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This question came from an exam on statistics from one of our clients (we are a tutorial center).

According to USA Today (March 17, 1997), women made up 33.7% of the editorial staff at a local TV station in 1990 and 36.2% in 1994. Assume 20 new employees were hired as editorial staff.

Question: Estimate the number that would have been women in each year respectively.

*This is the exact question from the exam. Is there a solution for this or is the given not suffice?

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    There is definitely not enough information. Had they told you how many of the new hires were women, there would be.2012-07-11

1 Answers 1

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If all $20$ new hires were women, then we could set up this system of equations, where $n$ is the number of women in 1990 and $N$ is the total number of employees in 1990: $\left\{ \begin{array}{rcl} \frac{n}{N}&=&0.337\\ \frac{n+20}{N+20}&=&0.362 \end{array} \right.$ which is the same as the system: $\left\{ \begin{array}{rcl} n&=&0.337N\\ n+20&=&0.362N+7.24 \end{array} \right.$ which has solution $N=510.4$, $n\approx172$.

The percentage of women did go up, so at the other extreme, maybe $\lceil0.337\cdot20\rceil=8$ of the new hires were women. In that case, we have the system $\left\{ \begin{array}{rcl} \frac{n}{N}&=&0.337\\ \frac{n+8}{N+20}&=&0.362 \end{array} \right.$ which is the same as the system: $\left\{ \begin{array}{rcl} n&=&0.337N\\ n+8&=&0.362N+7.24 \end{array} \right.$ which has solution $N=30.4$, $n\approx10.24$. As you can see, there are many possibilities depending upon how many of those $20$ are women. Seemingly, there are $13$ possibilities stemming from the $13$ choices (8 through 20) that we have for the number of newly hired women. If you have time, and preferably a spreadsheet program, you could investigate how many of these options are not really compatible with the given decimal precision. For example, if $8$ new hires were women then $N$ is somewhere in the neighborhood of $30$ and $n$ is somewhere in the neighborhood of $10$. But no combination of numerators in $\left\{9, 10, 11, 12\right\}$ and denominators in $\left\{29, 30, 31, 32\right\}$ yields a decimal that rounds to $0.337$.

But realistically, my guess is that the instructor accidentally left out some critical piece of information.

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    My guess is that they meant to say 20 women were hired between 1990 and 1994 and no men.2012-07-11