Sampling & statistical investigations

Year 8 core - answers

Fluency

Census (everyone in the population).
Systematic.
Sample. Reason: census of every $13$ -year-old in Australia is impractical and costly.
Selection bias toward shoppers; non-shoppers are under-represented.
Population: all Year 8 students at our school. Sample: simple random sample of at least $30$ from the Year 8 roll.

Fluency

Stratified.
Question bias (loaded or leading wording).
Self-selection: only motivated listeners call in, and they may hold strong or particular views.
Year 7: $30$ . Year 8: $28$ . Year 9: $26$ . Method: $10\%$ of each.

Reasoning

No. $20$ out of $500$ is $4\%$ ; random variation alone can shift results by $\pm 10$ percentage points. A bigger sample is needed for confidence.
Each sample contains different individuals; small differences in who’s in the sample translate to small differences in the statistics.
“How would you rate the coach’s performance this season on a scale from $1$ (poor) to $5$ (excellent)?” - avoids leading wording.
Are the $500$ readers a random sample of all readers, or self-selected? Is the poll reflective of the newspaper’s audience only? What’s the margin of error?

Problem solving

Population: all Year 8 students. Method: stratified random sample across classes. Sample size: $\geq 30$ . Ask: “How many hours did you sleep last school night?”. Display: dot plot or column graph. Report mean, median, range, and acknowledge uncertainty.
Mean cycling percentage $= \dfrac{18 + 22 + 16 + 21}{4 \times 50} \times 100 = \dfrac{77}{200} \times 100 = 38.5\%$ . Variability: range $16$ to $22$ per sample of $50$ , i.e. $\pm 6$ percentage points - modest.
$800$ is typically large enough for industrial QC at $1\%$ sampling. Factors: is the sample random across shifts and machines? Is $1\%$ enough given the tolerance required?
Station B (more days = more data to average, less random day-to-day noise) - provided both stations are in the same area and used comparable instruments.