{"id":1160,"date":"2015-11-12T18:35:31","date_gmt":"2015-11-12T18:35:31","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1160"},"modified":"2017-03-31T22:16:04","modified_gmt":"2017-03-31T22:16:04","slug":"find-the-line-of-best-fit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/find-the-line-of-best-fit\/","title":{"raw":"Find the line of best fit","rendered":"Find the line of best fit"},"content":{"raw":"

Once we recognize a need for a linear function to model the\u00a0data in \"Draw and interpret scatter plots<\/a>,\" the natural follow-up question is \"what is that linear function?\" 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 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>\r\n\r\n

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Example 2: 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<\/colgroup>
Chirps<\/strong><\/td>\r\n44<\/td>\r\n35<\/td>\r\n20.4<\/td>\r\n33<\/td>\r\n31<\/td>\r\n35<\/td>\r\n18.5<\/td>\r\n37<\/td>\r\n26<\/td>\r\n<\/tr>
Temperature<\/strong><\/td>\r\n80.5<\/td>\r\n70.5<\/td>\r\n57<\/td>\r\n66<\/td>\r\n68<\/td>\r\n72<\/td>\r\n52<\/td>\r\n73.5<\/td>\r\n53<\/td>\r\n<\/tr><\/tbody><\/table><\/div>\r\n
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Solution<\/h3>\r\n

On a graph, we could try sketching a line.<\/p>\r\n

Using the starting and ending points of our hand drawn line, points (0, 30) and (50, 90), this graph has a slope of<\/p>\r\n\r\n

[latex]m=\\frac{60}{50}=1.2[\/latex]<\/div>\r\n

and a y<\/em>-intercept at 30. This gives an equation of<\/p>\r\n\r\n

[latex]T\\left(c\\right)=1.2c+30[\/latex]<\/div>\r\n

where c<\/em>\u00a0is the number of chirps in 15 seconds, and T<\/em>(c<\/em>)\u00a0is the temperature in degrees Fahrenheit. The resulting equation is represented in the graph below.<\/p>\r\n\r\n

\r\n<\/span><\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"ScatterFigure 3<\/b>[\/caption]\r\n\r\n<\/figure><\/div>\r\n
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Analysis of the Solution<\/h3>\r\n

This linear equation can then be used to approximate answers to various questions we might ask about the trend.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Recognizing Interpolation or Extrapolation<\/h2>\r\n

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 interpolation<\/strong> when we predict a value inside the domain and range of the data. The process of extrapolation<\/strong> is used when we predict a value outside the domain and range of the data.<\/p>\r\n

The graph below compares the two processes for the cricket-chirp data addressed in Example 2. 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>\r\n

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 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 x<\/em> = 50, that is in 50 years, the model would not apply because we could not account for factors fifty years in the future.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"ScatterFigure 4.<\/b> Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.[\/caption]\r\n\r\n

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A General Note: Interpolation and Extrapolation<\/h3>\r\n

Different methods of making predictions are used to analyze data.<\/p>\r\n\r\n