordingly.\r\n\r\n<img class=\"alignnone wp-image-1180\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12030227\/Picture122-300x300.jpg\" alt=\"A grid\" width=\"848\" height=\"848\" \/>\r\n\r\n[reveal-answer q=\"585466\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"585466\"]We tend to graph the explanatory variable such as time on the horizontal axis, and the response variable on the vertical axis. Recall that a linear relationship should reveal a linear pattern in your graph.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\n5)\u00a0 What do your horizontal and vertical axes represent?\r\n\r\n[reveal-answer q=\"419293\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"419293\"]What did you label the axes?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n6) Explain the ordered pair (5 , 163) in the context of Kai\u2019s situation.\r\n\r\n[reveal-answer q=\"525905\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"525905\"]State the relationship between the variables in words.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\n7) What pattern is made when you have a constant rate of change (two pounds lost per week)?\r\n\r\n[reveal-answer q=\"306814\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"306814\"]What shape does your graph make?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Making a Prediction From a Graph<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\n8) At this constant rate of weight loss, how much will Kai weigh after 10 weeks? After 25 weeks?\r\n\r\n[reveal-answer q=\"339590\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"339590\"]What do <em>you<\/em> think? Can you use arithmetic to predict these values?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\n9) Describe how you arrived at your answers to Questions 8. Use mathematical operations in your description.\r\n\r\n[reveal-answer q=\"584848\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"584848\"]Briefly describe <em>your<\/em> process.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>\u00a0Writing an Equation to Describe a Model<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\n10) Use your description in Question 9 to write a linear equation that describes Kai\u2019s situation. Make sure to explain what your variables mean.\r\n\r\n[reveal-answer q=\"761252\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"761252\"]You can use [latex]x[\/latex] and [latex]y[\/latex] or more descriptive variables to write your equation. See the recall box above for the slope-intercept form of a linear equation.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Linear Equations in the Context of Statistics<\/h2>\r\nYou may recall the slope intercept form of a linear equation from algebra as [latex]y=mx+b[\/latex], where [latex]m[\/latex] represents slope and [latex]b[\/latex] represents the y-intercept. While the components of the equation\u00a0 are the same in statistics, the letters representing slope and the y-intercept are different. Tip: don't let the different use of the letter [latex]b[\/latex] in the statistics formula trip you up! See the explanation below for the form of the equation used in statistics: [latex]y = b + ax[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nLet's practice converting an algebraic linear equation from the form [latex]y=mx+b[\/latex] into a form commonly used in statistics, [latex]y = b + ax[\/latex].\r\n<p style=\"padding-left: 30px;\">Ex. A linear equation representing the relationship between two variables is given as<\/p>\r\n<p style=\"text-align: center;\">[latex]y = 1.5x + 12[\/latex].<\/p>\r\n\r\n<ol>\r\n \t<li>What is the slope of the equation? What is the y-intercept?<\/li>\r\n \t<li>Rewrite the equation in the form [latex]y = b + ax[\/latex].<\/li>\r\n \t<li>In the context of statistics, what does the letter [latex]b[\/latex] represent? How about [latex]a[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"762886\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"762886\"]\r\n<ol>\r\n \t<li>The slope is [latex]1.5[\/latex] and the y-intercept is [latex]12[\/latex], or the point [latex](0, 12)[\/latex].<\/li>\r\n \t<li>In statistics, we would write the equation as [latex]y = 12 + 1.5x[\/latex].<\/li>\r\n \t<li>The letter [latex]b[\/latex] in statistics represents the y-intercept, while the letter [latex]a[\/latex] represents the slope.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn statistics, we use specific letters to represent variables in linear equations. The following are the common naming conventions:\r\n<ul>\r\n \t<li aria-level=\"1\">x represents the <strong>explanatory variable<\/strong> represented on the horizontal axis.<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">y represents the <strong>response variable<\/strong> represented on the vertical axis.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">a represents the <strong>y-intercept<\/strong>, which is the y-coordinate of the point where a line intersects the y-axis.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">b represents the <strong>constant rate of change<\/strong> or <strong>slope<\/strong> of the line.<\/li>\r\n<\/ul>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\n11) For Kai\u2019s situation, identify the following:\r\n\r\n&nbsp;\r\n\r\nPart A: Explanatory variable\r\n\r\n[reveal-answer q=\"799106\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"799106\"]This is the independent variable, graphed on the horizontal axis.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: Response variable\r\n\r\n[reveal-answer q=\"96589\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"96589\"]This is the dependent variable, graphed on the vertical axis.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart C: Slope\r\n\r\n[reveal-answer q=\"78203\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"78203\"]This is the constant rate of change.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart D: y-intercept\r\n\r\n[reveal-answer q=\"923405\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"923405\"]This is the location on the y-axis where the line intersects it, usually where the explanatory variable equals zero.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart E: Rewrite your equation from Question 10 using this new naming convention.\r\n\r\n[reveal-answer q=\"441516\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"441516\"]Put the parts together into the equation form [latex]y=b+ax[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nHopefully, you feel comfortable modeling linearly related bivariate data by creating a table and graph and writing an equation to describe the relationship. It's time to move on to the course section and activity now.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>what you&#8217;ll need to know:<\/h3>\n<p>In this support activity you\u2019ll become familiar with the following:<\/p>\n<ul>\n<li>Use a given trend to create an appropriately labeled graph from a data table.<\/li>\n<li>Interpret an ordered pair (x, y) that represents a data point on a graph.<\/li>\n<li>Use a given graph to make a linear prediction.<\/li>\n<li>Write a linear equation to describe a given scenario.<\/li>\n<li>Identify the explanatory and response variables of a scenario.<\/li>\n<li>Interpret the slope and y-intercept in the context of statistics given the equation of linearly related variables.<\/li>\n<\/ul>\n<\/div>\n<p>In the next preview assignment and in the next class, you will need to know how to write a linear equation that describes a situation where there is a constant rate of change. In this activity, you&#8217;ll prepare for that by following steps to model data from two variables using a linear equation. You will create a table of variable values then use that data to plot points on a graph. Then you&#8217;ll use the graph and data to write a linear equation that describes the scenario. You may have seen linear equations before. See the recall box below for a refresher on the slope-intercept form of a linear equation and how to represent linearly related variable values as ordered pairs in order to create a graph that models the data.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall<\/h3>\n<p>When modeling data from bivariate data (two variables) with a linear relationship, we need to understand how to write and graph a linear equation.<\/p>\n<ul>\n<li>Core skill:\u00a0\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q10624\">Identify the parts of a linear equation in slope-intercept form<\/span><\/p>\n<div id=\"q10624\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Given two variables, let&#8217;s name them [latex]x[\/latex] and [latex]y[\/latex], that have a relationship in which the value of [latex]y[\/latex] is dependent upon the value of [latex]x[\/latex], we can describe the relationship using a\u00a0<strong>linear equation<\/strong>.<\/li>\n<li>Ex. If a certain tree grows two inches in a year, we can say that a 7 inch tall sapling will grow 2 inches each year thereafter by letting the number of years be represented by [latex]x[\/latex] and the height of the tree in inches be represented by [latex]y[\/latex].\n<ul>\n<li>&#8220;The height [latex]y[\/latex] is equivalent to the starting height plus 2 inches per year.&#8221;<\/li>\n<li>Mathematically, we can write [latex]y = 7 + 2x[\/latex].<\/li>\n<li>Equivalently, we can write\u00a0[latex]y = 2x + 7[\/latex] which is in a form we call &#8220;slope-intercept form.&#8221;<\/li>\n<\/ul>\n<\/li>\n<li>The familiar slope-intercept form of a linear equation you probably learned about in algebra is [latex]y=mx+b[\/latex], where\n<ul>\n<li>[latex]x[\/latex] and [latex]y[\/latex] are, respectively, the explanatory and response variables,<\/li>\n<li>[latex]m[\/latex] represents the constant rate of change (<strong><span style=\"background-color: #ffff99;\">the slope<\/span><\/strong>), and<\/li>\n<li>[latex]b[\/latex] represents the initial value in a given scenario (which can be zero or negative).\u00a0<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Core skill:\u00a0\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q143779\">Graph a linear equation<\/span><\/p>\n<div id=\"q143779\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>To graph a linear equation in the <strong><span style=\"background-color: #ffff99;\">coordinate plane<\/span><\/strong>, identify a paired set of values,\u00a0[latex]x[\/latex] and [latex]y[\/latex] and write them as an ordered pair [latex]\\left(x, y\\right)[\/latex].<\/li>\n<li>For example, using the sapling growth rate from the core skill above, a 7 inch sapling is predicted to grow 2 inches per year thereafter.\n<ul>\n<li>At the start (zero years thereafter), it is 7 inches tall. That information can be represented by the ordered pair [latex]\\left(0, 7\\right)[\/latex].<\/li>\n<li>One year later, it will have grown 2 inches, giving the ordered pair\u00a0[latex]\\left(1, 9\\right)[\/latex].<\/li>\n<li>The predicted height of the sapling five years thereafter would be represented by the ordered pair\u00a0[latex]\\left(5, 17\\right)[\/latex]. That is, at a steady growth rate of 2 inches per year, the sapling would have picked up [latex]5\\ast2=10[\/latex] inches of growth, making it 17 inches tall. <span style=\"background-color: #ffff99;\">See the student resource on <strong>slope<\/strong> for more information about constant rates of change<\/span>.<\/li>\n<\/ul>\n<\/li>\n<li>To graph an ordered pair such as\u00a0[latex]\\left(5, 17\\right)[\/latex] on a plane, locate the point on the plane where 5 on the horizontal axis intersects with 17 on the vertical axis, and place a dot there. Once you have places several such points on the plane for two linearly related variables, you will be able to draw a straight line between them.\u00a0\u00a0<span style=\"background-color: #ffff99;\">See the student resource on the coordinate plane for more information about graphing ordered pairs<\/span>.\u00a0<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>[<span style=\"background-color: #ffff99;\">Coordinate Plane<\/span>], [<span style=\"background-color: #ffff99;\">Slope<\/span>] [&lt;&#8211; link to these from DC Student Resources please, in the places indicated above]<\/p>\n<\/div>\n<p>In this activity, we&#8217;ll follow several steps to see how to model bivariate data that is linearly related. Let&#8217;s begin by watching the story of a special dog&#8217;s journey to a healthier life. The video below follows the dog Kai on a one year journey to lose 100 pounds. As you watch the video, note how Kai&#8217;s small, constant changes eventually add up to a big change.<\/p>\n<h2>Life Lessons from Kai<\/h2>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-HFFSYcEMCI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A small and constant change can create a straight path to a happier life!<\/p>\n<p>The first two questions below may help you identify and process any emotions related to watching the video. Watching Kail&#8217;s journey may stir thoughts about something in your life that you would like to change. It could be anything at all, not just a physical characteristic like Kai&#8217;s. Maybe you would like to work on punctuality or study skills, getting more sleep, or hitting the gym more regularly.\u00a0 Whatever it is, reframe your thoughts and answer Question 1\u00a0 below individually. You don&#8217;t have to write down or share your answer with anyone.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>1) Reflect on how the video made you feel. Is there something in your life that you would like to change? [You are not required to write this answer down if you would prefer to keep it to yourself.]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q295305\">Hint<\/span><\/p>\n<div id=\"q295305\" class=\"hidden-answer\" style=\"display: none\">Examples include increased GPA, faster run time, greater personal best weight lifted, etc. You don&#8217;t need to answer this if you would prefer to keep it to yourself!<\/div>\n<\/div>\n<\/div>\n<p>Now, answer Question 2 on your own then compare your answers with a partner.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>2) List some phrases or ideas from this video that might help you meet a goal or life challenge. [This is required.]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q400432\">Hint<\/span><\/p>\n<div id=\"q400432\" class=\"hidden-answer\" style=\"display: none\">These don&#8217;t need to be word-for-word from the video. Just list some ideas that stood out to you as helpful for any person to achieve a goal.<\/div>\n<\/div>\n<\/div>\n<h3>Modeling and Interpreting a Linear Relationship<\/h3>\n<p>When Kai met Pam, he was almost 100 pounds overweight, and for this reason alone, his owner wanted him euthanized. Kai\u2019s life was in danger and he was miserable, but Pam rescued him. Here are the details of Kai\u2019s journey:<\/p>\n<p><strong>Initial Weight:<\/strong> Kai weighed 173 pounds before Pam agreed to foster him. She modified his diet and began walking him three times daily.<\/p>\n<p><strong>Constant Rate of Change:<\/strong> A reasonable goal was for Kai to lose about two pounds each week.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3)\u00a0 Assuming Kai meets his goal of losing two pounds per week, let\u2019s track his weight from week to week for the first six weeks. Complete the following chart.<\/p>\n<div style=\"margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td>Week<\/td>\n<td>Kai\u2019s weight<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>173<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>171<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q877164\">Hint<\/span><\/p>\n<div id=\"q877164\" class=\"hidden-answer\" style=\"display: none\">Complete the chart by filling in the weight that results from each decrease of 2 pounds per week.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Question 4<\/h3>\n<p>4) Create a graph to keep track of Kai\u2019s weight loss for the first six weeks, assuming he manages to consistently lose two pounds each week. Label your axes accordingly.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1180\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12030227\/Picture122-300x300.jpg\" alt=\"A grid\" width=\"848\" height=\"848\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q585466\">Hint<\/span><\/p>\n<div id=\"q585466\" class=\"hidden-answer\" style=\"display: none\">We tend to graph the explanatory variable such as time on the horizontal axis, and the response variable on the vertical axis. Recall that a linear relationship should reveal a linear pattern in your graph.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>5)\u00a0 What do your horizontal and vertical axes represent?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q419293\">Hint<\/span><\/p>\n<div id=\"q419293\" class=\"hidden-answer\" style=\"display: none\">What did you label the axes?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p>6) Explain the ordered pair (5 , 163) in the context of Kai\u2019s situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525905\">Hint<\/span><\/p>\n<div id=\"q525905\" class=\"hidden-answer\" style=\"display: none\">State the relationship between the variables in words.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p>7) What pattern is made when you have a constant rate of change (two pounds lost per week)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q306814\">Hint<\/span><\/p>\n<div id=\"q306814\" class=\"hidden-answer\" style=\"display: none\">What shape does your graph make?<\/div>\n<\/div>\n<\/div>\n<h3>Making a Prediction From a Graph<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p>8) At this constant rate of weight loss, how much will Kai weigh after 10 weeks? After 25 weeks?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q339590\">Hint<\/span><\/p>\n<div id=\"q339590\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think? Can you use arithmetic to predict these values?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p>9) Describe how you arrived at your answers to Questions 8. Use mathematical operations in your description.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q584848\">Hint<\/span><\/p>\n<div id=\"q584848\" class=\"hidden-answer\" style=\"display: none\">Briefly describe <em>your<\/em> process.<\/div>\n<\/div>\n<\/div>\n<h3>\u00a0Writing an Equation to Describe a Model<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p>10) Use your description in Question 9 to write a linear equation that describes Kai\u2019s situation. Make sure to explain what your variables mean.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q761252\">Hint<\/span><\/p>\n<div id=\"q761252\" class=\"hidden-answer\" style=\"display: none\">You can use [latex]x[\/latex] and [latex]y[\/latex] or more descriptive variables to write your equation. See the recall box above for the slope-intercept form of a linear equation.<\/div>\n<\/div>\n<\/div>\n<h2>Linear Equations in the Context of Statistics<\/h2>\n<p>You may recall the slope intercept form of a linear equation from algebra as [latex]y=mx+b[\/latex], where [latex]m[\/latex] represents slope and [latex]b[\/latex] represents the y-intercept. While the components of the equation\u00a0 are the same in statistics, the letters representing slope and the y-intercept are different. Tip: don&#8217;t let the different use of the letter [latex]b[\/latex] in the statistics formula trip you up! See the explanation below for the form of the equation used in statistics: [latex]y = b + ax[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Let&#8217;s practice converting an algebraic linear equation from the form [latex]y=mx+b[\/latex] into a form commonly used in statistics, [latex]y = b + ax[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">Ex. A linear equation representing the relationship between two variables is given as<\/p>\n<p style=\"text-align: center;\">[latex]y = 1.5x + 12[\/latex].<\/p>\n<ol>\n<li>What is the slope of the equation? What is the y-intercept?<\/li>\n<li>Rewrite the equation in the form [latex]y = b + ax[\/latex].<\/li>\n<li>In the context of statistics, what does the letter [latex]b[\/latex] represent? How about [latex]a[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q762886\">Show Answer<\/span><\/p>\n<div id=\"q762886\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The slope is [latex]1.5[\/latex] and the y-intercept is [latex]12[\/latex], or the point [latex](0, 12)[\/latex].<\/li>\n<li>In statistics, we would write the equation as [latex]y = 12 + 1.5x[\/latex].<\/li>\n<li>The letter [latex]b[\/latex] in statistics represents the y-intercept, while the letter [latex]a[\/latex] represents the slope.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In statistics, we use specific letters to represent variables in linear equations. The following are the common naming conventions:<\/p>\n<ul>\n<li aria-level=\"1\">x represents the <strong>explanatory variable<\/strong> represented on the horizontal axis.<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">y represents the <strong>response variable<\/strong> represented on the vertical axis.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">a represents the <strong>y-intercept<\/strong>, which is the y-coordinate of the point where a line intersects the y-axis.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">b represents the <strong>constant rate of change<\/strong> or <strong>slope<\/strong> of the line.<\/li>\n<\/ul>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p>11) For Kai\u2019s situation, identify the following:<\/p>\n<p>&nbsp;<\/p>\n<p>Part A: Explanatory variable<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q799106\">Hint<\/span><\/p>\n<div id=\"q799106\" class=\"hidden-answer\" style=\"display: none\">This is the independent variable, graphed on the horizontal axis.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: Response variable<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96589\">Hint<\/span><\/p>\n<div id=\"q96589\" class=\"hidden-answer\" style=\"display: none\">This is the dependent variable, graphed on the vertical axis.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part C: Slope<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q78203\">Hint<\/span><\/p>\n<div id=\"q78203\" class=\"hidden-answer\" style=\"display: none\">This is the constant rate of change.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part D: y-intercept<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q923405\">Hint<\/span><\/p>\n<div id=\"q923405\" class=\"hidden-answer\" style=\"display: none\">This is the location on the y-axis where the line intersects it, usually where the explanatory variable equals zero.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part E: Rewrite your equation from Question 10 using this new naming convention.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q441516\">Hint<\/span><\/p>\n<div id=\"q441516\" class=\"hidden-answer\" style=\"display: none\">Put the parts together into the equation form [latex]y=b+ax[\/latex].<\/div>\n<\/div>\n<\/div>\n<p>Hopefully, you feel comfortable modeling linearly related bivariate data by creating a table and graph and writing an equation to describe the relationship. It&#8217;s time to move on to the course section and activity now.<\/p>\n","protected":false},"author":428269,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3839","chapter","type-chapter","status-publish","hentry"],"part":4241,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3839","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/428269"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3839\/revisions"}],"predecessor-version":[{"id":4789,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3839\/revisions\/4789"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4241"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3839\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3839"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3839"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3839"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}