Lesson 11: Sampling Distributions of Sample ProportionsAP

Take a sample, compute a statistic. Take another sample, get a slightly different statistic. Sampling distributions describe that variation: the distribution of a statistic over all possible samples of the same size. This is the conceptual engine of the entire second half of the course — every standard error, margin of error and p-value you will ever compute is a statement about a sampling distribution. CED Unit 5 builds them for two statistics: the sample proportion p̂ and the sample mean x̄.

Setup: parameters, statistics, and sampling variability

A parameter (p, μ, σ) is a fixed number describing the population — usually unknown. A statistic (p̂, x̄, s) describes a sample and varies from sample to sample; that variation is sampling variability, and it is not a defect — it is quantifiable. A statistic is an unbiased estimator if the mean of its sampling distribution equals the parameter: the estimates are centred on the truth, with no systematic over- or under-shooting. (Biased designs from Unit 3 break this before any mathematics starts — no formula repairs a voluntary-response sample.)

The sampling distribution of p̂

Imagine repeatedly drawing random samples of size n from a population with proportion p, computing p̂ = (number of successes)/n each time. The sampling distribution of p̂ has:

• Centre: μ_p̂ = p. Sample proportions are unbiased — they aim at the true p.
• Spread: σ_p̂ = √(p(1−p)/n). Larger samples → smaller spread, shrinking like 1/√n: quadruple n and the standard deviation is *halved*.
• Shape: approximately normal when the Large Counts condition holds: np ≥ 10 and n(1−p) ≥ 10 — enough expected successes and failures for the distribution to smooth out.
• Fine print: the spread formula assumes sampling with replacement (perfect independence). For sampling without replacement from a finite population, we need the 10% condition: n ≤ 10% of the population, so draws are nearly independent.

Worked example 1. Suppose 60% of adults in a city favour a transit proposal (p = 0.60), and we survey a random sample of n = 100. Then μ_p̂ = 0.60 and σ_p̂ = √(0.60 × 0.40/100) = √0.0024 ≈ 0.049. Large Counts: np = 60 ≥ 10 and n(1−p) = 40 ≥ 10, so p̂ is approximately normal. What is the probability the sample shows p̂ above 0.68 — a headline "two-thirds support"? Standardise: z = (0.68 − 0.60)/0.049 ≈ 1.63, and P(Z > 1.63) ≈ 0.052. Even with true support at only 60%, a poll has about a 5% chance of reporting 68%+ purely by sampling luck. That number is the whole logic of the margin of error, one lesson early.

Worked example 2 (the 1/√n law in action). For p = 0.5, σ_p̂ at three sample sizes: n = 25 → √(0.25/25) = 0.100; n = 100 → √(0.25/100) = 0.050; n = 400 → √(0.25/400) = 0.025. Each quadrupling of n halves the spread. This is why serious polls use n ≈ 1,000 (σ ≈ 0.016, margin of error about ±3%) and why doubling a poll from 100 to 200 respondents buys disappointingly little precision. News consumers who know this curve cannot be fooled by a poll of 80 people with a dramatic headline.

Real-world application

Election forecasting is applied sampling-distribution theory: aggregators combine polls by weighting sample sizes precisely because σ_p̂ shrinks with n. In manufacturing, a line with true defect rate p = 0.03 is audited with samples of n = 200 daily; counts of defects far in the tail of the sampling distribution trigger a shutdown investigation. The question is always the same shape: "If the parameter were really p, how surprising is this p̂?" — which is, verbatim, the logic of the significance test in Lesson 15.

Common mistakes

• Using p̂ in the standard deviation when describing the sampling distribution model. The model is built from the *parameter* p: σ_p̂ = √(p(1−p)/n). (Later, confidence intervals will use p̂ because p is unknown — a deliberate, examinable difference.)
• Checking Large Counts with observed counts only in the model setting: the condition is np and n(1−p), computed from the parameter.
• Confusing σ_p̂ with the population's own spread. Individuals are Bernoulli (all-or-nothing); the *proportion* varies far less than individuals do.
• Forgetting the 10% condition when sampling a substantial fraction of a small population without replacement.

AP Exam Experience

Question 1. A population has p = 0.5. For random samples of size 64, the standard deviation of the sampling distribution of p̂ is:

• (A) 0.5
• (B) 0.25
• (C) 0.0625
• (D) 0.0078
• (E) 0.125

Answer: (C). σ_p̂ = √(0.5 × 0.5/64) = √0.00390625 = 0.0625. Choice (A) is the spread of a single observation; (B) is p(1−p) before dividing by n and taking the root — the two arithmetic steps students drop.

Question 2. Which sample size is large enough for the sampling distribution of p̂ to be approximately normal if p = 0.08?

• (A) n = 30
• (B) n = 60
• (C) n = 100
• (D) n = 125
• (E) n = 200

Answer: (D). Large Counts requires np ≥ 10: n ≥ 10/0.08 = 125. (At n = 125: np = 10 and n(1−p) = 115, both fine.) Choice (C) fails with np = 8 — close-but-not-enough is exactly how the exam writes this trap; "n ≥ 30" folklore does not apply to proportions.

Question 3. A statistic is described as an unbiased estimator of a parameter. This means:

• (A) Every sample gives exactly the parameter value.
• (B) The sampling distribution of the statistic is centred at the parameter.
• (C) The statistic's standard deviation is zero.
• (D) The statistic always comes from an SRS.
• (E) The sample value equals the population value whenever n is large.

Answer: (B). Unbiasedness is a statement about the centre of the sampling distribution — the long-run average of the statistic equals the parameter. Individual samples still vary (A, C, E are wrong), and unbiasedness comes from the estimator's design, not merely from using an SRS (D).

Free-response question. In a large city, 35% of households own a dog (p = 0.35). A researcher takes a random sample of 200 households. (a) Describe the sampling distribution of p̂ (centre, spread and shape), verifying the conditions you use. (b) Find the probability that the sample proportion of dog-owning households is within 0.05 of the true proportion.

Scoring guideline.

• Part (a): Essentially correct if all three features are correct and conditions verified: centre μ_p̂ = 0.35; spread σ_p̂ = √(0.35 × 0.65/200) ≈ 0.0337; shape approximately normal BECAUSE np = 200(0.35) = 70 ≥ 10 and n(1−p) = 130 ≥ 10 (and the 10% condition is addressed or the population is described as large). Partially correct if one feature or the condition check is missing. Incorrect if the spread uses σ = √(p(1−p)) ≈ 0.477 (the individual-level SD) — a signature confusion readers see constantly.
• Part (b): Essentially correct if it standardises both ends: z = ±0.05/0.0337 ≈ ±1.48 and gives P(−1.48 < Z < 1.48) ≈ 0.862, about an 86% chance. Partially correct if it computes a one-sided probability (about 0.93) — the phrase "within 0.05" makes the question two-sided, and readers flag the missed side.

Deeper dive: unbiased is not the same as accurate

Two words the course keeps separate: a statistic is unbiased if its sampling distribution is centred on the parameter, and low-variability if that distribution is tightly spread. Accuracy in estimation needs both — an unbiased estimator with huge spread (tiny samples) is honest but useless, and a precise-but-biased estimator (a voluntary poll with n = 100,000) is confidently wrong, which is worse. Picture the sampling distribution as a target pattern: unbiased = centred on the bullseye; small σ = tight grouping. The four combinations all appear in exam stems: centred and tight (ideal SRS, decent n), centred and scattered (SRS, n = 10), off-centre and tight (Literary Digest), off-centre and scattered (a small convenience sample — the worst quadrant, and the most common in the wild). When a question asks whether increasing n fixes a described study, this picture answers it: n shrinks spread only. Centre problems are design problems, and they survive every sample size.

A note on the word "distribution" — three different objects

Unit 5 confusion often reduces to one word used for three things. The population distribution describes individual values (for proportions, individuals are just success/failure). The sample distribution describes the values in one drawn sample — for p̂ there is almost nothing to see: some successes, some failures. The sampling distribution describes the statistic across *all possible samples* of the size — an imaginary but precisely characterised object with mean p and standard deviation √(p(1−p)/n). Exam questions deliberately blur these ("the distribution is approximately normal" — *which* distribution?). Discipline: whenever the word appears, ask "distribution of what, over what collection?" If the collection is samples and the "what" is a statistic, you are in Unit 5 territory and the centre/spread/shape machinery applies.

Second worked example: a modest proportion, honestly spread

Suppose the true proportion of left-handed students in a school system is p = 0.20, and a researcher samples n = 50 students at random. The sampling distribution of p̂ has centre 0.20 and spread σ_p̂ = √(0.20 × 0.80/50) = √0.0032 ≈ 0.0566; Large Counts holds (np = 10 ≥ 10 — right at the boundary — and n(1−p) = 40 ≥ 10), so approximate normality is defensible though not luxurious. The probability the sample lands within five points of the truth: z = 0.05/0.0566 ≈ 0.88, and P(−0.88 < Z < 0.88) ≈ 0.62. So even a perfectly executed random sample of 50 misses the ±0.05 window nearly 40% of the time. That is not a flaw in the researcher; it is the mathematics of n = 50, and it calibrates expectations for every small poll you will ever read: a sample of 50 carries an inherent wobble of roughly ±11 percentage points at the 95% level (1.96 × 0.0566 ≈ 0.111). When a headline trumpets a 4-point shift measured on 50 people, the sampling distribution of this lesson is the rebuttal.