Questions: Graphs and Functions Computing residuals a table and scatter plot show the average monthly temperature, x, and a family's monthly heating cost, y, for 11 different months. The equation of the line of best fit is y=-1.2x+91.54. Average monthly temperature, x (in °F) Monthly heating cost, y (in dollars) ------------------------------------------------------------------------------ 23.0 51.00 30.7 68.00 35.7 67.00 37.6 40.64 40.3 29.77 43.4 52.01 48.1 49.73 50.1 30.14 53.0 15.43 54.3 22.78 57.0 30.50 Use the equation of the line of best fit to fill in the blanks below. Give exact answers, not rounded approximations. Average monthly temperature (in °F) Observed monthly heating cost (in dollars) Predicted monthly heating cost (in dollars) Residual (in dollars) ---------------------------------------------------------------------------------------------------------------------------------------------------- 23.0 35.7

Graphs and Functions Computing residuals a table and scatter plot show the average monthly temperature, x, and a family's monthly heating cost, y, for 11 different months. The equation of the line of best fit is y=-1.2x+91.54. Average monthly temperature, x (in °F) Monthly heating cost, y (in dollars) ------------------------------------------------------------------------------ 23.0 51.00 30.7 68.00 35.7 67.00 37.6 40.64 40.3 29.77 43.4 52.01 48.1 49.73 50.1 30.14 53.0 15.43 54.3 22.78 57.0 30.50 Use the equation of the line of best fit to fill in the blanks below. Give exact answers, not rounded approximations. Average monthly temperature (in °F) Observed monthly heating cost (in dollars) Predicted monthly heating cost (in dollars) Residual (in dollars) ---------------------------------------------------------------------------------------------------------------------------------------------------- 23.0 35.7
Transcript text: Graphs and Functions Computing residuals a table and scatter plot show the average monthly temperature, $x$, and a family's monthly heating cost, $y$, for 11 different months. The equation of the line of best fit is $y=-1.2 x+91.54$. \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Average \\ monthly \\ temperature, $\boldsymbol{x}$ \\ (in ${ }^{\circ} \mathrm{F}$ ) \end{tabular} & \begin{tabular}{c} Monthly \\ heating cost, $\boldsymbol{y}$ \\ (in dollars) \end{tabular} \\ \hline 23.0 & 51.00 \\ \hline 30.7 & 68.00 \\ \hline 35.7 & 67.00 \\ \hline 37.6 & 40.64 \\ \hline 40.3 & 29.77 \\ \hline 43.4 & 52.01 \\ \hline 48.1 & 49.73 \\ \hline 50.1 & 30.14 \\ \hline 53.0 & 15.43 \\ \hline 54.3 & 22.78 \\ \hline 57.0 & 30.50 \\ \hline \end{tabular} Use the equation of the line of best fit to fill in the blanks below. Give exact answers, not rounded approximations. \begin{tabular}{|c|c|c|c|} \hline \begin{tabular}{c} Average monthly \\ temperature \\ (in ${ }^{\circ} \mathrm{F}$ ) \end{tabular} & \begin{tabular}{c} Observed monthly \\ heating cost \\ (in dollars) \end{tabular} & \begin{tabular}{c} Predicted monthly \\ heating cost \\ (in dollars) \end{tabular} & \begin{tabular}{c} Residual \\ (in dollars) \end{tabular} \\ \hline 23.0 & $\square$ & $\square$ & $\square$ \\ \hline 35.7 & $\square$ & $\square$ & $\square$ \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Identify the given data and equation

The given data includes:

  • Average monthly temperature (\(x\)) in °F
  • Monthly heating cost (\(y\)) in dollars

The equation of the line of best fit is: \[ y = -1.2x + 91.54 \]

Step 2: Calculate the predicted monthly heating cost

Use the equation of the line of best fit to calculate the predicted monthly heating cost for the given temperatures.

For \(x = 23.0\): \[ y = -1.2(23.0) + 91.54 \] \[ y = -27.6 + 91.54 \] \[ y = 63.94 \]

For \(x = 35.7\): \[ y = -1.2(35.7) + 91.54 \] \[ y = -42.84 + 91.54 \] \[ y = 48.70 \]

Step 3: Calculate the residuals

The residual is the difference between the observed monthly heating cost and the predicted monthly heating cost.

For \(x = 23.0\): Observed monthly heating cost = 51.00 Predicted monthly heating cost = 63.94 Residual = Observed - Predicted \[ \text{Residual} = 51.00 - 63.94 \] \[ \text{Residual} = -12.94 \]

For \(x = 35.7\): Observed monthly heating cost = 67.00 Predicted monthly heating cost = 48.70 Residual = Observed - Predicted \[ \text{Residual} = 67.00 - 48.70 \] \[ \text{Residual} = 18.30 \]

Final Answer

| Average monthly temperature (in °F) | Observed monthly heating cost (in dollars) | Predicted monthly heating cost (in dollars) | Residual (in dollars) | |-------------------------------------|--------------------------------------------|--------------------------------------------|-----------------------| | 23.0 | 51.00 | 63.94 | -12.94 | | 35.7 | 67.00 | 48.70 | 18.30 |

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