Statistical investigation and survey design for HSC Maths Standard 2: questionnaire design, ethics and privacy, sampling methods and whether a sample is representative (Year 11 Data analysis, 2024 syllabus)

Sampling fraction

$\frac{60}{600} = \frac{1}{10}$.

Each stratum

Year 10: $240 \times \frac{1}{10} = 24$. Year 11: $200 \times \frac{1}{10} = 20$. Year 12: $160 \times \frac{1}{10} = 16$.

Check

$24 + 20 + 16 = 60$.

Marker's note: one mark for the sampling fraction (or the proportion method) and one for all three correct numbers.

Interval. $\frac{840}{35} = 24$, so every $24$th person is chosen.

Method. Pick a random starting number from $1$ to $24$, then choose every $24$th person after it (for example, starting at $7$: persons $7, 31, 55, \ldots$).

Marker's note: one mark for the interval $24$ and one for a random start followed by every $24$th.

(a) Response rate

$\frac{312}{1200} = 0.26 = 26\%$.

(b) Very satisfied

$25\% \times 312 = 0.25 \times 312 = 78$ households.

(c) Bias

People who feel strongly are more likely to reply, so the $74\%$ who did not reply may hold different views, and the sample may not represent the population.

Marker's note: one mark per part. For (c), link the low response rate to the people who did not reply, not just to the small number.

(a) Percentage. $\frac{3150}{4500} = 0.70 = 70\%$.

(b) Not justified. $70\% \times 60\,000 = 42\,000$ is the reporter's calculation, but the vote is a self-selected sample: only listeners who chose to vote are included, they may feel strongly about the issue, and non-listeners are missing. The sample is probably not representative of all residents, so the result should not be scaled to the whole population.

Marker's note: one mark for $70\%$, one for naming self-selection, one for linking it to a lack of representativeness.

(a) Faults

(i) It is a leading question: the wording "Don't you agree" and "lazy" pushes the respondent towards "Yes". (ii) It asks about two ideas at once (staying up late and laziness) and the choices Yes/No cannot capture degrees of opinion or the facts.

(b) Better question

"On a school night, what time do you usually go to sleep? (Before 9 pm / 9 pm to 10 pm / 10 pm to 11 pm / After 11 pm)". The wording is neutral, has one idea and gives clear, non-overlapping options.

(c) Concern

Collecting names and addresses identifies the respondents, so answers are not anonymous. Students may not answer honestly and their personal information could be misused if it is not stored securely, so only collect the information the investigation needs.

Marker's note: one mark each for two faults, one for a neutral rewritten question with distinct options, one for the privacy concern.

(a) Target population

All residents of the town who travel to work.

(b) Method and bias

This is a convenience sample: people at the station at peak time. It over-represents train commuters and leaves out drivers, cyclists, and people who work from home or at other hours.

(c) Better method

Use a random (or systematic) sample from the electoral roll. For a systematic sample of $150$ from $6000$, the interval is $\frac{6000}{150} = 40$, so choose a random start from $1$ to $40$ and then every $40$th person. Contact each by a method that gives everyone an equal chance to respond.

Marker's note: one mark for the population, one for method and bias, one for a better method, one for the interval $40$.