{"id":814,"date":"2025-07-15T19:38:08","date_gmt":"2025-07-15T19:38:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=814"},"modified":"2026-04-01T08:38:04","modified_gmt":"2026-04-01T08:38:04","slug":"linear-models-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-models-learn-it-4\/","title":{"raw":"Linear Models: Learn It 4","rendered":"Linear Models: Learn It 4"},"content":{"raw":"

Finding the Line of Best Fit<\/h2>\r\nOnce we recognize a need for a linear function to model that data, the natural follow-up question is \u201cwhat is that linear function?\u201d\r\n\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 [latex]y[\/latex]-intercept. We can approximate the slope of the line by extending it until we can estimate the [latex]\\dfrac{\\text{rise}}{\\text{run}}[\/latex].\r\n\r\n
The table below shows the number of cricket chirps in [latex]15[\/latex] seconds, for several different air temperatures, in degrees Fahrenheit.[footnote]Selected data from http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/. Retrieved Aug 3, 2010.[\/footnote]\r\n<\/colgroup>\r\n\r\n\r\n\r\n
Chirps<\/strong><\/td>\r\n[latex]44[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n[latex]20.4[\/latex]<\/td>\r\n[latex]33[\/latex]<\/td>\r\n[latex]31[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n[latex]18.5[\/latex]<\/td>\r\n[latex]37[\/latex]<\/td>\r\n[latex]26[\/latex]<\/td>\r\n<\/tr>\r\n
Temperature<\/strong><\/td>\r\n[latex]80.5[\/latex]<\/td>\r\n[latex]70.5[\/latex]<\/td>\r\n[latex]57[\/latex]<\/td>\r\n[latex]66[\/latex]<\/td>\r\n[latex]68[\/latex]<\/td>\r\n[latex]72[\/latex]<\/td>\r\n[latex]52[\/latex]<\/td>\r\n[latex]73.5[\/latex]<\/td>\r\n[latex]53[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\"APlotting this data suggests that there may be a positive linear trend, though certainly not perfectly so. We can see from the trend in the data that the number of chirps increases as the temperature increases.<\/span>\r\n\r\nIn the plotted data, we have sketched a line that seems to best fit the data.\r\n\r\nWhat is the estimated linear function?\r\n\r\n[reveal-answer q=\"674835\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"674835\"]Note: We can only try to estimate the slope of this line by observing its steepness and direction.\r\n\r\nSteps to Estimate the Slope:\r\n
    \r\n \t
  1. Pick two points on the line: Choose points that are easy to read. For example, the first point [latex](18.5, 52)[\/latex] and the last point [latex](44, 80.5)[\/latex] are close to the fitted line.<\/li>\r\n \t
  2. Calculate the rise and run, and estimate the slope:<\/li>\r\n<\/ol>\r\n
    [latex]\\begin{align*} \\text{Rise} &= 80.5 - 52 = 28.5 \\\\ \\text{Run} &= 44 - 18.5 = 25.5 \\\\ \\text{Slope} &= \\frac{\\text{Rise}}{\\text{Run}} = \\frac{28.5}{25.5} \\approx 1.12 \\end{align*}[\/latex]<\/center>By using the points, we estimated that slope [latex]\\approx 1.12[\/latex].\r\n\r\nTo find the equation of the line using the last point [latex](44, 80.5)[\/latex], and the slope [latex]\\approx 1.12[\/latex] we previously estimated, we can use the point-slope form of the equation of a line.\r\n\r\n
    [latex]\\begin{align*} y - y_1 = m(x - x_1) \\\\ y - 80.5 &= 1.12(x - 44) \\\\ y - 80.5 &= 1.12x - 1.12 \\cdot 44 \\\\ y - 80.5 &= 1.12x - 49.28 \\\\ y &= 1.12x - 49.28 + 80.5 \\\\ y &= 1.12x + 31.22 \\end{align*}[\/latex]<\/center>So, the estimated linear function is: [latex]y = 1.12x + 31.22[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm_question hide_question_numbers=1]318767[\/ohm_question]<\/section>
    \r\n

    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 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\r\n
    Find the least squares regression line using the cricket-chirp data in the table below.Use an online graphing calculator.\r\n<\/colgroup>\r\n\r\n\r\n\r\n
    Chirps<\/strong><\/td>\r\n[latex]44[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n[latex]20.4[\/latex]<\/td>\r\n[latex]33[\/latex]<\/td>\r\n[latex]31[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n[latex]18.5[\/latex]<\/td>\r\n[latex]37[\/latex]<\/td>\r\n[latex]26[\/latex]<\/td>\r\n<\/tr>\r\n
    Temperature<\/strong><\/td>\r\n[latex]80.5[\/latex]<\/td>\r\n[latex]70.5[\/latex]<\/td>\r\n[latex]57[\/latex]<\/td>\r\n[latex]66[\/latex]<\/td>\r\n[latex]68[\/latex]<\/td>\r\n[latex]72[\/latex]<\/td>\r\n[latex]52[\/latex]<\/td>\r\n[latex]73.5[\/latex]<\/td>\r\n[latex]53[\/latex]<\/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
      \r\n \t
    1. Click the plus button (add item) in the upper left corner and select table.<\/li>\r\n \t
    2. Enter chirps data in the [latex]x_1[\/latex] column.<\/li>\r\n \t
    3. Enter temperature data in the [latex]y_1[\/latex] column.\r\n<\/colgroup>\r\n\r\n\r\n\r\n
      Chirps<\/strong><\/td>\r\n[latex]44[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n[latex]20.4[\/latex]<\/td>\r\n[latex]33[\/latex]<\/td>\r\n[latex]31[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n[latex]18.5[\/latex]<\/td>\r\n[latex]37[\/latex]<\/td>\r\n[latex]26[\/latex]<\/td>\r\n<\/tr>\r\n
      Temperature<\/strong><\/td>\r\n[latex]80.5[\/latex]<\/td>\r\n[latex]70.5[\/latex]<\/td>\r\n[latex]57[\/latex]<\/td>\r\n[latex]66[\/latex]<\/td>\r\n[latex]68[\/latex]<\/td>\r\n[latex]72[\/latex]<\/td>\r\n[latex]52[\/latex]<\/td>\r\n[latex]73.5[\/latex]<\/td>\r\n[latex]53[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
    4. 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 [latex]x_1[\/latex] to [latex]60[\/latex] and [latex]y_1[\/latex] to [latex]100[\/latex]<\/li>\r\n \t
    5. In the empty cell below the table you created, enter the expression [latex]y_1\u223cmx_1+b[\/latex]<\/li>\r\n \t
    6. 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\"\"\r\n\r\nAnalysis of the Solution<\/strong>\r\n\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 [latex]30[\/latex] chirps in [latex]15[\/latex] seconds from [latex]66[\/latex] degrees to:\r\n

      [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<\/section>

      Steps to obtain the equation of the regression line and equation:\r\n[latex]\\\\[\/latex]\r\nStep 1: <\/strong>Under \"Enter Data\",<\/strong> select the\u00a0\u201cEnter Own\u201d<\/strong>.\r\nStep 2: <\/strong>Change the name of the [latex]x[\/latex]- and [latex]y[\/latex]-variable accordingly.\r\nStep 3: <\/strong>Enter the input ([latex]x[\/latex]Var) and output ([latex]y[\/latex]Var) accordingly.\r\nStep 4: \"Submit<\/strong> Data\"<\/strong> and you will see the scatterplot on the right side of the statistical tool.\r\nStep 5:<\/strong> Under Plot Options: click on <\/span>\"Regression Line\"<\/strong> and you will see that the statistical tool will draw the line that best fit your data in your scatterplot. Right above the scatterplot, you will also see the equation of that line.<\/span>