\authorandnumber{Carlos C. Rodriguez}
\problemdate{Thu Mar 11 15:18:50 EDT 1999}
\textbook{Statistics by Freedman, Pisani and Purves}
\partname{5 Using the Normal Curve in a vertical Strip}
\chaptername{11 The R.M.S. Error for Regression}
\begin{problem}
\score{1}
 A law school finds the following relationship between
LSAT scores and first-year scores.

\[ average LSAT score = 650, SD = 80 \]
\[ average first-year score = 65, SD = 10, r = 0.60 \]

The scatter diagram is football-shaped.

 Consider the students who scored about 750 on the LSAT. The
regression method shows that the average of their first
year scores was over seventy points. The
percentage of these students (i.e. those with LSAT 750)
with first year scores between 70 and 80 is closest to:

\begin{choice}
\gooditem{45%}
\baditem{90%}
\baditem{80%}
\baditem{20%}
\baditem{55%}
\end{choice}

\begin{longanswer}
The average first year score for those with LSAT of 750
is estimated by the regression line. LSAT of 750 is:
\[ (750-650)/80 = 1.25 \]
standard units. Thus, the regression estimate is
\[ r 1.25 = (0.6)*1.25 = 0.75 \]
standard units of first year scores. This corresponds to
\[ 65 + 0.75 * 10 = 72.5 \]
first year score points.
The list of FY scores of those with LSAT 750, follows the
normal curve with an average of 72.5 and an SD given by
the RMS error for the regression of FY on LSAT. We have
\[ RMS = \sqrt{1 - r^{2}} * 10 = 8 \]
The problem is now reduced to estimating the percentage of
entries between 70 and 80 in a list that follows the normal
curve with average 72.5 and SD of 8. This is estimated by
looking up the area under the normal curve from -0.31 (70 in SUs)
and 0.94 (80 in SUs). This is given by:
\[ \frac{A(z=0.31)+A(z=0.94)}{2} = \frac{24\% + 66\%}{2} = 45\% \]

\end{longanswer}
\end{problem}
