Questions: Populations, Sampling Methods, and Representativeness

Question 1

The 1936 Literary Digest poll surveyed 10 million people but predicted the wrong presidential winner by a wide margin. What was the primary cause of this failure?

• A. The sample was too small to detect the true preferences of the electorate
• B. The sampling frame (telephone and car registration lists) systematically oversampled wealthier, Republican-leaning voters
• C. The poll was conducted too far in advance of the election
• D. The questions used were ambiguously worded, confusing respondents

Answer: (B)

The Literary Digest failure is a textbook case of sampling bias, not sample size. With 10 million respondents, size was not the problem. The sampling frame — people with telephones and cars in 1936 — systematically excluded lower-income voters who favored Roosevelt. The bias in *who* was selected overwhelmed the precision gained from large numbers. This illustrates the core principle: representativeness is about selection method, not volume.

Question 2

A researcher wants to estimate the average anxiety level of adults in a large city. Which approach provides the strongest statistical basis for inference to the full population?

• A. Survey 5,000 volunteers who respond to a public Facebook post
• B. Survey every patient at the city's three largest mental health clinics
• C. Draw a simple random sample of 800 adults from city registration records
• D. Survey 10,000 university students at local campuses

Answer: (C)

Simple random sampling from a complete sampling frame gives every adult a known, equal probability of inclusion — the condition required for standard inferential statistics to be valid. Options A and D are convenience samples with systematic biases (social media users, students). Option B oversamples people already seeking mental health treatment, severely biasing the estimate upward. A smaller random sample (800) outperforms a larger biased sample (5,000 or 10,000) for accurate population inference.

Question 3

A convenience sample of 10,000 participants is necessarily more representative of the population than a random sample of 1,000 participants.

• T. True
• F. False

Answer: False

Representativeness depends on *how* participants are selected, not *how many* are selected. A large convenience sample can be systematically biased — overrepresenting certain demographic groups — in ways that a smaller random sample avoids. The Literary Digest poll exemplifies this: 10 million biased respondents produced a worse estimate than a well-designed smaller random sample. Sample size increases precision (reduces sampling error) only if the sample is unbiased in the first place.

Question 4

Non-probability sampling methods (e.g., convenience samples, purposive samples) can be appropriate and valid for some research purposes.

• T. True
• F. False

Answer: True

Non-probability samples are not inherently invalid — they are inappropriate for making precise population-level statistical inferences, but they serve many legitimate research purposes. Exploratory research, hypothesis generation, studies of rare or hard-to-reach populations, qualitative research, and initial pilot testing often rely on purposive or convenience sampling. The key is honesty: results should be explicitly bounded to the sample or similar populations, rather than overgeneralized to groups not represented in the sample.

Question 5

Why can a well-drawn random sample of 1,000 people produce more accurate population estimates than a convenience sample of 100,000 people?

Model answer: A random sample gives every member of the population a known, equal chance of selection, so the resulting sample's statistics are unbiased estimates of population parameters. A convenience sample systematically over- or under-represents certain groups, introducing bias that does not shrink as sample size grows — more observations from a biased process just give a more precise estimate of the wrong thing.

Statistical inference theory assumes random selection: confidence intervals, significance tests, and sampling distributions are derived under the assumption that each observation is drawn randomly from the population. A convenience sample violates this assumption, so the inferential machinery doesn't apply. Large biased samples are like measuring with a miscalibrated ruler many times — you get more precise measurements, but they're all wrong in the same systematic direction. Randomization ensures the errors are unsystematic and average out.