% Problem: Final Exam Statistics
% Topics: Markov Bounds
% Variants of this problem appeared in:
% SP 2000, PS 10.5, problem 2
% FA 2000, PS 11, problem 5
% FA 2001, Practice Problems 12
\begin{problem} 
One hundred twenty students take the 6.042 final exam.  The mean on the exam is
$90$ and the lowest score was 30.  You have no other information about the
students and the exam, e.g., you should not assume that the final is worth
100 points.

\begin{problemparts}

\problempart State the best possible upper bound on the number of students who
scored at least $180$.  

\solution{Let $R$ be the score of a student chosen at random.  
Apply Markov's Bound to $R-30$:
\[
\Pr(R \geq 180) = \Pr(R-30 \geq 150) 
        \leq \frac{\expect{R-30}}{150} 
        = \frac{60}{150} = \frac{2}{5}.
\]
So at most $\frac{2}{5} \cdot 120 = 48$ students scored 
greater than or equal to $180$.}

\problempart Give an example set of scores which achieve your bound.

\solution{Of the 120 students, 48 score 180 and 72 score 30.  
The mean is $\frac{48 \cdot 180 + 72 \cdot 30}{120} = 90$, as required.}

\problempart If the maximum score on the exam was $100$, 
give the best possible upper bound on the number of students 
who scored \emph{at most} $50$. 

\solution{Apply Markov's Bound to $100-R$: 
\[
\Pr(R \leq 50) = \Pr(100-R \geq 50)
        \leq \frac{\expect{100-R}}{50}
        = \frac{10}{50}
        = \frac{1}{5}.
\]
So at most $\frac{1}{5} \cdot 120 = 24$ students scored $50$ or less.} 

\end{problemparts}
\end{problem}
