{"id":250,"date":"2019-07-15T22:43:45","date_gmt":"2019-07-15T22:43:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/fitting-linear-models-to-data\/"},"modified":"2021-11-22T03:35:58","modified_gmt":"2021-11-22T03:35:58","slug":"fitting-linear-models-to-data","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/fitting-linear-models-to-data\/","title":{"raw":"4.3: Fitting Linear Models to Data","rendered":"4.3: Fitting Linear Models to Data"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Draw and interpret scatter plots<\/li>\r\n \t<li>Find the line of best fit using a calculator<\/li>\r\n \t<li>Distinguish between linear and nonlinear relations<\/li>\r\n \t<li>Use a linear model to make predictions<\/li>\r\n<\/ul>\r\n<\/div>\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<div class=\"textbox examples\">\r\n<h3>recall ordered pairs as data points<\/h3>\r\nWhen 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.\r\n\r\n<\/div>\r\nA <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=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014335\/CNX_Precalc_Figure_02_04_0012.jpg\" 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=\"487\" height=\"337\" \/> A scatter plot of age and final exam score variables.[\/caption]\r\n\r\nNotice this scatter plot does <em>not<\/em> indicate a <strong>linear relationship<\/strong>. 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<div class=\"textbox exercises\">\r\n<h3>\u00a0Example: Using a Scatter Plot to Investigate Cricket Chirps<\/h3>\r\nThe table below\u00a0shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit.[footnote]Selected data from <a href=\"http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/\" target=\"_blank\" rel=\"noopener\">http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/<\/a>. Retrieved Aug 3, 2010[\/footnote] Plot this data, and determine whether the data appears to be linearly related.\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>44<\/td>\r\n<td>35<\/td>\r\n<td>20.4<\/td>\r\n<td>33<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<td>18.5<\/td>\r\n<td>37<\/td>\r\n<td>26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>80.5<\/td>\r\n<td>70.5<\/td>\r\n<td>57<\/td>\r\n<td>66<\/td>\r\n<td>68<\/td>\r\n<td>72<\/td>\r\n<td>52<\/td>\r\n<td>73.5<\/td>\r\n<td>53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"579142\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"579142\"]\r\n\r\nPlotting this data\u00a0suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so.<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014337\/CNX_Precalc_Figure_02_04_0022.jpg\" alt=\"Scatter plot, titled 'Cricket Chirps Vs Air Temperature'. The x-axis is the Cricket Chirps in 15 Seconds, and the y-axis is the Temperature (F). The line regression is generally positive.\" width=\"487\" height=\"386\" \/>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Finding the Line of Best Fit<\/h2>\r\nOne way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the <em>y<\/em>-intercept. We can approximate the slope of the line by extending it until we can estimate the [latex]\\frac{\\text{rise}}{\\text{run}}[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Line of Best Fit<\/h3>\r\nFind a linear function that fits the data in the table below\u00a0by \"eyeballing\" a line that seems to fit.\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> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>Chirps<\/strong><\/td>\r\n<td>44<\/td>\r\n<td>35<\/td>\r\n<td>20.4<\/td>\r\n<td>33<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<td>18.5<\/td>\r\n<td>37<\/td>\r\n<td>26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>80.5<\/td>\r\n<td>70.5<\/td>\r\n<td>57<\/td>\r\n<td>66<\/td>\r\n<td>68<\/td>\r\n<td>72<\/td>\r\n<td>52<\/td>\r\n<td>73.5<\/td>\r\n<td>53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"768322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"768322\"]\r\n\r\nOn a graph, we could try sketching a line.\r\n\r\nUsing the starting and ending points of our hand drawn line, points (0, 30) and (50, 90), this graph has a slope of [latex]m=\\frac{60}{50}=1.2[\/latex] and a <em>y<\/em>-intercept at 30. This gives an equation of [latex]T\\left(c\\right)=1.2c+30[\/latex]\r\n\r\nwhere <em>c<\/em>\u00a0is the number of chirps in 15 seconds, and <em>T<\/em>(<em>c<\/em>)\u00a0is the temperature in degrees Fahrenheit. The resulting equation is represented in the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014339\/CNX_Precalc_Figure_02_04_0032.jpg\" alt=\"Scatter plot, showing the line of best fit. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"432\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nThis linear equation can then be used to approximate answers to various questions we might ask about the trend.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3681&amp;theme=oea&amp;iframe_resize_id=mom200[\/embed]\r\n\r\n<\/div>\r\n<h2>Recognizing Interpolation or Extrapolation<\/h2>\r\nWhile the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as <strong>interpolation<\/strong> when we predict a value inside the domain and range of the data. The process of <strong>extrapolation<\/strong> is used when we predict a value outside the domain and range of the data.\r\n\r\nThe graph below compares the two processes for the cricket-chirp data addressed in the previous example. We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.\r\n\r\nThere is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called <strong>model breakdown<\/strong>. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when [latex]x=50[\/latex], that is, in 50 years, the model would not apply because we could not account for factors fifty years in the future.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014341\/CNX_Precalc_Figure_02_04_0042.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"430\" \/> Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Interpolation and Extrapolation<\/h3>\r\nDifferent methods of making predictions are used to analyze data.\r\n<ul>\r\n \t<li>The method of <strong>interpolation<\/strong> involves predicting a value inside the domain and\/or range of the data.<\/li>\r\n \t<li>The method of <strong>extrapolation<\/strong> involves predicting a value outside the domain and\/or range of the data.<\/li>\r\n \t<li><strong>Model breakdown<\/strong> occurs at the point when the model no longer applies.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Understanding Interpolation and Extrapolation<\/h3>\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>44<\/td>\r\n<td>35<\/td>\r\n<td>20.4<\/td>\r\n<td>33<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<td>18.5<\/td>\r\n<td>37<\/td>\r\n<td>26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>80.5<\/td>\r\n<td>70.5<\/td>\r\n<td>57<\/td>\r\n<td>66<\/td>\r\n<td>68<\/td>\r\n<td>72<\/td>\r\n<td>52<\/td>\r\n<td>73.5<\/td>\r\n<td>53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUse the cricket data above\u00a0to answer the following questions:\r\n<ol>\r\n \t<li>Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\r\n \t<li>Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"882447\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"882447\"]\r\n<ol>\r\n \t<li>The number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is inside the domain of our data so would be interpolation. Using our model:\r\n[latex]\\begin{array}{l}T\\left(30\\right)=30+1.2\\left(30\\right)\\hfill \\\\ T\\left(30\\right)=66\\text{ degrees}\\hfill \\end{array}[\/latex]\r\nBased on the data we have, this value seems reasonable.<\/li>\r\n \t<li>The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation because 40 is outside the range of our data. Using our model:\r\n[latex]\\begin{array}{l}40=30+1.2c\\hfill \\\\ 10=1.2c\\hfill \\\\ c\\approx 8.33\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nWe can compare the regions of interpolation and extrapolation using the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014345\/CNX_Precalc_Figure_02_04_0052.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"485\" height=\"429\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nOur model predicts the crickets would chirp 8.33 times in 15 seconds. While this might be possible, we have no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether at or below 50 degrees.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nAccording to the data from the table in the cricket-chirp example, what temperature can we predict if we counted 20 chirps in 15 seconds?\r\n\r\n[reveal-answer q=\"271439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"271439\"]\r\n\r\n[latex]54^\\circ \\text{F}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Finding the Line of Best Fit Using a Graphing Utility<\/h2>\r\nWhile eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values.[footnote]Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.[\/footnote] One such technique is called <strong>least squares regression<\/strong> and can be computed by many graphing calculators as well as both spreadsheet and statistical software. Least squares regression is also called linear regression, and we can use an online graphing calculator to perform linear regressions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Least Squares Regression Line<\/h3>\r\nFind the least squares regression line using the cricket-chirp data in the table below.\r\n\r\nUse an online graphing calculator.\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).\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Chirps<\/strong><\/td>\r\n<td>44<\/td>\r\n<td>35<\/td>\r\n<td>20.4<\/td>\r\n<td>33<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<td>18.5<\/td>\r\n<td>37<\/td>\r\n<td>26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>80.5<\/td>\r\n<td>70.5<\/td>\r\n<td>57<\/td>\r\n<td>66<\/td>\r\n<td>68<\/td>\r\n<td>72<\/td>\r\n<td>52<\/td>\r\n<td>73.5<\/td>\r\n<td>53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"374127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"374127\"]\r\n\r\nThe following instructions are for Desmos, and other online graphing tools may be slightly different.\r\n<ol>\r\n \t<li>Click the plus button (add item) in the upper left corner and select table.<\/li>\r\n \t<li>Enter chirps data in the x1 column.<\/li>\r\n \t<li>Enter temperature data in the y1 column.\r\n<table summary=\"Two rows and ten columns. The first row is labeled, 'L1'. The second row is labeled is labeled, 'L2'. Reading the remaining rows as ordered pairs (i.e., (L2, L2), 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).\">\r\n<tbody>\r\n<tr>\r\n<td><strong>x1<\/strong><\/td>\r\n<td>44<\/td>\r\n<td>35<\/td>\r\n<td>20.4<\/td>\r\n<td>33<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<td>18.5<\/td>\r\n<td>37<\/td>\r\n<td>26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><b>y1<\/b><\/td>\r\n<td>80.5<\/td>\r\n<td>70.5<\/td>\r\n<td>57<\/td>\r\n<td>66<\/td>\r\n<td>68<\/td>\r\n<td>72<\/td>\r\n<td>52<\/td>\r\n<td>73.5<\/td>\r\n<td>53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>If you can't see the points on the grid, use the plus and minus buttons in the upper right hand corner to zoom in or out on the grid, or click on the wrench and change the upper bound of x1 to 60 and y1 to 100<\/li>\r\n \t<li>In the empty cell below the table you created, enter the expression y1\u223cmx1+b<\/li>\r\n \t<li>You can add labels to your graph by clicking on the wrench in the upper right hand corner and typing them into the cells that say \"add a label\"<\/li>\r\n<\/ol>\r\nHere is an example of how your graph may look:\r\n\r\n<img class=\"alignnone size-full wp-image-6803\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09215809\/Screen-Shot-2019-07-09-at-2.56.40-PM.png\" alt=\"\" width=\"1420\" height=\"1002\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nNotice that this line is quite similar to the equation we \"eyeballed\" but should fit the data better. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}T\\left(30\\right)=30.281+1.143\\left(30\\right)\\hfill \\\\ \\text{}T\\left(30\\right)=64.571\\hfill \\\\ \\text{}T\\left(30\\right)\\approx 64.6\\text{ degrees}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Will there ever be a case where two different lines will serve as the best fit for the data?<\/strong>\r\n\r\n<em>No. There is only one best fit line.<\/em>\r\n\r\n<\/div>\r\n<h2>Distinguish Between Linear and Nonlinear Relations<\/h2>\r\nAs we saw in the cricket-chirp example, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <strong>correlation coefficient<\/strong>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a \"diagnostic on\" selection to find the correlation coefficient, which mathematicians label as <em>r<\/em>. The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.\r\n\r\nWe should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense of the relationship between the value of <em>r<\/em>\u00a0and the graph of the data, the image below\u00a0shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"901\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014349\/CNX_Precalc_Figure_02_04_0072.jpg\" alt=\"A series of scatterplot graphs. Some are linear and some are not.\" width=\"901\" height=\"401\" \/> Plotted data and related correlation coefficients. (credit: \"DenisBoigelot,\" Wikimedia Commons)[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Correlation Coefficient<\/h3>\r\nThe <strong>correlation coefficient<\/strong> is a value, <em>r<\/em>, between \u20131 and 1.\r\n<ul>\r\n \t<li><em>r<\/em> &gt; 0 suggests a positive (increasing) relationship<\/li>\r\n \t<li><em>r<\/em> &lt; 0 suggests a negative (decreasing) relationship<\/li>\r\n \t<li>The closer the value is to 0, the more scattered the data.<\/li>\r\n \t<li>The closer the value is to 1 or \u20131, the less scattered the data is.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Correlation Coefficient<\/h3>\r\nCalculate the correlation coefficient for cricket-chirp data in the table below.\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).\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Chirps<\/strong><\/td>\r\n<td>44<\/td>\r\n<td>35<\/td>\r\n<td>20.4<\/td>\r\n<td>33<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<td>18.5<\/td>\r\n<td>37<\/td>\r\n<td>26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>80.5<\/td>\r\n<td>70.5<\/td>\r\n<td>57<\/td>\r\n<td>66<\/td>\r\n<td>68<\/td>\r\n<td>72<\/td>\r\n<td>52<\/td>\r\n<td>73.5<\/td>\r\n<td>53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"520385\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"520385\"]\r\n\r\nOnline graphing calculators\u00a0provide you with the correlation coefficient when you use it to calculate a linear regression. The correlation coefficients is labeled as <em>r\u00a0<\/em>= 0.951 for this dataset. This value is very close to 1 which suggests a strong increasing linear relationship. Below is an example of what your graph will look like if you choose to use Desmos.\r\n\r\n<img class=\"alignnone size-full wp-image-6805\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09220143\/Screen-Shot-2019-07-09-at-3.01.03-PM.png\" alt=\"\" width=\"1806\" height=\"1008\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Use a Linear Model to Make Predictions<\/h2>\r\nOnce we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned previously, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Regression Line to Make Predictions<\/h3>\r\nGasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in the table below.[footnote]<a href=\"http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html<\/a>[\/footnote] Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.Is this an interpolation or an extrapolation?\r\n<table id=\"Table_02_04_03\" style=\"height: 30px;\" summary=\"Two rows and twelve columns. The first row is labeled, 'Year'. The second row is labeled is labeled, 'Consumption (billions of gallons)'. Reading the remaining rows as ordered pairs (i.e., (Year, Consumption), we have the following values: ('94, 113), ('95, 116), ('96, 118), ('97, 119), ('98, 123), ('99, 125), ('00, 126), ('01, 128), ('02, 131), ('03, 133), and ('04, 136).\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><strong>Year<\/strong><\/td>\r\n<td style=\"height: 15px;\">'94<\/td>\r\n<td style=\"height: 15px;\">'95<\/td>\r\n<td style=\"height: 15px;\">'96<\/td>\r\n<td style=\"height: 15px;\">'97<\/td>\r\n<td style=\"height: 15px;\">'98<\/td>\r\n<td style=\"height: 15px;\">'99<\/td>\r\n<td style=\"height: 15px;\">'00<\/td>\r\n<td style=\"height: 15px;\">'01<\/td>\r\n<td style=\"height: 15px;\">'02<\/td>\r\n<td style=\"height: 15px;\">'03<\/td>\r\n<td style=\"height: 15px;\">'04<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><strong>Consumption (billions of gallons)<\/strong><\/td>\r\n<td style=\"height: 15px;\">113<\/td>\r\n<td style=\"height: 15px;\">116<\/td>\r\n<td style=\"height: 15px;\">118<\/td>\r\n<td style=\"height: 15px;\">119<\/td>\r\n<td style=\"height: 15px;\">123<\/td>\r\n<td style=\"height: 15px;\">125<\/td>\r\n<td style=\"height: 15px;\">126<\/td>\r\n<td style=\"height: 15px;\">128<\/td>\r\n<td style=\"height: 15px;\">131<\/td>\r\n<td style=\"height: 15px;\">133<\/td>\r\n<td style=\"height: 15px;\">136<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"671301\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"671301\"]\r\n\r\nWe can introduce an input variable, <em>t<\/em>, representing years since 1994. This makes entering the data into online graphing calculator easier.\r\n\r\nRead the value for b and the value for the slope, m, from the online graphing calculator to create the equation for the regression line:\r\n<p style=\"text-align: center;\">[latex]C\\left(t\\right)=113.318+2.209t[\/latex]<\/p>\r\nThe correlation coefficient was calculated to be 0.997, suggesting a very strong increasing linear trend.\r\n\r\nUsing this to predict consumption in 2008, which is 14 years after 1994 [latex]\\left(t=14\\right)[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}C\\left(14\\right)=113.318+2.209\\left(14\\right)\\hfill \\\\ C\\left(14\\right)=144.244\\hfill \\end{array}[\/latex]<\/p>\r\nThe model predicts 144.244 billion gallons of gasoline consumption in 2008. This is an extrapolation because there is not a datapoint whose x1 value is 2008.\r\nThe scatter plot of the data, including the least squares regression line, is shown below in Desmos, but you can use any online graphing calculator. Note how we changed the viewing window for the y-axis to [latex]80 &lt; y &lt; 150[\/latex].\r\n\r\n<img class=\"alignnone size-full wp-image-6807\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09221444\/Screen-Shot-2019-07-09-at-3.13.36-PM.png\" alt=\"\" width=\"1802\" height=\"1064\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse an online graphing calculator to find a linear regression for the following data, which represents the amount of time a scuba diver can spend underwater as a function of the depth of the water.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Depth (feet)<\/td>\r\n<td>Time (minutes)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>50<\/td>\r\n<td>80<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>60<\/td>\r\n<td>55<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>70<\/td>\r\n<td>45<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>80<\/td>\r\n<td>35<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>90<\/td>\r\n<td>25<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>100<\/td>\r\n<td>22<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n1) Write the equation for the least squares regression line.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n2) According to the regression line, how long can a diver spend at a depth of 110 feet?\r\n\r\n3)How about 120 feet? Why doesn't this make sense?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n4) At what depth would the dive time be zero?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"23398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23398\"]\r\n\r\nHere is a sample graph for this dataset.\r\n\r\n<img class=\"alignnone size-full wp-image-6810\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09221941\/Screen-Shot-2019-07-09-at-3.18.52-PM.png\" alt=\"\" width=\"1782\" height=\"1130\" \/>\r\n\r\n1) The equation for the regression line is [latex]y=-1.1143x+127.24[\/latex]\r\n2) A diver can spend [latex]y=-1.1143(110)+127.24=4.667[\/latex] minutes at a depth of 110 feet.\r\n3) A diver can spend [latex]y=-1.1143(120)+127.24=-6.48[\/latex] minutes at a depth of 120 feet. This doesn't make sense because a negative value for time doesn't have any meaning.\r\n4) To find at what depth the dive time would be zero, we need to set the regression equation equal to zero.\r\n[latex]\\begin{array}{l}0=-1.1143x+127.24\\\\-127.24=-1.1143x\\\\114.19 = x\\end{array}[\/latex]\r\n\r\nA diver, at a depth of 114.19 feet, would have a dive time of 0 minutes.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nHere are more data sets that you can plot using an online graphing calculator. \u00a0Try to find a linear regression for them then look at the correlation coefficient to determine whether there is a linear relationship.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Depth of the Columbia River<\/td>\r\n<td>Water Velocity<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0.66<\/td>\r\n<td>1.55<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1.98<\/td>\r\n<td>1.11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.64<\/td>\r\n<td>1.42<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.3<\/td>\r\n<td>1.39<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4.62<\/td>\r\n<td>1.39<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5.94<\/td>\r\n<td>1.14<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7.26<\/td>\r\n<td>0.91<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8.58<\/td>\r\n<td>0.59<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9.9<\/td>\r\n<td>0.59<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10.56<\/td>\r\n<td>0.41<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>11.22<\/td>\r\n<td>0.22<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"width: 412px;\">\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 197.75px; height: 30px;\">% of Mississippi River in Crops (By Basin)<\/td>\r\n<td style=\"width: 192.25px; height: 30px;\">Nitrate Concentration (mg\/ L)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14.3379px;\">\r\n<td style=\"width: 197.75px; height: 14.3379px;\">2.4<\/td>\r\n<td style=\"width: 192.25px; height: 14.3379px;\">0.647<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">1.3<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">1.062<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">14.3<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">1.432<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">0.5<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">0.579<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">45.6<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">3.561<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">46.6<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">3.938<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">1.5<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">0.927<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">53.6<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">2.549<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">4.1<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">0.357<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 197.75px; height: 14px;\">3.1<\/td>\r\n<td style=\"width: 192.25px; height: 14px;\">0.245<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"p1\">Dimensions of the Lava Dome in Mt. St. Helens, t = 0 on 18 October 1980 (eruption was 18 May 1980).<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Days<\/td>\r\n<td>Millions of Cubic Meters<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>2.9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>70<\/td>\r\n<td>13<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>109<\/td>\r\n<td>28<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>173<\/td>\r\n<td>40<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>242<\/td>\r\n<td>56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>322<\/td>\r\n<td>64<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>376<\/td>\r\n<td>75<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>547<\/td>\r\n<td>88<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>603<\/td>\r\n<td>100<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>699<\/td>\r\n<td>115<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>872<\/td>\r\n<td>152<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>922<\/td>\r\n<td>154<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1087<\/td>\r\n<td>173<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1343<\/td>\r\n<td>178<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1692<\/td>\r\n<td>212<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1858<\/td>\r\n<td>243<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h2>FYI<\/h2>\r\nDivers who want or need to descend to depths greater than 100 feet employ different techniques and equipment to help them safely navigate the depth. For example, different gas mixtures or rebreather equipment may be used. \u00a0Gas mixtures such as oxygen, helium, and nitrogen can help to mitigate the narcotic effects of breathing gas at great depths.[footnote]https:\/\/en.wikipedia.org\/wiki\/Trimix_(breathing_gas)[\/footnote]\r\n\r\n[caption id=\"attachment_2980\" align=\"aligncenter\" width=\"351\"]<img class=\"wp-image-2980\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/21212829\/Trevor_Jackson_returns_from_SS_Kyogle-300x225.jpg\" width=\"351\" height=\"263\" \/> A scuba diver using rebreather with open circuit bailout cylinders returning from a 600-foot (180 m) dive.[\/caption]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Draw and interpret scatter plots<\/li>\n<li>Find the line of best fit using a calculator<\/li>\n<li>Distinguish between linear and nonlinear relations<\/li>\n<li>Use a linear model to make predictions<\/li>\n<\/ul>\n<\/div>\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<div class=\"textbox examples\">\n<h3>recall ordered pairs as data points<\/h3>\n<p>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.<\/p>\n<\/div>\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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014335\/CNX_Precalc_Figure_02_04_0012.jpg\" 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=\"487\" height=\"337\" \/><\/p>\n<p class=\"wp-caption-text\">A scatter plot of age and final exam score variables.<\/p>\n<\/div>\n<p>Notice this scatter plot does <em>not<\/em> indicate a <strong>linear relationship<\/strong>. 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<div class=\"textbox exercises\">\n<h3>\u00a0Example: Using a Scatter Plot to Investigate Cricket Chirps<\/h3>\n<p>The table below\u00a0shows the number of cricket chirps in 15 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-250-1\" href=\"#footnote-250-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> Plot this data, and determine whether the data appears to be linearly related.<\/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>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/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=\"q579142\">Show Solution<\/span><\/p>\n<div id=\"q579142\" class=\"hidden-answer\" style=\"display: none\">\n<p>Plotting this data\u00a0suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so.<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014337\/CNX_Precalc_Figure_02_04_0022.jpg\" alt=\"Scatter plot, titled 'Cricket Chirps Vs Air Temperature'. The x-axis is the Cricket Chirps in 15 Seconds, and the y-axis is the Temperature (F). The line regression is generally positive.\" width=\"487\" height=\"386\" \/>\n<\/div>\n<\/div>\n<\/div>\n<h2>Finding the Line of Best Fit<\/h2>\n<p>One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the <em>y<\/em>-intercept. We can approximate the slope of the line by extending it until we can estimate the [latex]\\frac{\\text{rise}}{\\text{run}}[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Line of Best Fit<\/h3>\n<p>Find a linear function that fits the data in the table below\u00a0by &#8220;eyeballing&#8221; a line that seems to fit.<\/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> <\/colgroup>\n<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/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=\"q768322\">Show Solution<\/span><\/p>\n<div id=\"q768322\" class=\"hidden-answer\" style=\"display: none\">\n<p>On a graph, we could try sketching a line.<\/p>\n<p>Using the starting and ending points of our hand drawn line, points (0, 30) and (50, 90), this graph has a slope of [latex]m=\\frac{60}{50}=1.2[\/latex] and a <em>y<\/em>-intercept at 30. This gives an equation of [latex]T\\left(c\\right)=1.2c+30[\/latex]<\/p>\n<p>where <em>c<\/em>\u00a0is the number of chirps in 15 seconds, and <em>T<\/em>(<em>c<\/em>)\u00a0is the temperature in degrees Fahrenheit. The resulting equation is represented in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014339\/CNX_Precalc_Figure_02_04_0032.jpg\" alt=\"Scatter plot, showing the line of best fit. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"432\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This linear equation can then be used to approximate answers to various questions we might ask about the trend.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3681\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3681&#38;theme=oea&#38;iframe_resize_id=ohm3681&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Recognizing Interpolation or Extrapolation<\/h2>\n<p>While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as <strong>interpolation<\/strong> when we predict a value inside the domain and range of the data. The process of <strong>extrapolation<\/strong> is used when we predict a value outside the domain and range of the data.<\/p>\n<p>The graph below compares the two processes for the cricket-chirp data addressed in the previous example. We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.<\/p>\n<p>There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called <strong>model breakdown<\/strong>. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when [latex]x=50[\/latex], that is, in 50 years, the model would not apply because we could not account for factors fifty years in the future.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014341\/CNX_Precalc_Figure_02_04_0042.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"430\" \/><\/p>\n<p class=\"wp-caption-text\">Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Interpolation and Extrapolation<\/h3>\n<p>Different methods of making predictions are used to analyze data.<\/p>\n<ul>\n<li>The method of <strong>interpolation<\/strong> involves predicting a value inside the domain and\/or range of the data.<\/li>\n<li>The method of <strong>extrapolation<\/strong> involves predicting a value outside the domain and\/or range of the data.<\/li>\n<li><strong>Model breakdown<\/strong> occurs at the point when the model no longer applies.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Understanding Interpolation and Extrapolation<\/h3>\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>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Use the cricket data above\u00a0to answer the following questions:<\/p>\n<ol>\n<li>Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\n<li>Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q882447\">Show Solution<\/span><\/p>\n<div id=\"q882447\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is inside the domain of our data so would be interpolation. Using our model:<br \/>\n[latex]\\begin{array}{l}T\\left(30\\right)=30+1.2\\left(30\\right)\\hfill \\\\ T\\left(30\\right)=66\\text{ degrees}\\hfill \\end{array}[\/latex]<br \/>\nBased on the data we have, this value seems reasonable.<\/li>\n<li>The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation because 40 is outside the range of our data. Using our model:<br \/>\n[latex]\\begin{array}{l}40=30+1.2c\\hfill \\\\ 10=1.2c\\hfill \\\\ c\\approx 8.33\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<p>We can compare the regions of interpolation and extrapolation using the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014345\/CNX_Precalc_Figure_02_04_0052.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"485\" height=\"429\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Our model predicts the crickets would chirp 8.33 times in 15 seconds. While this might be possible, we have no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether at or below 50 degrees.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>According to the data from the table in the cricket-chirp example, what temperature can we predict if we counted 20 chirps in 15 seconds?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q271439\">Show Solution<\/span><\/p>\n<div id=\"q271439\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]54^\\circ \\text{F}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Finding the Line of Best Fit Using a Graphing Utility<\/h2>\n<p>While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values.<a class=\"footnote\" title=\"Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.\" id=\"return-footnote-250-2\" href=\"#footnote-250-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> One such technique is called <strong>least squares regression<\/strong> and can be computed by many graphing calculators as well as both spreadsheet and statistical software. Least squares regression is also called linear regression, and we can use an online graphing calculator to perform linear regressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Least Squares Regression Line<\/h3>\n<p>Find the least squares regression line using the cricket-chirp data in the table below.<\/p>\n<p>Use an online graphing calculator.<\/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<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/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=\"q374127\">Show Solution<\/span><\/p>\n<div id=\"q374127\" class=\"hidden-answer\" style=\"display: none\">\n<p>The following instructions are for Desmos, and other online graphing tools may be slightly different.<\/p>\n<ol>\n<li>Click the plus button (add item) in the upper left corner and select table.<\/li>\n<li>Enter chirps data in the x1 column.<\/li>\n<li>Enter temperature data in the y1 column.<br \/>\n<table summary=\"Two rows and ten columns. The first row is labeled, 'L1'. The second row is labeled is labeled, 'L2'. Reading the remaining rows as ordered pairs (i.e., (L2, L2), 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<tbody>\n<tr>\n<td><strong>x1<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><b>y1<\/b><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>If you can&#8217;t see the points on the grid, use the plus and minus buttons in the upper right hand corner to zoom in or out on the grid, or click on the wrench and change the upper bound of x1 to 60 and y1 to 100<\/li>\n<li>In the empty cell below the table you created, enter the expression y1\u223cmx1+b<\/li>\n<li>You can add labels to your graph by clicking on the wrench in the upper right hand corner and typing them into the cells that say &#8220;add a label&#8221;<\/li>\n<\/ol>\n<p>Here is an example of how your graph may look:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6803\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09215809\/Screen-Shot-2019-07-09-at-2.56.40-PM.png\" alt=\"\" width=\"1420\" height=\"1002\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that this line is quite similar to the equation we &#8220;eyeballed&#8221; but should fit the data better. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}T\\left(30\\right)=30.281+1.143\\left(30\\right)\\hfill \\\\ \\text{}T\\left(30\\right)=64.571\\hfill \\\\ \\text{}T\\left(30\\right)\\approx 64.6\\text{ degrees}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Will there ever be a case where two different lines will serve as the best fit for the data?<\/strong><\/p>\n<p><em>No. There is only one best fit line.<\/em><\/p>\n<\/div>\n<h2>Distinguish Between Linear and Nonlinear Relations<\/h2>\n<p>As we saw in the cricket-chirp example, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <strong>correlation coefficient<\/strong>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a &#8220;diagnostic on&#8221; selection to find the correlation coefficient, which mathematicians label as <em>r<\/em>. The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.<\/p>\n<p>We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense of the relationship between the value of <em>r<\/em>\u00a0and the graph of the data, the image below\u00a0shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.<\/p>\n<div style=\"width: 911px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014349\/CNX_Precalc_Figure_02_04_0072.jpg\" alt=\"A series of scatterplot graphs. Some are linear and some are not.\" width=\"901\" height=\"401\" \/><\/p>\n<p class=\"wp-caption-text\">Plotted data and related correlation coefficients. (credit: &#8220;DenisBoigelot,&#8221; Wikimedia Commons)<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Correlation Coefficient<\/h3>\n<p>The <strong>correlation coefficient<\/strong> is a value, <em>r<\/em>, between \u20131 and 1.<\/p>\n<ul>\n<li><em>r<\/em> &gt; 0 suggests a positive (increasing) relationship<\/li>\n<li><em>r<\/em> &lt; 0 suggests a negative (decreasing) relationship<\/li>\n<li>The closer the value is to 0, the more scattered the data.<\/li>\n<li>The closer the value is to 1 or \u20131, the less scattered the data is.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Correlation Coefficient<\/h3>\n<p>Calculate the correlation coefficient for cricket-chirp data in the table below.<\/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<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/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=\"q520385\">Show Solution<\/span><\/p>\n<div id=\"q520385\" class=\"hidden-answer\" style=\"display: none\">\n<p>Online graphing calculators\u00a0provide you with the correlation coefficient when you use it to calculate a linear regression. The correlation coefficients is labeled as <em>r\u00a0<\/em>= 0.951 for this dataset. This value is very close to 1 which suggests a strong increasing linear relationship. Below is an example of what your graph will look like if you choose to use Desmos.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6805\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09220143\/Screen-Shot-2019-07-09-at-3.01.03-PM.png\" alt=\"\" width=\"1806\" height=\"1008\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Use a Linear Model to Make Predictions<\/h2>\n<p>Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned previously, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Regression Line to Make Predictions<\/h3>\n<p>Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in the table below.<a class=\"footnote\" title=\"http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html\" id=\"return-footnote-250-3\" href=\"#footnote-250-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a> Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.Is this an interpolation or an extrapolation?<\/p>\n<table id=\"Table_02_04_03\" style=\"height: 30px;\" summary=\"Two rows and twelve columns. The first row is labeled, 'Year'. The second row is labeled is labeled, 'Consumption (billions of gallons)'. Reading the remaining rows as ordered pairs (i.e., (Year, Consumption), we have the following values: ('94, 113), ('95, 116), ('96, 118), ('97, 119), ('98, 123), ('99, 125), ('00, 126), ('01, 128), ('02, 131), ('03, 133), and ('04, 136).\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Year<\/strong><\/td>\n<td style=\"height: 15px;\">&#8217;94<\/td>\n<td style=\"height: 15px;\">&#8217;95<\/td>\n<td style=\"height: 15px;\">&#8217;96<\/td>\n<td style=\"height: 15px;\">&#8217;97<\/td>\n<td style=\"height: 15px;\">&#8217;98<\/td>\n<td style=\"height: 15px;\">&#8217;99<\/td>\n<td style=\"height: 15px;\">&#8217;00<\/td>\n<td style=\"height: 15px;\">&#8217;01<\/td>\n<td style=\"height: 15px;\">&#8217;02<\/td>\n<td style=\"height: 15px;\">&#8217;03<\/td>\n<td style=\"height: 15px;\">&#8217;04<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><strong>Consumption (billions of gallons)<\/strong><\/td>\n<td style=\"height: 15px;\">113<\/td>\n<td style=\"height: 15px;\">116<\/td>\n<td style=\"height: 15px;\">118<\/td>\n<td style=\"height: 15px;\">119<\/td>\n<td style=\"height: 15px;\">123<\/td>\n<td style=\"height: 15px;\">125<\/td>\n<td style=\"height: 15px;\">126<\/td>\n<td style=\"height: 15px;\">128<\/td>\n<td style=\"height: 15px;\">131<\/td>\n<td style=\"height: 15px;\">133<\/td>\n<td style=\"height: 15px;\">136<\/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=\"q671301\">Show Solution<\/span><\/p>\n<div id=\"q671301\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can introduce an input variable, <em>t<\/em>, representing years since 1994. This makes entering the data into online graphing calculator easier.<\/p>\n<p>Read the value for b and the value for the slope, m, from the online graphing calculator to create the equation for the regression line:<\/p>\n<p style=\"text-align: center;\">[latex]C\\left(t\\right)=113.318+2.209t[\/latex]<\/p>\n<p>The correlation coefficient was calculated to be 0.997, suggesting a very strong increasing linear trend.<\/p>\n<p>Using this to predict consumption in 2008, which is 14 years after 1994 [latex]\\left(t=14\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}C\\left(14\\right)=113.318+2.209\\left(14\\right)\\hfill \\\\ C\\left(14\\right)=144.244\\hfill \\end{array}[\/latex]<\/p>\n<p>The model predicts 144.244 billion gallons of gasoline consumption in 2008. This is an extrapolation because there is not a datapoint whose x1 value is 2008.<br \/>\nThe scatter plot of the data, including the least squares regression line, is shown below in Desmos, but you can use any online graphing calculator. Note how we changed the viewing window for the y-axis to [latex]80 < y < 150[\/latex].\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6807\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09221444\/Screen-Shot-2019-07-09-at-3.13.36-PM.png\" alt=\"\" width=\"1802\" height=\"1064\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use an online graphing calculator to find a linear regression for the following data, which represents the amount of time a scuba diver can spend underwater as a function of the depth of the water.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Depth (feet)<\/td>\n<td>Time (minutes)<\/td>\n<\/tr>\n<tr>\n<td>50<\/td>\n<td>80<\/td>\n<\/tr>\n<tr>\n<td>60<\/td>\n<td>55<\/td>\n<\/tr>\n<tr>\n<td>70<\/td>\n<td>45<\/td>\n<\/tr>\n<tr>\n<td>80<\/td>\n<td>35<\/td>\n<\/tr>\n<tr>\n<td>90<\/td>\n<td>25<\/td>\n<\/tr>\n<tr>\n<td>100<\/td>\n<td>22<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>1) Write the equation for the least squares regression line.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>2) According to the regression line, how long can a diver spend at a depth of 110 feet?<\/p>\n<p>3)How about 120 feet? Why doesn&#8217;t this make sense?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>4) At what depth would the dive time be zero?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23398\">Show Solution<\/span><\/p>\n<div id=\"q23398\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here is a sample graph for this dataset.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6810\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/09221941\/Screen-Shot-2019-07-09-at-3.18.52-PM.png\" alt=\"\" width=\"1782\" height=\"1130\" \/><\/p>\n<p>1) The equation for the regression line is [latex]y=-1.1143x+127.24[\/latex]<br \/>\n2) A diver can spend [latex]y=-1.1143(110)+127.24=4.667[\/latex] minutes at a depth of 110 feet.<br \/>\n3) A diver can spend [latex]y=-1.1143(120)+127.24=-6.48[\/latex] minutes at a depth of 120 feet. This doesn&#8217;t make sense because a negative value for time doesn&#8217;t have any meaning.<br \/>\n4) To find at what depth the dive time would be zero, we need to set the regression equation equal to zero.<br \/>\n[latex]\\begin{array}{l}0=-1.1143x+127.24\\\\-127.24=-1.1143x\\\\114.19 = x\\end{array}[\/latex]<\/p>\n<p>A diver, at a depth of 114.19 feet, would have a dive time of 0 minutes.\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Here are more data sets that you can plot using an online graphing calculator. \u00a0Try to find a linear regression for them then look at the correlation coefficient to determine whether there is a linear relationship.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Depth of the Columbia River<\/td>\n<td>Water Velocity<\/td>\n<\/tr>\n<tr>\n<td>0.66<\/td>\n<td>1.55<\/td>\n<\/tr>\n<tr>\n<td>1.98<\/td>\n<td>1.11<\/td>\n<\/tr>\n<tr>\n<td>2.64<\/td>\n<td>1.42<\/td>\n<\/tr>\n<tr>\n<td>3.3<\/td>\n<td>1.39<\/td>\n<\/tr>\n<tr>\n<td>4.62<\/td>\n<td>1.39<\/td>\n<\/tr>\n<tr>\n<td>5.94<\/td>\n<td>1.14<\/td>\n<\/tr>\n<tr>\n<td>7.26<\/td>\n<td>0.91<\/td>\n<\/tr>\n<tr>\n<td>8.58<\/td>\n<td>0.59<\/td>\n<\/tr>\n<tr>\n<td>9.9<\/td>\n<td>0.59<\/td>\n<\/tr>\n<tr>\n<td>10.56<\/td>\n<td>0.41<\/td>\n<\/tr>\n<tr>\n<td>11.22<\/td>\n<td>0.22<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 412px;\">\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"width: 197.75px; height: 30px;\">% of Mississippi River in Crops (By Basin)<\/td>\n<td style=\"width: 192.25px; height: 30px;\">Nitrate Concentration (mg\/ L)<\/td>\n<\/tr>\n<tr style=\"height: 14.3379px;\">\n<td style=\"width: 197.75px; height: 14.3379px;\">2.4<\/td>\n<td style=\"width: 192.25px; height: 14.3379px;\">0.647<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">1.3<\/td>\n<td style=\"width: 192.25px; height: 14px;\">1.062<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">14.3<\/td>\n<td style=\"width: 192.25px; height: 14px;\">1.432<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">0.5<\/td>\n<td style=\"width: 192.25px; height: 14px;\">0.579<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">45.6<\/td>\n<td style=\"width: 192.25px; height: 14px;\">3.561<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">46.6<\/td>\n<td style=\"width: 192.25px; height: 14px;\">3.938<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">1.5<\/td>\n<td style=\"width: 192.25px; height: 14px;\">0.927<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">53.6<\/td>\n<td style=\"width: 192.25px; height: 14px;\">2.549<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">4.1<\/td>\n<td style=\"width: 192.25px; height: 14px;\">0.357<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 197.75px; height: 14px;\">3.1<\/td>\n<td style=\"width: 192.25px; height: 14px;\">0.245<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"p1\">Dimensions of the Lava Dome in Mt. St. Helens, t = 0 on 18 October 1980 (eruption was 18 May 1980).<\/p>\n<table>\n<tbody>\n<tr>\n<td>Days<\/td>\n<td>Millions of Cubic Meters<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2.9<\/td>\n<\/tr>\n<tr>\n<td>70<\/td>\n<td>13<\/td>\n<\/tr>\n<tr>\n<td>109<\/td>\n<td>28<\/td>\n<\/tr>\n<tr>\n<td>173<\/td>\n<td>40<\/td>\n<\/tr>\n<tr>\n<td>242<\/td>\n<td>56<\/td>\n<\/tr>\n<tr>\n<td>322<\/td>\n<td>64<\/td>\n<\/tr>\n<tr>\n<td>376<\/td>\n<td>75<\/td>\n<\/tr>\n<tr>\n<td>547<\/td>\n<td>88<\/td>\n<\/tr>\n<tr>\n<td>603<\/td>\n<td>100<\/td>\n<\/tr>\n<tr>\n<td>699<\/td>\n<td>115<\/td>\n<\/tr>\n<tr>\n<td>872<\/td>\n<td>152<\/td>\n<\/tr>\n<tr>\n<td>922<\/td>\n<td>154<\/td>\n<\/tr>\n<tr>\n<td>1087<\/td>\n<td>173<\/td>\n<\/tr>\n<tr>\n<td>1343<\/td>\n<td>178<\/td>\n<\/tr>\n<tr>\n<td>1692<\/td>\n<td>212<\/td>\n<\/tr>\n<tr>\n<td>1858<\/td>\n<td>243<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h2>FYI<\/h2>\n<p>Divers who want or need to descend to depths greater than 100 feet employ different techniques and equipment to help them safely navigate the depth. For example, different gas mixtures or rebreather equipment may be used. \u00a0Gas mixtures such as oxygen, helium, and nitrogen can help to mitigate the narcotic effects of breathing gas at great depths.<a class=\"footnote\" title=\"https:\/\/en.wikipedia.org\/wiki\/Trimix_(breathing_gas)\" id=\"return-footnote-250-4\" href=\"#footnote-250-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a><\/p>\n<div id=\"attachment_2980\" style=\"width: 361px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2980\" class=\"wp-image-2980\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/21212829\/Trevor_Jackson_returns_from_SS_Kyogle-300x225.jpg\" width=\"351\" height=\"263\" alt=\"image\" \/><\/p>\n<p id=\"caption-attachment-2980\" class=\"wp-caption-text\">A scuba diver using rebreather with open circuit bailout cylinders returning from a 600-foot (180 m) dive.<\/p>\n<\/div>\n<\/div>\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-250\">\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>Revision and Adaptation. <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><li>Temperature as a Function of the Number of Cricket Chirps in a 15 Second Period Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/ruvzg6iy3o\">https:\/\/www.desmos.com\/calculator\/ruvzg6iy3o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Consumption of Gas as a Function of Year Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/lv27pmtdbh\">https:\/\/www.desmos.com\/calculator\/lv27pmtdbh<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Dive Time as a Function of Depth Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/zyrpta1uls\">https:\/\/www.desmos.com\/calculator\/zyrpta1uls<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><li>Scuba diver using rebreather with open circuit bailout cylinders returning from a 600-foot (180 m) dive. <strong>Authored by<\/strong>: Trevor Jackson. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/commons.wikimedia.org\/w\/index.php?curid=25988843\">https:\/\/commons.wikimedia.org\/w\/index.php?curid=25988843<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-250-1\">Selected data from <a href=\"http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/\" target=\"_blank\" rel=\"noopener\">http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/<\/a>. Retrieved Aug 3, 2010 <a href=\"#return-footnote-250-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-250-2\">Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values. <a href=\"#return-footnote-250-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-250-3\"><a href=\"http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html<\/a> <a href=\"#return-footnote-250-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-250-4\">https:\/\/en.wikipedia.org\/wiki\/Trimix_(breathing_gas) <a href=\"#return-footnote-250-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":17533,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"cc\",\"description\":\"Scuba diver using rebreather with open circuit bailout cylinders returning from a 600-foot (180 m) dive\",\"author\":\"Trevor Jackson\",\"organization\":\"\",\"url\":\"https:\/\/commons.wikimedia.org\/w\/index.php?curid=25988843\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"original\",\"description\":\"Temperature as a Function of the Number of Cricket Chirps in a 15 Second Period Interactive\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/www.desmos.com\/calculator\/ruvzg6iy3o\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Consumption of Gas as a Function of Year Interactive\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/www.desmos.com\/calculator\/lv27pmtdbh\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Dive Time as a Function of Depth Interactive\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/www.desmos.com\/calculator\/zyrpta1uls\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"b27fb45b-0ddf-429b-beb2-ac9c996b3288","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-250","chapter","type-chapter","status-publish","hentry"],"part":232,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/250","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/250\/revisions"}],"predecessor-version":[{"id":1015,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/250\/revisions\/1015"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/parts\/232"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/250\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/media?parent=250"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=250"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/contributor?post=250"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/license?post=250"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}