Random Sampling Methods

Unit 3 · Collecting Data

What AP Stats asks here

Random sampling is what makes a sample's results generalize to a population. AP problems hinge on two recognitions: identify which named method a procedure uses, and notice when a non-random sample (especially voluntary-response with huge n) masquerades as evidence about the broader population.

Sampling methods at a glance

SRS

\text{every group of size } n \text{ equally likely}

Stratified

\text{split into strata, SRS within each}

Cluster

\text{randomly pick whole clusters; survey all in each}

Systematic

\text{every } k\text{-th unit from random start}

AP Tip: Stratify when strata correlate with the variable of interest. If neighborhoods differ wildly in income but residents within a neighborhood are similar, stratifying by neighborhood shrinks the variance of the income estimate.

Caution: Voluntary response and convenience samples are biased no matter how big. The 1936 Literary Digest poll had n = 2.4 million and called the election wrong; Gallup's random n = 50,000 called it right.

Type 1

Identify the sampling method

Memory aid: stratified = SRS within every group; cluster = sample entire selected groups; systematic = every k-th unit from a random start.

Example 1

A researcher randomly selects 10 of the 50 dorms on campus and surveys EVERY student in those dorms. Which sampling method is this? (A) Simple Random Sample (B) Stratified sampling (C) Cluster sampling (D) Convenience sampling

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Type 2

Stratify-or-not decision

Stratification helps when the candidate strata are correlated with the variable of interest. It adds nothing when the strata are unrelated.

Example 2

Which stratification would most likely improve precision over a plain SRS? (A) Estimate average resting heart rate; stratify by age group. (B) Estimate proportion who prefer dark chocolate; stratify by birth month. (C) Estimate average commute time; stratify by zodiac sign. (D) Estimate income; stratify by alphabetical position of last name.

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Type 3

Voluntary response trap

Self-selected respondents are not representative of the broader population, no matter how many of them respond. Large n reduces variance, not bias.

Example 3

A talk-show host asks listeners to call and vote; 50,000 people respond, with 75% favoring Option A. Which conclusion is best supported? (A) Option A is preferred by about 75% of the general public — n is so large the result must be accurate. (B) Option A is preferred by about 75% of self-selected listeners who chose to call; not generalizable to the general public. (C) Option A is preferred by exactly 75% of every demographic. (D) Without a confidence interval, no conclusion is possible.

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