L-DS-1: Solutions and Reasoning Guides — Statistical Vocabulary and Sampling

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — The Four Elements (C1 + C2)

Step 1 — Population: All professional basketball players. The 60 NBA players are the sample actually measured. Because the stated target is professional basketball broadly, the population is not limited to the NBA.

Step 2 — Statistic: minutes per week. This mean was computed from the 60 sampled players.

Step 3 — Parameter: , the population mean weekly strength-training time. This population mean is the intended target, but an NBA-only sample does not by itself justify treating as representative of every professional basketball league.

Problem 2 — Notation Match (C1 + C2)

(a) . All 4,500 employees were measured, so the value is a population mean.

(b) . The 80 apartments are a sample, so their mean rent is a sample statistic.

Problem 3 — Bias Identification and Direction (C5)

Bias type: Voluntary-response bias. Viewers decide for themselves whether to text, so people with strong opinions are more likely to enter the sample.

Direction: Cannot be determined from the information given. Strong opponents, strong supporters, or a group mobilized by the broadcaster could dominate. Identifying the selection mechanism does not by itself reveal the direction of its effect.

Problem 4 — Variable-Type Rematch (C3)

Classify the meaning of the variable, not the formatting or code used to record it:

• Nominal: category labels with no natural order.
• Ordinal: ordered categories whose gaps are not guaranteed equal.
• Discrete: countable numerical values.
• Continuous: measurements modelled along a continuum, even when recorded with rounding.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Choose the Sampling Design (C4)

The generator draws from school, hospital, factory, and retail settings.

• SRS: every size- sample from the complete list is equally likely.
• Stratified: divide into mutually exclusive strata and randomly sample from every stratum. Equal allocation guarantees representation but is proportional only when stratum sizes are equal.
• Systematic: choose a random start and an exact fixed interval. Generated cases satisfy , so every start from 1 through gives exactly selections. Periodicity aligned with can make a realized sample badly unrepresentative and inflate sampling variance, even though a genuinely random start preserves equal inclusion probabilities in this exact case.
• Cluster: randomly select some groups and include every unit in the selected groups. When outcomes are positively correlated within clusters, this often has higher sampling variance than an equally sized SRS.
• Convenience: choose units because they are easiest to reach.
• Multistage: use more than one nested selection stage.

Problem 2 — Parameter or Statistic? (C2)

This generator alternates between a census and a sample.

• A number calculated from the whole defined population is a parameter.
• A number calculated from a subset is a statistic.
• Use and for means.
• Use and for proportions.

Problem 3 — Bias in a News Story (C4 + C5)

The claim that the survey “proves” remote work raises productivity is not justified.

• Only remote employees at five technology companies are described, so the sample does not support a conclusion about all workers.
• Productivity is self-reported rather than measured objectively.
• There is no comparable in-office group.
• The excerpt does not state how employees within the companies were selected, so the exact sampling method cannot be identified.
• Social-desirability, undercoverage, and non-response concerns are plausible, but their directions should not be asserted without evidence.

An honest conclusion is: “Among the surveyed remote employees, 84% reported feeling more productive at home.”

The follow-up generator distinguishes voluntary response, convenience selection, undercoverage, nonresponse, and leading-question bias from their mechanics.

Problem 4 — Variable Type (C3)

The current bank includes age, education level, call counts, store department, temperature, letter grades, package weight, and meal ratings.

• Nominal: category labels with no natural order.
• Ordinal: ordered categories whose gaps are not guaranteed equal.
• Discrete: countable numerical values.
• Continuous: values modelled along a continuum, even when recorded with rounding.

A numerical code does not automatically make a variable quantitative, and a rounded measurement does not automatically make it discrete.

Problem 5 — Population or Sample? (C1)

This generator changes the setting and may ask for either group.

• Population: the entire group the researcher wants the conclusion to describe.
• Sample: the subset actually measured.
• A variable or a computed number is neither group.

Use the sentence before the question: the population follows “wants to know,” while the sample follows “can only measure.”

Problem 6 — Survey Question Critique (C6 · Enrichment)

(a) “Don’t you agree…” is leading. Fix: “Do you think the school’s sports facilities need improvement? Yes / No / No opinion.”

(b) Price and speed are combined in one double-barrelled question. Fix: ask one satisfaction question about price and a separate one about speed.

(c) “Regularly,” “sometimes,” and “often” are ambiguous. Fix: ask for the number of days in an average week with at least 30 minutes of exercise.

(d) The expert-consensus preamble leads respondents. Fix: “Do you support the introduction of carbon taxes in Canada? Yes / No / Unsure.”

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Question 1 — Feynman Test

A complete explanation should say:

• the population is the full target group;
• the sample is the subset measured;
• samples are used because a census may be impractical;
• a parameter describes a population;
• a statistic is computed from a sample and can estimate a parameter.

Question 2 — Apply (C3 + C4)

Plan 1: An unstratified two-stage cluster sample. Stores are selected first; transactions are then sampled within those stores. Its main weakness is that a random set of 30 stores could include too few stores from a small region.

Plan 2: A region-stratified two-stage cluster sample. Exactly one quarter of each region’s stores is selected, and the same transaction sampling fraction is used within each selected store. This preserves proportional regional representation at the store stage without giving low-traffic stores the same customer weight as high-traffic stores.

Satisfaction rating: Qualitative — Ordinal. Report category frequencies and a median or mode when appropriate. A simple numerical mean assumes equal gaps that the labels do not guarantee.

Question 3 — Find the Error (C4 + C5)

Each generated instance contains exactly one corrupted claim. Possible errors are:

1. Calling a voluntary-response poll random.
2. Claiming its large response count makes it a representative estimate.
3. Using for a proportion instead of or .
4. Generalizing from the respondents to a broader population.

The other three claims in that instance are intentionally correct.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

🔬 Path A — The Analyst: Dissecting a Real Survey

1. The professional groups were formed after responses arrived, so they are not sampling strata; the newsletter frame and voluntary response also cannot establish a national population mean.
2. A defensible report says that the 1,847 respondents had hours and treats the unknown Canadian-adult mean as a target the design cannot estimate representatively.
3. Newsletter undercoverage and nonresponse have unknown direction from the stated evidence; the loaded word “excessive” plausibly encourages underreporting.
4. Underlying screen-time duration is quantitative and continuous; the recorded whole-hour response from 0 to 24 is quantitative and discrete. Whole-hour recording loses within-hour detail. A neutral question defines the time frame and devices, removes “excessive,” and accepts a suitable numerical response.

🏗️ Path B — The Architect: Design the Study

1. Define the population by public institution, regular-education status, and reference term; define as the population mean composite score from the pre-specified validated instrument and use a weighted sample mean as the estimate.
2. Stratify by institution size from a dated roster, guarantee a minimum comparison sample per stratum, allocate the remainder proportionally, randomly select students, and weight by known inclusion probabilities.
3. Address nonresponse with consistent follow-ups and frame comparisons; adjust weights only when available frame variables justify the adjustment.
4. Preserve the validated instrument’s wording and scoring procedure. Classify each ordered item response as ordinal, but treat the documented composite score as quantitative only because the validated scoring rule supports that use. Derive the ordinal institution-size stratum from the dated sampling roster; do not ask for an unrelated demographic without a pre-specified, approved purpose. Minimize sensitive data, separate identifiers, suppress small cells, report aggregates, and complete the applicable ethics and Quebec privacy review.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Challenge 1 — The “Perfect” Sample? (C2)

The population mean for the eight wages is:

For the sampled wages of $20 and $80:

Equal values do not change their roles: the first is a parameter and the second is a statistic. The two-person sample matches the mean and range but omits six wage levels, so it does not reproduce the population distribution.

Four of the 28 possible pairs have mean $50:

The sample mean is unbiased because the average of all 28 possible pair means equals $50, not because this one pair happens to match. Each wage appears in seven pairs, so the sum of all pair totals is , the sum of their means is , and their average is .

Challenge 2 — Concentrated vs. Spread Multistage Designs (C4)

(a) students, which fits the stated cap of 5,000 completed questionnaires.

(b) Select 60 schools, then 50 students within each selected school.

(c) Both designs survey 3,000 students. Given the stated positive within-school correlation, the 60-school design should generally be more precise because it spreads the same number of observations across more school contexts. The 30-school design requires fewer school agreements. Both fit the questionnaire cap; broader cost claims require more information.

(d) Use for the population proportion and for the sample proportion.

Cold-transfer rematch: Read every constraint before choosing. Current generated cases require one of three complete designs:

• stratify and weight when subgroup comparisons must be guaranteed;
• sample groups and then units within groups when a fixed budget should be spread across natural clusters; or
• use a random start and exact interval when a complete ordered frame has no relevant periodic structure. If relevant periodicity exists, the main danger is an unstable or badly unrepresentative realized sample—not automatic design bias when the start is genuinely random.

Challenge 3 — When Convenience Sampling Creates Bias (C5)

The proof uses a fixed-size model: registration closes after exactly people enroll. If is person ‘s inclusion probability, then and:

Therefore:

The sign-up mechanism alone does not determine the direction: frequent exercisers might see the gym posting more often, while low exercisers seeking support or an incentive might be more motivated to volunteer. Direction requires evidence about exposure and response by prior exercise level. For the remaining parts, the problem explicitly assumes inclusion probability is positively associated with exercise time; under that condition, the association pushes the expected unweighted mean above the population mean. Unequal inclusion probabilities alone do not force bias in this fixed-size model. If they are unrelated to the measured values, the terms can balance; if they are known, an unequal-probability design may use appropriate weights. Increasing the number of volunteers through the same gym mechanism reduces random variation around the reachable-pool mean but does not repair undercoverage of non-gym-goers.