{"id":1164,"date":"2015-11-12T18:35:31","date_gmt":"2015-11-12T18:35:31","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1164"},"modified":"2017-03-31T22:16:45","modified_gmt":"2017-03-31T22:16:45","slug":"use-a-linear-model-to-make-predictions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/use-a-linear-model-to-make-predictions\/","title":{"raw":"Use a linear model to make predictions","rendered":"Use a linear model to make predictions"},"content":{"raw":"

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>\r\n\r\n

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Example 6: Using a Regression Line to Make Predictions<\/h3>\r\n

Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in the table below.[footnote]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.<\/p>\r\n\r\n
Year<\/strong><\/td>\r\n'94<\/td>\r\n'95<\/td>\r\n'96<\/td>\r\n'97<\/td>\r\n'98<\/td>\r\n'99<\/td>\r\n'00<\/td>\r\n'01<\/td>\r\n'02<\/td>\r\n'03<\/td>\r\n'04<\/td>\r\n<\/tr>
Consumption (billions of gallons)<\/strong><\/td>\r\n113<\/td>\r\n116<\/td>\r\n118<\/td>\r\n119<\/td>\r\n123<\/td>\r\n125<\/td>\r\n126<\/td>\r\n128<\/td>\r\n131<\/td>\r\n133<\/td>\r\n136<\/td>\r\n<\/tr><\/tbody><\/table>\r\nThe scatter plot of the data, including the least squares regression line, is shown in Figure 8.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"ScatterFigure 8<\/b>[\/caption]\r\n\r\n<\/div>\r\n
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Solution<\/h3>\r\n

We can introduce new input variable, t<\/em>, representing years since 1994.<\/p>\r\n

The least squares regression equation is:<\/p>\r\n\r\n

[latex]C\\left(t\\right)=113.318+2.209t[\/latex]<\/div>\r\n

Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.<\/p>\r\n

Using this to predict consumption in 2008 [latex]\\left(t=14\\right)[\/latex],<\/p>\r\n\r\n

[latex]\\begin{cases}C\\left(14\\right)=113.318+2.209\\left(14\\right)\\hfill \\\\ \\text{ }=144.244\\hfill \\end{cases}[\/latex]<\/div>\r\n

The model predicts 144.244 billion gallons of gasoline consumption in 2008.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Try It 2<\/h3>\r\n

Use the model we created using technology in Example 6\u00a0to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

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

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Example 6: Using a Regression Line to Make Predictions<\/h3>\n

Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in the table below.[1]<\/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.<\/p>\n\n\n\n\n
Year<\/strong><\/td>\n’94<\/td>\n’95<\/td>\n’96<\/td>\n’97<\/td>\n’98<\/td>\n’99<\/td>\n’00<\/td>\n’01<\/td>\n’02<\/td>\n’03<\/td>\n’04<\/td>\n<\/tr>\n
Consumption (billions of gallons)<\/strong><\/td>\n113<\/td>\n116<\/td>\n118<\/td>\n119<\/td>\n123<\/td>\n125<\/td>\n126<\/td>\n128<\/td>\n131<\/td>\n133<\/td>\n136<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The scatter plot of the data, including the least squares regression line, is shown in Figure 8.<\/p>\n

\"Scatter<\/p>\n

Figure 8<\/b><\/p>\n<\/div>\n<\/div>\n

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Solution<\/h3>\n

We can introduce new input variable, t<\/em>, representing years since 1994.<\/p>\n

The least squares regression equation is:<\/p>\n

[latex]C\\left(t\\right)=113.318+2.209t[\/latex]<\/div>\n

Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.<\/p>\n

Using this to predict consumption in 2008 [latex]\\left(t=14\\right)[\/latex],<\/p>\n

[latex]\\begin{cases}C\\left(14\\right)=113.318+2.209\\left(14\\right)\\hfill \\\\ \\text{ }=144.244\\hfill \\end{cases}[\/latex]<\/div>\n

The model predicts 144.244 billion gallons of gasoline consumption in 2008.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Try It 2<\/h3>\n

Use the model we created using technology in Example 6\u00a0to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?<\/p>\n

Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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