Understand · explore · practise
Original cumulative questions on sampling designs, bias, data types, grouped measurements and weather records, with separate hints and worked solutions.
Before you startThe preceding data collection lessons.
01 / Choose a method deliberately
Try each question before opening its hint or solution. For a design question, state the population and frame, explain precisely how units enter the sample and identify a limitation linked to that context.
The scenario selector is a warm-up. A method can be appropriate in one situation and impractical in another.
A complete numbered list; choose 20 with every possible 20-person subset equally likely.
Simple random sampling: The defining condition concerns complete subsets, not just each individual’s inclusion probability.
02 / Population and frame
A frame that misses part of the target population can distort conclusions even when selection from the frame is random.
A library wants the proportion of all registered members who favour Sunday opening. It surveys 100 members using the library on Tuesday morning. Identify the target population, the people accessible to the survey and one specific risk.
Distinguish membership from attendance at that time.
Target: all registered members. Accessible group: Tuesday-morning visitors. Members who work or study then may be excluded and may value Sunday opening differently.
A simple random sample of 200 members is emailed; only 80 reply. Does the initial random selection guarantee representative replies?
The final responding group is selected in two stages.
No. Willingness or ability to reply may be related to opinions. Report 80 responses from 200 selected (40%) and consider non-response bias.
03 / Work through sampling designs
Numbers and randomisation are part of the answer. Do not stop after naming a method.
A list has 840 numbered records. Select 40 systematically. Give the interval, permissible random starts and first three labels if the start is 8.
Divide population size by sample size.
Interval 840/40 = 21. Choose the start fairly from 1–21. With start 8, labels begin 8, 29, 50 and end at 8 + 39×21 = 827.
Every 21st record is a supervisor, and a systematic sample uses interval 21. Explain the risk even with a fair random start.
The sample repeatedly hits the same position in each block.
One start may select only supervisors and others none. This can make individual sample estimates very unrepresentative and increase sampling variability. A fair random start does not necessarily create bias in the mean over all possible starts.
Three groups contain 180, 120 and 60 records. Allocate a proportional stratified sample of 48, then state how to choose the records.
Use 48 times each group size divided by 360.
Allocate 24, 16 and 8. Randomly select that many distinct records within each group from an appropriate frame.
A researcher uses those same group targets but asks the first willing people encountered. What changed?
Matching group totals is not sufficient.
The procedure is quota sampling rather than stratified random sampling. Selection within groups depends on availability and willingness.
04 / Read the meaning of a value
Separate the underlying variable from its recording precision and coding.
Classify postcode, number of enquiries, exact duration and satisfaction coded 1 = poor, 2 = fair, 3 = good.
Ask whether arithmetic on the values represents quantities.
Postcode: qualitative. Number of enquiries: discrete quantitative. Exact duration: continuous quantitative. Satisfaction: ordinal qualitative; its numerical codes do not establish equal gaps.
Mass is recorded to the nearest 0.1 kg. Give the true-value interval for a recorded mass of 63.2 kg, using half-up rounding for positive measurements.
Use half of 0.1 on each side.
63.15 ≤ mass < 63.25 kg. The recorded grid is discrete, but underlying mass is continuous.
05 / Boundaries and representative values
For rounded integer observations, use the precision to infer boundaries. For explicitly stated intervals, use those actual endpoints.
Lengths are recorded to the nearest centimetre. A class contains recorded values 40–49 inclusive. Find its true boundaries, width and midpoint.
The displayed labels are rounded measurements.
39.5 ≤ length < 49.5 cm; width 10 cm; midpoint 44.5 cm. A midpoint is a representative class value, not a known class mean.
A class is 40 ≤ x < 50. Should 0.5 be added to either endpoint? Give its midpoint.
These are already interval boundaries.
No. Use 40 and 50 as stated. Midpoint = (40 + 50)/2 = 45.
06 / Interpret weather records
The following rainfall values are invented. They are not observations copied from the large data set.
Ten selected days contain six numerical rainfall values totalling 9.6 mm, three trace entries and one unavailable entry. Approximate traces by zero. Find the observed-day mean.
Keep trace days in the denominator.
Nine observed days contribute: mean 9.6/9 = 16/15 ≈ 1.067 mm/day. The unavailable day is excluded, so this is not the known mean across all ten days.
For that nine-day observed sample, bound the amount by which treating traces as zero can underestimate the exact mean, given each trace is below 0.05 mm.
There are three trace amounts.
Less than 3×0.05/9 = 1/60 ≈ 0.01667 mm/day. This says nothing about the unavailable day.
A visibility reading is 900 decametres. Convert it to kilometres. Why should a July comparison of a UK station with Perth mention season?
One decametre is ten metres; the hemispheres differ.
900×10 = 9000 m = 9 km. July is summer in the UK and winter in Perth, so a comparison must account for seasonal context.
A student compares selected May–October records in 1987 and 2015 and claims a steady yearly trend throughout the intervening period. Explain the problem.
The intervening years have not been observed in that selection.
Two selected years and months do not show every intervening year or the whole annual cycle. Describe the observed comparison and avoid claiming a steady trend without suitable additional evidence.
07 / Evaluate a complete report
Pause, replay or seek freely. The notes explain the same idea and stay in view.
A credible report states who was eligible, how selection worked, how records were cleaned and what the result can support.
A report says: “We selected 50 dates. Forty-two had numerical readings, five were traces and three were missing. The numerical total was 18.8 mm, so the mean was 18.8/42.” Correct it under the trace-as-zero convention.
The five traces are observed days.
There are 47 observed days. The mean is 18.8/47 = 0.4 mm/day. State the trace approximation and the three missing dates. Dividing by 42 incorrectly discards the trace days.
A census records all 600 eligible items but a miscalibrated scale adds 2 g to every mass. Is its mean necessarily correct?
Completeness and accurate measurement are different.
No. Every item is included, but the systematic measurement error raises the recorded mean by 2 g. A census does not eliminate measurement error.
08 / Check your reasoning
Define → select → record → calculate → qualify.
Every stage can change what the final number means.
If you struggled with an item, revisit the corresponding lesson and try its independent practice before returning. The aim is to explain why a procedure works and where its conclusions stop.
Section 1 of 8 · Choose a method deliberately