Question 14
A factory produces cans of juice. Each can has a labelled volume of 500 mL.
The factory manager conducted a random sample of 100 cans to assess whether the labelled volume was being met satisfactorily in production.
The mean volume of the sample was 498.9 mL with a sample standard deviation of 4.0 mL.
Based on this sample and using a $z$-value of 2, determine an approximate confidence interval for the population mean volume.
[3 marks]Industry regulation requires that the mean volume must meet or exceed the labelled volume.
Use your result from Question 14a) to state whether the current production meets the regulation. Justify your decision using mathematical reasoning.
[2 marks]The whole interval sits below the label
Watch the interval grow out from the sample mean, then see where 500 mL falls.
The interval gives the plausible values for the true mean of all the cans. None of them reaches 500 mL.
Change the sample and see whether the decision changes
Keep $z=2$ and change the sample mean, the sample standard deviation or the sample size. The decision depends on where 500 mL falls against the interval.
Work through the solution one mark at a time
Try each step yourself before you reveal it.
Write down what you have been given, with the right symbol for each.
$$n=100,\quad \bar{x}=498.9,\quad s=4.0,\quad z=2$$
You only have the sample standard deviation, so $s$ stands in for $\sigma$. That is fine for a sample this large. This mark can be implied by your substitution.
Marker: correctly translates given information into relevant variables
Substitute into the confidence interval from the formula book.
$$\left(\bar{x}-z\frac{s}{\sqrt{n}},\ \bar{x}+z\frac{s}{\sqrt{n}}\right)=\left(498.9-2\times\frac{4}{\sqrt{100}},\ 498.9+2\times\frac{4}{\sqrt{100}}\right)$$
Working out the margin of error first is also accepted: $E=2\times\frac{4}{10}=0.8$.
Marker: substitutes into confidence interval formula
Since $\sqrt{100}=10$, the margin of error is $0.8$.
$$(498.9-0.8,\ 498.9+0.8)$$
Marker: determines required confidence interval
Compare the labelled volume with the interval. Say exactly where it falls.
The labelled volume of 500 mL lies above the upper value of the confidence interval, because $500\gt499.7$.
Other wordings are accepted, such as “the interval lies entirely below the labelled volume” or “we are about 95% confident the true mean is below 500 mL”.
Marker: discusses the labelled juice volume in relation to the confidence interval values
Now state the decision the question asked for.
You can also say the industry regulation is not being met, or that a mean of 500 mL is questionable. The decision has to follow from your reasoning in the step before.
Marker: comments on the reasonableness of the current production
Each phrase of the question gave you something
This is the standard confidence interval routine, with the $z$-value given and numbers chosen so you can work without a calculator. Part b) is the usual judgement: find the claimed value, then see whether it is inside the interval.
If I had to guess where you might lose marks, it would be part b). Saying “498.9 is less than 500” is not enough. A sample mean can come out below 500 mL even when the true mean is exactly 500 mL. It is the whole interval sitting below 500 mL that lets you decide.
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 2025 QCAA Specialist Mathematics 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.
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