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In addition to quadratic and exponential models, another common type of model is called a power model. Power models are models in the form hat{y}=a cdot x^{p}. Here are data on

Exponential models
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asked 2021-01-06

In addition to quadratic and exponential models, another common type of model is called a power model. Power models are models in the form \(\displaystyle\hat{{{y}}}={a}\ \cdot\ {x}^{{{p}}}\). Here are data on the eight planets of our solar system. Distance from the sun is measured in astronomical units (AU), the average distance Earth is from the sun. \(\begin{array}{|c|c|}\hline \text{Planet} & \text {Distance from sun}\text {(astronomical units)} & \text{Period of revolution}\text{(Earth years)} \\ \hline \text{Mercury} & 0.387 & 0.241 \\ \hline \text { Venus } & 0.723 & 0.615 \\ \hline \text { Earth } & 1.000 & 1.000 \\ \hline \hline \text { Mars } & 1.524 & 1.881 \\ \hline \text { Jupiter } & 5.203 & 11.862 \\ \hline \text { Saturn } & 9.539 & 29.456 \\ \hline \text { Uranus } & 19.191 & 84.070 \\ \hline \text { Neptune } & 30.061 & 164.810 \\ \hline \end{array}\) Calculate and interpret the residual for Neptune.

Answers (1)

2021-01-07
Step 1 Note: The solution gives the commands for the calculation using aTi83/84-calculator. If you use a different type of technology, then the commands will differ. Press on STAT and then select 1:Edit ... Enter the data of distance from the sun in the list \(\displaystyle{L}_{{{1}}}\ \text{and enter the data of period of revolution in the list}\ {L}_{{{2}}}\). Next, press on STAT, select CALC and then select \(\displaystyle{P}_{{{W}{T}}}\ {R}_{{{e}{g}}}.\ \text{Next, we need to finish the command by entering}\ {L}_{{{1}}},\ {L}_{{{2}}}.\)
\(\displaystyle{P}_{{{W}{T}}}\ {R}_{{{e}{g}}}\ {L}_{{{1}}},\ {L}_{{{2}}}\) Finally, pressing on ENTER then gives us the following result: \(\displaystyle{y}={a}\ \cdot\ {x}^{{{b}}}\)
\(\displaystyle{a}={1.003}\)
\(\displaystyle{b}={1.4998}\)
\(\displaystyle{r}^{{{2}}}={1}\) This then implies that the regression line is: \(\displaystyle\hat{{{y}}}={a}\ \cdot\ {x}^{{{b}}}={1.0003}\ \cdot\ {x}^{{{1.4998}}}\) where x represents the distance from sun and y represents the period of revolution. Step 2 Given: \(\displaystyle{x}={30.061}\)
\(\displaystyle{y}={164.810}\) Evaluate the equation of the regression line at \(\displaystyle{x}={30.061}:\)
\(\displaystyle\hat{{{y}}}={1.0003}\ \cdot\ {30.061}^{{{1.4998}}}\ \approx\ {164.7631}\) The residual is the difference between the observed y-value and the predicted y-value. \(\displaystyle\text{Residual}\ ={y}\ -\ \hat{{{y}}}\ ={164.810}\ -\ {164.7631}={0.0469}\) Neptune's period of revolution is 0.0469 earth years longer than expected.
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