Question

# 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

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.

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.