Exercise 1.2.33

Question

The table shows world average daily oil consumption from 1985 to 2015, measured in thousands of barrels per day. 
    \begin{enumerate}[label=(\alph*)]
        \item Make a scatter plot and decide whether a linear model is appropriate.
        \item Find and graph the regression line.
        \item Use the linear model to estimate the oil consumption in 2002 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=32, ymax=100000, ymin=54000, xlabel={Years since 1985}, ylabel={Thousands of barrels of oil per day}, 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}](0,60083);
         \addplot[mark=*, mark options={scale=1, fill=black}](5,66533);
         \addplot[mark=*, mark options={scale=1, fill=black}](10,70099);
         \addplot[mark=*, mark options={scale=1, fill=black}](15,76784);
         \addplot[mark=*, mark options={scale=1, fill=black}](20,84077);
         \addplot[mark=*, mark options={scale=1, fill=black}](25,87302);
         \addplot[mark=*, mark options={scale=1, fill=black}](30,94071);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    As we can see, the linear model is appropriate for this graph.

\item Using Python, we calculate the regression function
    \begin{minted}{python}
    import numpy as np 
    X = np.array([0, 5, 10, 15, 20, 25, 30]) 
    Y = np.array([60083, 66533, 70099, 76784, 84077, 87302, 94071]) 
 
    result = np.polyfit(X, Y, 1)  # 1 indicates linear regression 
    slope = result[0] 
    intercept = result[1] 
 
    print("Slope (m):", slope) 
    print("Intercept (b):", intercept)
    \end{minted}
    (Alternatively, we can use \href{https://www.wolframalpha.com/input?i=linear+regression+%280%2C60083%29+%285%2C66533%29+%2810%2C70099%29+%2815%2C76784%29+%2820%2C84077%29+%2825%2C87302%29+%2830%2C94071%29}{WolframAlpha}). In other words, we get the regression formula:
    \begin{equation*}
        c:[0, +\infty) 	o [0, +\infty), \quad c(t) = 1124.86 x + 60119.9.
    \end{equation*}
    The regression line then looks as follows: \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=34, ymax=100000, ymin=54000, xlabel={Years since 1985}, ylabel={Thousands of barrels of oil per day}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:34, samples=2, mark=none, color=green!60!black, ultra thick] {1124.86*x + 60119.9};
         \addplot[mark=*, mark options={scale=1, fill=black}](0,60083);
         \addplot[mark=*, mark options={scale=1, fill=black}](5,66533);
         \addplot[mark=*, mark options={scale=1, fill=black}](10,70099);
         \addplot[mark=*, mark options={scale=1, fill=black}](15,76784);
         \addplot[mark=*, mark options={scale=1, fill=black}](20,84077);
         \addplot[mark=*, mark options={scale=1, fill=black}](25,87302);
         \addplot[mark=*, mark options={scale=1, fill=black}](30,94071);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}

\item We should find the values $c(17)$ and $c(32)$ using the graph in (b):\begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=34, ymax=100000, ymin=54000, xlabel={Years since 1985}, ylabel={Thousands of barrels of oil per day}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:34, samples=2, mark=none, color=green!60!black, ultra thick] {1124.86*x + 60119.9};
         \addplot[mark=*, mark options={scale=1, fill=black}](0,60083);
         \addplot[mark=*, mark options={scale=1, fill=black}](5,66533);
         \addplot[mark=*, mark options={scale=1, fill=black}](10,70099);
         \addplot[mark=*, mark options={scale=1, fill=black}](15,76784);
         \addplot[mark=*, mark options={scale=1, fill=black}](20,84077);
         \addplot[mark=*, mark options={scale=1, fill=black}](25,87302);
         \addplot[mark=*, mark options={scale=1, fill=black}](30,94071);
         \addplot +[dashed,violet,mark=none] coordinates {(17,0) (17,79242.52) (0,79242.52)};
         \addplot +[dashed,green!60!black,mark=none] coordinates {(32,0) (32,96115.42) (0,96115.42)};
         \addplot[mark=*, mark options={scale=1, fill=yellow!70!black}](32,96115.42);
         \addplot[mark=*, mark options={scale=1, fill=orange}](17,79242.52);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    As we can see from the graph, $c(17)\approx 79,000$ thousands of barrels per day and $c(32)\approx 96,000$ thousands of barrels per day.
\end{enumerate}

Exercise 1.2.33

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Exercise 1.2.33

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