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\title{\sc Math 20 -- Homework 6}
\author{due Wednesday, August 9}
\date{}
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\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. Staple your work together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive full credit.\\
\emph{You may use Wolfram Alpha to compute any necessary sums or integrals. If you have trouble with this, let me know.}\\
\textbf{If you're using facts about distributions to answer the questions, be very clear about which distribution you're using to model that problem and why that distribution is appropriate.}
\begin{enumerate}
\item A school wishes to accept 2000 students for their freshman class, and they expect 20,000 applications. In order to make their admissions decisions very easy, the only criterion they will use is SAT score. So, their goal is to accept a student if and only if their SAT score is in the top 10\%. However, because their computer system is so old, the applications only come in one at a time, and they must decide whether to accept or reject before moving on to the next application. Assuming that SAT scores are normally distributed with a mean of $1000$ and a standard deviation of $200$, how should they set the score threshold to end up with as close to 2000 students as possible? Give your answer first symbolically (in terms of a pdf, cdf, etc), then use a normal distribution table\footnote{For example: \href{http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf}{http://www.stat.ufl.edu/$\sim$athienit/Tables/Ztable.pdf}. The numbers in the leftmost column represent the number of standard deviations from the mean up to 1 decimal place, and the numbers along to top row then refine to the second decimal place. Please ask me if you have questions about this.} to provide a numerical answer.
\item The density function $f(x)$ of a continuous random variable $X$ is given by
\[
f(x) = \left\{ \begin{array}{ll} A + Bx^2,&\text{ if $x \in [0,2]$}\\0,&\text{ otherwise}\end{array}\right..
\]
If $\E[X] = 1/2$, find $A$ and $B$.
\item Since 1851, exactly $116$ hurricanes have hit Florida (this includes the years $1851$ and $2016$, but not $2017$---only direct hits by \emph{hurricanes} are counted, not tropical storms). In 2005, Florida was hit by four hurricanes: Cindy, Dennis, Katrina, and Wilma. If the probability of hurricane strikes has remained the same since $1851$, what is the probability of Florida being struck by four or more hurricanes in the same year?
\item In the solutions manual to a Calculus textbook, there is about one faulty solution per fifty questions. In a book with ten chapters, each with one hundred questions, what is the probability that there are at least 15 faulty solutions in the whole book? Give your answer two ways: first with a binomial distribution, then with a Poisson approximation. Use Wolfram Alpha or some other tools to find both answers numerically, and compare them.
\end{enumerate}
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