DS-1: Solutions — Statistical Vocabulary and Sampling

Problem 1 — The Four Elements (C1 + C2)

NBA strength training scenario: basketball analytics company surveys 60 NBA players, average 280 min/week.

Step 1 — Population: All professional basketball players.
Not just the 60 contacted (that’s the sample), and not just NBA players — the analytics company wants to draw conclusions about professional basketball broadly. Population = the whole target group.

Step 2 — Statistic: minutes/week.
The 280 was computed from the 60-player sample — it’s a statistic. The unknown average for all professional players (which we never measured) is the parameter.

Step 3 — Parameter notation: — population mean strength-training time.
is the statistic (sample mean). is what we’re trying to estimate. is sample standard deviation — a different statistic from a different question.

Problem 2 — Notation Match (C2)

2a — 67,200$. All 4,500 employees were measured — the whole population. When you measure everyone, the resulting number is a parameter. No sampling occurred.

**2b — 1,4501,450 was computed from this sample — it’s a statistic estimating the population mean rent .

Problem 3 — Classify the Variable (C3) — Variant Bank

Correct answers for all 5 variants:

• Variant 0 (Membership tier: Bronze/Silver/Gold/Platinum) → Qualitative — Ordinal. Tiers have a natural order but unequal gaps.
• Variant 1 (Number of defects per batch) → Quantitative — Discrete. Whole-number counts with equal gaps.
• Variant 2 (Blood type: A/B/AB/O) → Qualitative — Nominal. Categories with no ranking.
• Variant 3 (Marathon finishing time) → Quantitative — Continuous. Measured time; any decimal value possible.
• Variant 4 (Cafeteria rating: Terrible/Poor/Okay/Good/Excellent) → Qualitative — Ordinal. Ordered categories with unequal gaps.

Problem 4 — Identify the Sampling Method (C4) — Variant Bank

Correct answers for all 5 variants:

• Variant 0 (Select 4 of 15 neighbourhoods; survey all residents in chosen neighbourhoods) → Cluster sampling. Whole groups selected; all members of selected groups surveyed.
• Variant 1 (Inspect item #3, then every 20th item) → Systematic sampling. Random start + fixed interval.
• Variant 2 (Divide by year of study; randomly select 75 from each year) → Stratified sampling. Homogeneous groups (years); random sample from every group.
• Variant 3 (Interview first 50 people exiting mall) → Convenience sampling. Whoever is easiest to reach.
• Variant 4 (Randomly select cities → tracts → households at three levels) → Multistage sampling. Multiple sequential stages of random selection.

Problem 5 — Identify the Bias (C5)

Type of bias: Voluntary response bias.
Viewers who feel strongly about the tax issue are far more likely to text in than indifferent viewers. Self-selected responses systematically overrepresent extreme opinions.

Direction: Toward “No”.
Tax opponents tend to feel more urgently motivated to act. A tax increase hurts people economically in a direct, immediate way — that kind of tangible cost motivates stronger responses than the more diffuse benefits of public spending. The “Yes” side (those who support the tax) is likely less motivated to call a radio station to express it.

Problem 1 — The Full Picture (C1 + C2)

Pharmaceutical drug trial: 180 patients from 12 clinics, mean blood pressure drop of 14.2 mmHg.

• (a) Population: All patients with high blood pressure who would be candidates for this drug (broadly: all adults with hypertension).
• (b) Sample: The 180 patients enrolled from 12 Canadian clinics.
• (c) Parameter: = the true mean blood pressure reduction for all high-blood-pressure patients who take this drug. Unknown — we only measured 180.
• (d) Statistic: mmHg = sample mean reduction, computed from 180 patients.
• (e) Concern: Volunteer enrollment introduces self-selection bias — patients who enroll may be more health-motivated and adherent to treatment than the general hypertensive population, potentially inflating the apparent benefit.

Problem 2 — Is It or ? (C1 + C2)

• (a) Parameter — years. All 847 teachers were surveyed — the entire district = the population.
• (b) Statistic — (sample proportion). The 500 polled Canadians are a sample; the 62% estimates the population proportion of all Canadians who purchased online.
• (c) Parameter — . The registrar has all students’ records — the entire enrolled population was measured.
• (d) Statistic — bpm. 30 mice is a sample from the relevant population of lab mice.

Problem 3 — Variable Type Generator (C3)

This problem generates new scenarios randomly. The correct classification and explanation appear after you answer. General keys:

• Nominal: Category labels, no order (blood type, department, colour)
• Ordinal: Ordered categories, gaps may be unequal (ratings, education levels, military ranks)
• Discrete: Countable whole numbers (number of children, defects, goals)
• Continuous: Measured, any decimal possible (height, weight, time, temperature)

Problem 4 — Sampling Method Critique (C4) — Variant Bank

Core answers for all 5 variants:

• Variant 0 (City website survey): Voluntary response / convenience. Bias: strongly-opinionated residents (pro or con park) over-respond. Fix: mail survey to SRS of registered residents.
• Variant 1 (First 10 morning customers): Convenience sampling (time-of-day bias). Fix: systematic sampling of transaction receipts across all hours.
• Variant 2 (Number each of 6,200 students, randomly select 300): Simple random sampling. Strength: unbiased. Weakness: by chance, some schools may be over/underrepresented. Improvement: stratify by school.
• Variant 3 (Randomly select cities → tracts → households): Multistage sampling. Advantage: practical for dispersed populations without a complete national household list. Tradeoff: clustering effect reduces effective sample size.
• Variant 4 (Select record #12, then every 20th): Systematic sampling. Bias risk: periodicity in filing order. Good choice when: records are in random order and a complete list exists.

Problem 5 — Bias in a News Story (C5)

Remote work productivity study: 1,200 remote employees at 5 tech companies; 84% self-report higher productivity.

• (a) The conclusion is not justified. No control group, no objective productivity measure, no random sample, and the claim “proves” is too strong for any survey study.
• (b) Biases: social desirability (remote workers have incentive to claim productivity), undercoverage (only tech workers at remote-friendly companies), and convenience sampling (no random selection from a defined population).
• (c) Excluded: workers in non-remote industries, workers who tried remote work and returned to the office, workers in non-tech sectors. These exclusions bias the population estimate upward (remote-friendly tech workers are probably the most suitable candidates for remote work).

Problem 6 — Survey Question Critique (C6)

• (a) “Don’t you agree that our school needs better sports facilities?” → Leading question. Fix: “Do you think our school’s sports facilities need improvement? [Yes / No / No opinion]”
• (b) “Are you satisfied with the price and speed?” → Double-barrelled. Fix: two separate questions — one for price, one for speed.
• (c) “How often do you exercise regularly?” → Ambiguous. Fix: “In an average week, on how many days do you exercise for at least 30 minutes? [0 / 1–2 / 3–4 / 5–6 / 7]”
• (d) “Given that most experts agree climate change is serious, do you support carbon taxes?” → Leading premise. Fix: “Do you support the introduction of carbon taxes in Canada? [Yes / No / Unsure]“

Question 1 — Feynman Test

Model answer (there is no single correct response — evaluate your own answer against this):

Checklist for your own answer:

• Did you explain why samples are used (not just that they are)?
• Did you make clear that parameters are usually unknown and statistics are computed?
• Did you use (or explain) the symbols and ?
• Did you explain that the statistic estimates the parameter?

Question 2 — Apply (C3 + C4)

Coffee chain with 240 locations in 4 regions: Plan 1 (cluster, 30 stores) vs. Plan 2 (stratified by region).

• (a) Plan 1 = Cluster sampling. Weakness: with only 30 of 240 stores, Atlantic (20 stores, only ~2–3 selected) may be barely represented. Opinions in small regions could be washed out.
• (b) Plan 2 = Stratified sampling by region. Fix: guarantees each region is represented proportionally to its store count — Atlantic always gets 5 surveyed stores regardless of random fluctuation.
• (c) Satisfaction rating = Qualitative — Ordinal. Ordered categories (Very Dissatisfied through Very Satisfied) with unequal gaps. Implication: report mode and percentage per category, not a simple mean. Computing “average satisfaction = 3.7” treats ordinal data as interval, which overclaims precision.

Question 3 — Find the Error (C4 + C5)

Food blogger Instagram poll: 847 responses, 91% prefer traditional bagels. Report calls it stratified sampling, calls 91% ”,” calls it quantitative-continuous.

Four errors in the student’s analysis:

1. “Huge random sample”: Voluntary response / convenience sampling — not random. Followers who engage with food content are not representative of all Montrealers.
2. “Reliable estimate”: A biased large sample is less reliable than an unbiased small one. The 91% is skewed by the food-enthusiast audience.
3. "" for a proportion: Population proportion is written , not . The sample proportion is . is reserved for means.
4. “Stratified sampling”: Stratification requires deliberate a priori division of the population into strata before sampling. Post-hoc grouping of respondents is not stratification. This was voluntary response sampling.

Path A — The Analyst: Screen Time Survey

Full solutions appear in the lesson’s Boss Fight section. Summary of errors found:

1. Misidentified as stratified sampling (actually voluntary response / convenience)
2. Misused for a sample-computed value (should be )
3. Misclassified screen time as qualitative-ordinal (it’s quantitative-continuous)
4. Used a leading question (“excessive hours”) that biases responses downward
5. Non-response bias and undercoverage from the newsletter subscriber convenience sample

Path B — The Architect: CEGEP Well-Being Study

Key design decisions:

• Method: Stratified sampling by institution size (small/medium/large), proportional allocation to enrollment, random selection within strata
• Parameter/Statistic: = true mean well-being score; = sample mean from 3,000 students
• Survey questions: Neutral wording, one concept per question, bounded time frame
• Residual bias: Non-response from students in greatest distress (hardest to mitigate)

Challenge 1 — Can a Statistic Equal a Parameter? (Variant Bank)

Core insight for all variants: can happen by coincidence, but it’s rare and doesn’t mean the sample “perfectly represents” the population. The key concept is sampling variability — different samples give different statistics. A statistic that happens to equal the parameter is still just one observation from the distribution of all possible sample means.

In Variant 1, none of the 6 possible size-2 samples produced . But the average of all 6 sample means = 60 = . This illustrates unbiasedness: is an unbiased estimator of (correct on average, not necessarily in any given sample).

Challenge 2 — Multistage vs. Cluster Design

• (a) 30 × 900 = 27,000 students. Not practical.
• (b) Two-stage: Stage 1 — randomly select 60 schools; Stage 2 — randomly select 50 students per school. Total = 3,000.
• **(c) ** Multistage gives more geographic/demographic diversity per student surveyed. Tradeoff: requires agreements with 60 schools vs. 30, but each school provides only 50 participants.
• (d) Population proportion of students spending >3h/day on social media is written p; the sample estimate is .

Challenge 3 — Why Convenience Sampling is Always Biased

• (a) Frequent exercisers have higher (selection probability); non-gym-goers have near-zero .
• (b) overestimates (exercise time inflated by overrepresentation of heavy exercisers).
• (c) A larger sample still draws from the biased pool. Bigger sample = more precise estimate of the wrong thing.
• (d) Proof sketch: unequal means individuals with systematically different values are over/underweighted in . This creates a systematic gap between and . Increasing sample size reduces variance but cannot eliminate the systematic deviation caused by the biased selection mechanism.