Exercise 1.2.31

Question

Anthropologists use a linear model that relates human femur (thighbone) length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. Here we find the model by analyzing the data on femur length and height for the eight males given in the table.
    \begin{enumerate}[label=(\alph*)]
        \item Make a scatter plot of the data.
        \item Find and graph the regression line that models the data.
        \item An anthropologist finds a human femur of length 53 cm. How tall was the person?
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item 
    \begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=30, xmax=54, ymax=190, ymin=0, xlabel={Femur length (cm)}, ylabel={Height (cm)}, 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}](50.1,178.5);
         \addplot[mark=*, mark options={scale=1, fill=black}](48.3,173.6);
         \addplot[mark=*, mark options={scale=1, fill=black}](45.2,164.8);
         \addplot[mark=*, mark options={scale=1, fill=black}](44.7,163.7);
         \addplot[mark=*, mark options={scale=1, fill=black}](44.5,168.3);
         \addplot[mark=*, mark options={scale=1, fill=black}](42.7,165);
         \addplot[mark=*, mark options={scale=1, fill=black}](39.5,155.4);
         \addplot[mark=*, mark options={scale=1, fill=black}](38, 155.8);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    \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%7B50.1%2C178.5%7D%2C%7B48.3%2C173.6%7D%2C%7B45.2%2C164.8%7D%2C%7B44.7%2C163.7%7D%2C+%7B44.5%2C168.3%7D%2C+%7B42.7%2C165%7D%2C+%7B39.5%2C155.4%7D%2C+%7B38%2C+155.8%7D%7D%22}{WolframAlpha}, we get the following regression line function:
    $$\forall l\in (0,+\infty):\quad h(l)=1.88074 x + 82.6497.$$
    We sketch the graph of the function $h$ as follows:
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=32, xmax=54, ymax=190, ymin=0, xlabel={Femur length (cm)}, ylabel={Height (cm)}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=32:54, samples=2, mark=none, color=blue, ultra thick] {1.88074*x + 82.6497};
         \addplot[mark=*, mark options={scale=1, fill=black}](50.1,178.5);
         \addplot[mark=*, mark options={scale=1, fill=black}](48.3,173.6);
         \addplot[mark=*, mark options={scale=1, fill=black}](45.2,164.8);
         \addplot[mark=*, mark options={scale=1, fill=black}](44.7,163.7);
         \addplot[mark=*, mark options={scale=1, fill=black}](44.5,168.3);
         \addplot[mark=*, mark options={scale=1, fill=black}](42.7,165);
         \addplot[mark=*, mark options={scale=1, fill=black}](39.5,155.4);
         \addplot[mark=*, mark options={scale=1, fill=black}](38, 155.8);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
    \item For $l=53$ cm, we have:
    $$h(53)=1.88074\cdot 53 + 82.6497\approx 182\mathrm{~cm.}$$
\end{enumerate}

Exercise 1.2.31

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Exercise 1.2.31

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