# Exam P Practice Problem 82 – Estimating the Median Weight of Bears

Problem 82-A

A wildlife biologist wished to estimate the median weight of bears in Alaska. The weights of the bear population he studied follow a continuous distribution with an unknown median $M$. He captured a sample of 15 bears. Let $Y_5$ be the weight (in pounds) of the fifth smallest bear in the sample of 15 captured bears. Let $Y_{11}$ be the weight (in pounds) of the fifth largest bear in the sample.

Calculate the probability that the median $M$ is between $Y_5$ and $Y_{11}$, i.e., $P(Y_5.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.5000$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.7899$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.8218$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8815$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9232$

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Problem 82-B

The wildlife biologist in Problem 82-A also wishes to estimate $\tau_{75}$, the seventy fifth percentile of the weights of bear population he studied. Let $Y_{10}$ be the weight of the tenth smallest bear in the sample of 15 captured bears. Let $Y_{14}$ be the weight of the second largest bear in the sample of 15 bears.

Calculate the probability that $\tau_{75}$ is between $Y_{10}$ and $Y_{14}$, i.e., $P(Y_{10}<\tau_{75}.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.6155$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.7500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.7715$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8383$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9232$

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$\copyright \ 2014 \ \text{ Dan Ma}$