2025 Paper 2, Q18
The results of an employee satisfaction survey of 500 employees at a large company are presented to board members. The results include a 95% confidence interval for the proportion of satisfied employees. The lower end of the confidence interval is 0.648.
A board member would like to use the survey results to make the claim that the proportion of the satisfied employees in the entire company is larger than 75%.
Evaluate the reasonableness of the claim.
Rebuild the interval from one end of it
The board was handed a single number. Because the interval is symmetric about the sample proportion, that one number is enough to rebuild the whole interval, and then to test the claim.
The formula, run backwards
Normally you are given $\hat p$ and $n$ and asked for the interval, using $\hat p\pm z\sqrt{\frac{\hat p(1-\hat p)}{n}}$ with $z=1.960$.
$0.648=\hat p-1.960\sqrt{\frac{\hat p(1-\hat p)}{500}}$. There is one equation and one unknown. But $\hat p$ appears twice, so it is a solver job, not an algebra job.
With $\hat p=0.6886$ the same margin runs upwards to 0.729. Only then can the 75% claim be judged.
The interval is symmetric about $\hat p$, so once the centre is known the far end is one subtraction away. No need to rebuild the margin of error.
How high would the survey have to come in?
Drag the lower end the board was given. The recovered sample proportion and the upper end both follow. And 75% only makes it inside once the lower end reaches about 0.670.
What does the interval let you say?
QCAA marking guide · 5 marks
$$z=-1.960\ \text{and}\ 1.960$$
Marker: correctly determines the $z$-score for a 95% confidence interval. Any number of decimal places, rounded or truncated, so 1.96 is fine.
$$0.648=\hat p-1.960\sqrt{\frac{\hat p(1-\hat p)}{500}}$$
Marker: determines an equation involving the lower cut-off of the CI formula. FT marks allowed for errors in prior working.
Solving with a GDC gives $\hat p=0.6886$, which is 344 of the 500 employees.
Marker: determines the sample proportion for the survey results. The unknown appears inside and outside the square root, so a numerical solver is the intended tool.
$$0.6886+1.960\sqrt{\frac{0.6886(1-0.6886)}{500}}=0.729$$
Marker: determines the upper end of the 95% confidence interval. The 95% CI is therefore $(0.648,\ 0.729)$.
The survey suggests we can be 95% confident the population proportion is between 64.8% and 72.9%.
75% is outside the interval, so the claim is not reasonable
Marker: determines if the claim is reasonable. This mark can only be awarded if prior working supports the claim. The sentence has to name the interval it is comparing 75% against.
Not reasonable, and by how much
75% misses the top of the interval by 2.1 percentage points. For the claim to survive, the survey would have needed about 355 satisfied employees instead of 344.
Doubling 0.648 to find the centre is wrong. The centre is $\hat p$, not twice the lower end. Once you know $\hat p$, though, upper $=2\hat p-$ lower is exact, and it saves you working out the margin again.
The claim is not impossible, but it is unsupported. The interval says the data are consistent with 64.8% to 72.9%, and saying more than 75% goes beyond what this survey can show.
Almost every confidence interval question you practise runs one way. You are given $\hat p$ and $n$, and you build the interval. This one gives you an end of the interval and expects you to run it backwards, which makes it an inversion. That is awkward, because $\hat p$ is both outside and inside the square root, so you cannot rearrange it. You have to set up the equation and let your calculator solve it. The second hidden step is that the question never asks for the interval at all. It asks whether a claim is reasonable, and the only way to answer is to work out the upper end that nobody mentioned. The 2025 subject report said students struggled to use a confidence interval to justify a claim, so I would practise this one until it is automatic.
Question wording and marking-guide steps are from the 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.