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\partie{Homework \#9}
The height (in cm) and weight (in kg) of 21 young girls are given in the table below.
\begin{center}
\begin{tabular}{| p{2cm} | *{11}{| c} |}\hline
Height $x$ & 166 & 160 & 163 & 165 & 155 & 169 & 171 & 160 & 162 & 165 & 153 \\ \hline
Weight $y$ & 59 & 57 & 56 & 58 & 54 & 60 & 61 & 53 & 54 & 56 & 51\\ \hline
\hline
Height $x$ & 158 & 176 & 168 & 150 & 167 & 164 & 166 & 161 & 158 & 170 &\\ \hline
Weight $y$ & 56 & 62 & 57 & 49 & 58 & 57 & 56 & 56 & 55 & 64 &\\ \hline
\end{tabular}
\end{center}
\begin{enumerate}
\item Show these data as a scatter plot with height on the $x$-axis and weight on the $y$-axis. The scale will be 2cm
for 5 units on both axes and the origin point will be $(145,40)$.
\item Compute independently the median height $h_0$ and the median weight $w_0$ of the 21 girls. Then, place the point
$G$ with coordinates $(h_0;w_0)$.
\item The points on the scatter plot seem to be roughly collinear. In this question, we will find the formula of a
linear function whose graph is a good approximation of the scatter plot. This is the \emph{median-median line}, also
referred to as the \emph{median fit line}.
\begin{enumerate}
\item To fit a median-median line to the points, divide the points into three groups. Do this by taking the set
of one-third of the points consisting of those with the smallest $x$-values, a middle group and a set of one-third of
the points with the largest $x$-values.
\item Consider each group of data separately and order the values of both variables. Ignore the data pairings
at this point.
\item Now create a summary point for each group of the data by using the median $x$-value and the median
$y$-value, and combining them to create an ordered pair. We have three summary points: $G_L$ for the leftmost data,
$G_M$ for the middle group, and $G_R$ for the right hand group. These summary points may, or may not, be actual data
points.
\item Now use the two outer summary points to determine the slope-intercept equation of the line $(G_LG_R)$.
\item Construct the line parallel to this line that is one-third of the way to the middle summary point. This
is the \emph{median-median line} $\mathscr{M}$. To do this:
\begin{itemize}
\itemb Find the $y$-coordinate of the point $P$ on the line with the same $x$-coordinate as the middle
summary point $G_M$.
\itemb Find the vertical distance between the middle summary point and the line by subtracting
$y$-values.
\itemb Find the coordinates of the point $Q$, one third of the way from the line to the middle summary
point.
\itemb Find the slope-intercept equation of the line parallel to $(G_LG_R)$ passing through $Q$.
\end{itemize}
\end{enumerate}
\item Is the point $G$ on the line $\mathscr{M}$ ? Answer the question graphically and with a computation.
\item In this question, we use the median-median line to estimate some values.
\begin{enumerate}
\item What should be the weight of a girl whose height is 1.80m ?
\item What should be the height of a girl whose weight is 55 kg ?
\end{enumerate}
\item The Lorenz law establishes a relation between the weight $W$ and the height $H$ for women :
$$\displaystyle{W=H-100-\frac{H-150}{2}}$$
\begin{enumerate}
\item Draw the line representing this function.
\item What do you notice about this line compared to $\mathscr{M}$ ?
\end{enumerate}
\end{enumerate}
\end{document}