Question 16
At council meetings in a particular town, new proposals are only discussed if more than 80% of the community are in favour of the proposal.
To discover community opinion on a new bus route proposal, the council conducted several surveys, each with a sample size of 120. The distribution of the sample proportions from the surveys had a standard deviation of 0.04.
Make a justified decision as to whether the new bus route proposal would be discussed at a council meeting.
[4 marks]One spread, two possible centres
The question only gives you the spread of the sample proportions. Watch how that spread points back to the proportion of the whole town in favour.
The spread of the sample proportions depends on $p(1-p)$, so two different town-wide proportions give a spread of 0.04. Neither one is above 0.8.
Find every proportion that gives a spread of 0.04
The black arch is $\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{120}}$ for every possible proportion $p$ in favour. Slide $p$ until the standard deviation is exactly 0.04. There is more than one place where that happens.
This proportion gives a standard deviation of exactly 0.04. About 26% of the town in favour fits the surveys, and it is well short of 80%.
This proportion gives a standard deviation of exactly 0.04 as well. About 74% in favour fits the surveys. It is much closer to 80%, but it is still below it.
To the right of $p=0.8$ is where the proposal would be discussed. From $p=0.8$ upwards the arch is never higher than about 0.0365, so it never reaches 0.04. No proportion of 80% or more fits the surveys.
This proportion gives a standard deviation bigger than 0.04, so it does not match the surveys.
This proportion gives a standard deviation smaller than 0.04, so it does not match the surveys.
Work through the solution one mark at a time
Try each step yourself before you reveal it. One mark sets up the equation, two marks are for the two values of $p$, and the last mark is the decision.
The standard deviation of the sample proportions is in the formula book. You know $\sigma$ and $n$, so the unknown is $p$, the proportion of the whole community in favour. Substitute, then square both sides to clear the square root.
$$\begin{aligned}\sigma&=\sqrt{\frac{p(1-p)}{n}}\\0.04&=\sqrt{\frac{p(1-p)}{120}}\\0.0016&=\frac{p(1-p)}{120}\\0.192&=p(1-p)\\0.192&=p-p^2\end{aligned}$$
$$p^2-p+0.192=0$$
For this mark the formula needs all three pieces: the given standard deviation, the given sample size and the population proportion $p$. An equivalent statement, such as $p(1-p)=0.192$ or $p^2-p+0.192=0$, is also accepted.
Marker: correctly substitutes the given information into the standard deviation formula for sample proportion
Use your calculator’s equation solver on the quadratic. You could also graph $y=0.04$ and $y=\sqrt{\frac{x(1-x)}{120}}$ and find where they intersect.
$$\begin{aligned}p&\approx0.2592\\&\approx26\%\end{aligned}$$
Appropriate rounding of the value or the percentage is accepted, such as 0.26. If you made a slip in step 1, you can still earn this mark by follow-through.
Marker: determines a possible value for the population proportion
It is a quadratic, so look for a second solution. Both roots are between 0 and 1, so both are possible proportions, and the information in the question cannot tell them apart.
$$\begin{aligned}p&\approx0.7408\\&\approx74\%\end{aligned}$$
The two values add to 1, because $p(1-p)$ is unchanged when you swap $p$ and $1-p$. That is a quick check on your calculator’s answers. Appropriate rounding is accepted, such as 0.74.
Marker: determines a second possible population proportion value
Now compare with the council’s rule. You do not know which value is the true proportion, so the sample response checks both of them. Each one is less than 80%.
The justification must compare at least one of your calculated proportions with the 80% requirement. Another valid justification is also accepted, but a bare “no” is not a justified decision. Comparing both values, as above, covers every case.
Marker: makes a justified decision regarding the proposal
Each phrase of the question gave you something
A simple familiar question gives you $p$ and $n$ and asks for the standard deviation. Here you have to run the formula backwards, and the unknown sits inside $p(1-p)$, so you get a quadratic with two answers. Then you carry both answers into a decision about the council’s rule. It is not complex unfamiliar, because the question names the distribution of the sample proportions and its standard deviation, and that points straight at the formula in the formula book.
If I had to guess where you might lose marks, it would be stopping at one value of $p$. Your solver may only show the first root, and the marking guide gives a separate mark for the second. For the decision, a comparison with 80% has to appear in writing. The guide needs at least one calculated value compared, and the sample response compares both, so do the same. Watch the algebra at the start too. Square 0.04 to get 0.0016 before you multiply by 120.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2024 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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