Question 10
A survey plans to draw conclusions based on a random sample of 1% of Queensland’s adult population.
To be regarded as a random sample, every
Not quite. Taking every 100th person on a list is systematic sampling. As written, the list order decides who is picked, not chance: the 100th, 200th, 300th and so on, every time. What makes a sample random is that every adult has an equal chance. In 2022, 7.0% of Queensland students chose (A).
Not quite. This is a self-selected sample. Only people who choose to take part can be in it, and the adults who never volunteer have no chance at all. In 2022, 5.0% of Queensland students chose (B).
Close. That describes a representative sample, which is what a random sample gives you on average. But you can match every subgroup without any chance involved, and one random sample won’t always match exactly. It isn’t what makes a sample random. In 2022, 6.5% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2022, 81.1% of Queensland students chose (D).
A made-up population of 1000 adults, so 1% is 10 people. The ages and who volunteers are invented, not real data.
Every 100th name gives the same 10 adults every time, so the other 990 have no chance. If the start were picked at random from the first 100 names, each adult would get a 1% chance. It is that equal chance, not the counting, that makes a sample random.
Only the 60 volunteers can ever be chosen. The other 940 adults have no chance, and in this made-up population the volunteers lean older.
The age groups match the population exactly, but the same 10 adults are chosen every time, the first names on the list in each group. Matching the subgroups doesn’t make a sample random.
Every adult has the same 1% chance. Take a new sample: the 10 change each time, and the age groups only match the population on average.
Start with the definition. In a random sample, chance does the choosing, and it treats every member of the population the same way.
So the test for each option is: does every adult in Queensland have the same chance of ending up in the sample?
For each option, ask who could be chosen.
(A) The list order decides: the 100th, 200th and so on. Nobody else can be picked. (B) Only people who volunteer can be picked. (C) describes how the finished sample looks, not how people are chosen.
Try each method in the figure and press “Take a new sample” a few times.
Only (D) states the defining property: every adult has an equal chance of being selected.
A random sample tends to represent each subgroup in about the right proportion, so (C) is a result you hope for, not the definition. One random sample can still miss.
A sample is random when every member of the population has an equal chance of being selected.
(A) as written lets the list order decide, not chance. (B) is self-selected. (C) describes a representative sample, which you can get without any chance at all.
How Queensland students answered in 2022
From the QCAA Mathematical Methods subject report for 2022. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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