Exercise 1.2.30

Question

When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The table lists the results of several experiments by different scientists.
    \begin{enumerate}[label=(\alph*)]
        \item Find the regression line for the data.
        \item Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data?
        \item What does the y-intercept of the regression line represent?
    \end{enumerate}

\begin{table}[ht]
\centering \footnotesize
\begin{tabular}{|c|c|}
	oprule
Asbestos exposure (fibers/mL) & Percent of mice that develop lung tumors \ \hline
\midrule
50   & 2 \ \hline
400  & 6 \ \hline
500  & 5 \ \hline
900  & 10 \ \hline
1100 & 26 \ \hline
1600 & 42 \ \hline
1800 & 37 \ \hline
2000 & 38 \ \hline
300  & 50 \ \hline
\bottomrule
\end{tabular}
\end{table}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \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%2C2%7D%2C%7B400%2C6%7D%2C%7B500%2C5%7D%2C%7B900%2C10%7D%2C%7B1100%2C26%7D%2C+%7B1600%2C42%7D%2C+%7B1800%2C37%7D%2C+%7B2000%2C38%7D%2C+%7B3000%2C50%7D%7D%22}{WolframAlpha}, we get the function:
    $$\forall x\in \mathbb{R}_+:\quad f(x)=0.304802 + 0.0187891\cdot x.$$
    \item The regression line then looks as follows: \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=3500, ymax=55, ymin=0, xlabel={Asbestos exposure (fibers/mL)}, ylabel={Percent of mice that develop lung tumors}, 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}](50,2);
         \addplot[domain=0:65000, samples=2, mark=none, color=blue, ultra thick] {0.304802 + 0.0187891*x};
         \addplot[mark=*, mark options={scale=1, fill=black}](400,6);
         \addplot[mark=*, mark options={scale=1, fill=black}](500,5);
         \addplot[mark=*, mark options={scale=1, fill=black}](900,10);
         \addplot[mark=*, mark options={scale=1, fill=black}](1100,26);
         \addplot[mark=*, mark options={scale=1, fill=black}](1600,42);
         \addplot[mark=*, mark options={scale=1, fill=black}](1800,37);
         \addplot[mark=*, mark options={scale=1, fill=black}](2000,38);
         \addplot[mark=*, mark options={scale=1, fill=black}](3000,50);
    \end{axis}    
    \end{tikzpicture}
    \end{figure}

\item The $y$-intercept represents the percent of mice that develop lung tumors when asbestos exposure is equal to zero. Then the percentage of mice that develop lung tumors is equal to $f(0)\approx 0.3\%.$
\end{enumerate}

Exercise 1.2.30

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Asbestos exposure (fibers/mL)	Percent of mice that develop lung tumors

50	2

400	6

500	5

900	10

1100	26

1600	42

1800	37

2000	38

300	50

Exercise 1.2.30

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