2020 Paper 2, Q20
Assuming the approximate normality of sample proportions ($\hat p_1$ and $\hat p_2$) and based on two independent samples, the approximate confidence interval for the difference of two proportions is given by
$$\left(\hat p_1-\hat p_2\right)\pm z\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}$$
If the approximate confidence interval for the difference between two proportions does not contain 0, this provides evidence that the two proportions are not equal.
The data in the table shows the observed frequencies of two drink preferences for independent samples of people who live in Town A and Town B.
Using the approximate 99% confidence interval for the difference of two proportions, determine if there is evidence to conclude that drink preference is associated with the town where the person lives.
The proportions are seven points apart, but the interval is wider than that
Town B does prefer tea more often than Town A, at least in the samples. The question is whether two samples this size can tell that apart from chance.
A formula you have never used, and a table of data
$\hat p_1=\frac{111}{216}=0.5139$ and $\hat p_2=\frac{150}{257}=0.5837$. These are the proportions who chose tea in each town. The totals are the sample sizes, not the grand total.
The standard error is $\sqrt{\frac{0.5139(0.4861)}{216}+\frac{0.5837(0.4163)}{257}}=0.0458$, and $z=2.576$ for 99%.
The stem tells you how to conclude. If the interval contains 0, there is no evidence the proportions are different. Quote that rule and apply it.
Choosing coffee instead of tea mirrors the interval to $(-0.048,\ 0.188)$. The width and the conclusion are both the same. QCAA marks both methods.
How confident would you have to be to see a difference?
Lower the confidence level and the interval shrinks. It stops containing zero only below about 87%, which is nowhere near the 99% the question demands.
Which two proportions go into the formula?
QCAA marking guide · 5 marks
Let $p_1$ be the proportion of Town A who prefer tea and $p_2$ the proportion of Town B who prefer tea.
$$\hat p_1=\frac{111}{216},\qquad \hat p_2=\frac{150}{257}$$
Marker: correctly determines the sample proportions. Defining the variables is part of the communication mark too.
$$\left(\frac{111}{216}-\frac{150}{257}\right)\pm 2.576\sqrt{\frac{\frac{111}{216}\left(1-\frac{111}{216}\right)}{216}+\frac{\frac{150}{257}\left(1-\frac{150}{257}\right)}{257}}$$
Marker: establishes the confidence interval for the difference of two proportions. FT marks allowed for incorrect sample proportions.
$$=(-0.188,\ 0.048)$$
Marker: determines the 99% confidence interval. Equivalent decimal values accepted. Keep the unrounded proportions in the calculator, because rounding them to 0.51 and 0.58 first shifts the ends.
This interval contains zero, so there is no evidence in the data that the two proportions are different.
Preference for tea does not depend on where the person lives
Marker: interprets the 99% confidence interval to determine equality of proportions. The stem hands you the decision rule. The mark is for applying it and saying what it means about towns and drinks.
Define $p_1$ and $p_2$ in words, distinguish $p$ from $\hat p$, show the substitution before the answer, and finish with a sentence about drink preference rather than about an interval.
Marker: shows logical organisation communicating key steps, proportion and sample-proportion notation, defining variables, substitution into the formula, setting up equations.
No evidence is not the same as no difference
The interval clears zero by 0.048 at the top. At 95% confidence it would be $(-0.160,\ 0.020)$, which still contains zero. Only below about 87% does the gap look real.
Coffee gives $(-0.048,\ 0.188)$: the mirror image, because a town’s coffee proportion is one minus its tea proportion. The conclusion cannot depend on which column you pick.
“No evidence of an association” is the claim. The towns may well be different. Two samples of about 230 people are just not big enough to pick up a seven-point gap at 99% confidence.
The formula is printed for you, the decision rule is printed for you, and the data is in a table, so the whole difficulty is choosing what goes where. The table gives you four numbers and two totals. You need exactly two proportions, both for the same drink, each divided by its own town total. I have seen students compare tea against coffee inside one town, or divide by 473. Neither of those is what the formula was built for. Your final sentence then has to turn the interval back into the language of the question. Do not just write “the interval contains zero”. Write “drink preference is not associated with the town”.
Every official complex unfamiliar question, 2020 to 2025
That is all 26 questions and 133 marks. The habit that runs through every one of them is finding the sentence that is secretly an equation. Go back to the index and try them in a random order, because they are much harder to recognise when they are mixed up.
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Question wording and marking-guide steps are from the 2020 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.