\documentclass[12pt]{article}\usepackage{ammaths}\begin{document}



\partie{Homework \#3}

The SAT Reasoning Test is a standardized test for college admissions in the United States. SAT consists of three major sections: Critical Reading, Mathematics, and Writing. Each section receives a score on the scale of 200-800. 

In the table below are shown the SAT Maths scores for 173 students.

\begin{center}
\begin{tabular}{cccccccccccc}
368	&	440	&	447	&	457	&	472	&	485	&	501	&	520	&	537	&	551	&	575	&	650	\\ 
370	&	441	&	448	&	459	&	474	&	486	&	501	&	521	&	538	&	551	&	575	&	654	\\ 
375	&	441	&	449	&	459	&	474	&	487	&	502	&	524	&	540	&	553	&	583	&	660	\\ 
382	&	442	&	449	&	460	&	474	&	489	&	505	&	525	&	540	&	554	&	585	&	670	\\
393	&	442	&	450	&	464	&	476	&	490	&	506	&	526	&	542	&	557	&	585	&	700	\\
403	&	444	&	450	&	466	&	476	&	490	&	506	&	526	&	542	&	557	&	589	&	736	\\
404	&	444	&	450	&	467	&	477	&	492	&	507	&	528	&	543	&	559	&	590	&	740	\\
408	&	444	&	450	&	468	&	479	&	495	&	509	&	529	&	544	&	560	&	590	&	750	\\
410	&	445	&	450	&	468	&	479	&	495	&	510	&	529	&	545	&	560	&	590	&		\\
413	&	445	&	451	&	470	&	480	&	495	&	512	&	530	&	547	&	560	&	592	&		\\
428	&	445	&	452	&	470	&	480	&	496	&	513	&	531	&	547	&	564	&	596	&		\\
430	&	445	&	456	&	470	&	480	&	496	&	513	&	532	&	548	&	570	&	616	&		\\
439	&	446	&	456	&	470	&	480	&	499	&	514	&	533	&	549	&	570	&	621	&		\\
440	&	446	&	457	&	471	&	482	&	499	&	515	&	533	&	549	&	571	&	642	&		\\
440	&	446	&	457	&	472	&	485	&	500	&	515	&	536	&	550	&	574	&	644	&		\\
\end{tabular}
\end{center}

In this exercise, we will discover a method to find an approximate value of the median and quartiles when the data is
grouped in intervals.

\begin{description}
\item[\sf Preliminary question] Explain why the median is the 87th value in this ordered data. Deduce its value.
Compute, in the same way, the exact values of the quartiles of these data.
\end{description}

\begin{enumerate}
\item Group the data in intervals of width 25, starting at 343.5. Show the results in a frequency table (with the
absolute frequencies).
\item Represent the data as a histogram.
\item \begin{enumerate}
	\item Add a new row or column in the table of question 1 where you'll show the cumulative absolute frequencies.
	\item How many students had a score between 593.5 and 618.5 ?
	\item How many students had a score less than 593.5 ?
	\end{enumerate}
\item \begin{enumerate}
	\item In what interval is the median ? This interval will be called $I_\textrm{Med}$ in the following questions
	\item What do you think of the middle of this interval as an approximate value of the median ?
	\item How many students had a score in the interval $I_\textrm{Med}$ ?
	\item We've seen in the preliminary that the median is the 87th value of the data. Deduce from the previous
questions the exact rank of the median in the interval $I_\textrm{Med}$.
	\item Assuming that the values are equally space inside the interval, we can deduce that an approximative value of the median is given by the formula :
	$$Med \approx 493.5+\frac{5}{23}\times25.$$
	Explain every element of this formula and use it to compute the approximate value.
	\item What do you think of this new value as an approximate value of the median ?
	\end{enumerate}
\item Use the same method to find approximate values of the first and third quartiles from the grouped data.
\end{enumerate}





\end{document}
