Exercise 1.2.32

Question

The table shows average US retail residential prices of electricity from 2000 to 2016, measured in cents per kilowatt hour.
    \begin{enumerate}[label=(\alph*)]
        \item Make a scatter plot. Is a linear model appropriate?
        \item Find and graph the regression line.
        \item Use your linear model from part (b) to estimate the average retail price of electricity in 2005 and 2017.
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item We make the scatter plot as follows:\begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=19, ymax=14, ymin=0, xlabel={Years since 2000}, ylabel={Cents/kWh}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[mark=*, mark options={scale=1, fill=black}](0,8.24);
         \addplot[mark=*, mark options={scale=1, fill=black}](2,8.44);
         \addplot[mark=*, mark options={scale=1, fill=black}](4,8.95);
         \addplot[mark=*, mark options={scale=1, fill=black}](6,10.40);
         \addplot[mark=*, mark options={scale=1, fill=black}](8,11.26);
         \addplot[mark=*, mark options={scale=1, fill=black}](10,11.54);
         \addplot[mark=*, mark options={scale=1, fill=black}](12,11.88);
         \addplot[mark=*, mark options={scale=1, fill=black}](14,12.52);
         \addplot[mark=*, mark options={scale=1, fill=black}](16,12.90);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    As we can see, the linear model is appropriate for this graph.

\item Using the Python code from \href{https://math.solverer.com/library/james_stewart/calculus_early_transcendentals/exercise_1-2-29}{Exercise 1.2.29 (c)} or \href{https://www.wolframalpha.com/input?i=regression+linear+&assumption=%7B%22F%22%2C+%22LinearFitCalculator%22%2C+%22data2%22%7D+-%3E%22%7B%7B0%2C8.24%7D%2C%7B2%2C8.44%7D%2C%7B4%2C8.95%7D%2C%7B6%2C10.4%7D%2C+%7B8%2C11.26%7D%2C+%7B10%2C11.54%7D%2C+%7B12%2C11.88%7D%2C+%7B14%2C12.52%7D%2C+%7B16%2C12.90%7D%7D%22}{WolframAlpha}, we get the following regression line function:
    $$\forall y\in (0,+\infty):\quad p(y)=8.15578 + 0.315667 x.$$
    We sketch the graph of the function $p$ as follows:
    \begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=19, ymax=14, ymin=0, xlabel={Years since 2000}, ylabel={Cents/kWh}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[mark=*, mark options={scale=1, fill=black}](0,8.24);
         \addplot[mark=*, mark options={scale=1, fill=black}](2,8.44);
         \addplot[mark=*, mark options={scale=1, fill=black}](4,8.95);
         \addplot[mark=*, mark options={scale=1, fill=black}](6,10.40);
         \addplot[mark=*, mark options={scale=1, fill=black}](8,11.26);
         \addplot[mark=*, mark options={scale=1, fill=black}](10,11.54);
         \addplot[mark=*, mark options={scale=1, fill=black}](12,11.88);
         \addplot[mark=*, mark options={scale=1, fill=black}](14,12.52);
         \addplot[mark=*, mark options={scale=1, fill=black}](16,12.90);
         \addplot[domain=0:19, samples=2, mark=none, color=blue, ultra thick] {8.15578 + 0.315667*x};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}

\item We should find the values $p(5)$ and $p(17)$ using the graph in (b):
    \begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=19, ymax=14.5, ymin=0, xlabel={Years since 2000}, ylabel={Cents/kWh}, xtick={0,5,10, 15,17},x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, ytick={0,5,9,10,13,14},y label style={anchor=north, above=3mm}]
         \addplot[mark=*, mark options={scale=1, fill=black}](0,8.24);
         \addplot[mark=*, mark options={scale=1, fill=black}](2,8.44);
         \addplot[mark=*, mark options={scale=1, fill=black}](4,8.95);
         \addplot[mark=*, mark options={scale=1, fill=black}](6,10.40);
         \addplot[mark=*, mark options={scale=1, fill=black}](8,11.26);
         \addplot[mark=*, mark options={scale=1, fill=black}](10,11.54);
         \addplot[mark=*, mark options={scale=1, fill=black}](12,11.88);
         \addplot[mark=*, mark options={scale=1, fill=black}](14,12.52);
         \addplot[mark=*, mark options={scale=1, fill=black}](16,12.90);
         \addplot +[dashed,violet,mark=none] coordinates {(17,0) (17,13.522119) (0,13.522119)};
         \addplot +[dashed,green!60!black,mark=none] coordinates {(5,0) (5,9.734115) (0,9.734115)};
         \addplot[mark=*, mark options={scale=1, fill=green!60!black}](5,9.734115);
         \addplot[mark=*, mark options={scale=1, fill=orange}](17,13.522119);
         \addplot[domain=0:19, samples=2, mark=none, color=blue, ultra thick] {8.15578 + 0.315667*x};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    As we can see from the graph, $p(5)\approx 9.7, \ p(17)\approx 13.5.$
\end{enumerate}

Exercise 1.2.32

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Exercise 1.2.32

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