# Sampling Distributions for Sample Proportions

> AP Statistics · AP 2024-2026 Statistics Curriculum
> Source: https://www.owlsprep.com/study/ap-statistics-u12-sampling-distributions-for-sample-proportions/

We will define the sampling distribution of sample proportions, verify required normality conditions, calculate its core parameters, and apply normal approximation to solve exam-aligned probability problems for categorical inference.

**Prerequisites:** [Basic stats terminology: population, sample, parameter vs statistic](https://www.owlsprep.com/study/ap-statistics-u1-intro-to-stats-terminology/); [Normal distribution probability and z-score calculations](https://www.owlsprep.com/study/ap-statistics-u2-normal-distributions/)

## Learning objectives

- Define the sampling distribution of sample proportion $\hat{p}$ and distinguish it from population and single-sample distributions
- Verify all three required conditions for normality of the sampling distribution of $\hat{p}$
- Calculate the exact mean and standard deviation of the sampling distribution of $\hat{p}$
- Apply normal approximation to compute probabilities for sample proportion outcomes

## Defining the Sampling Distribution of $\hat{p}$

It is critical to distinguish three distinct distribution types for categorical data: the population distribution (all individual outcomes for the full population), the sample distribution (outcomes from one single collected sample), and the sampling distribution of $\hat{p}$, which aggregates the sample proportion statistic across every possible random sample of the same fixed size.

**Sampling Distribution of Sample Proportion $\hat{p}$** — The full probability distribution of the sample proportion values calculated from all possible simple random samples of identical size n drawn from the same defined population.

*Notation:* $\hat{p} = \frac{\text{Number of successes in sample}}{n}$

*Example:* For a population of 2000 college students where 55% live on campus, all SRS of size 100 will produce $\hat{p}$ values that form this sampling distribution.

**Worked example:** A small town has 1500 registered voters, 62% of whom support a local ballot measure. You take repeated SRS of size 75 voters. Explain what the sampling distribution of $\hat{p}$ (proportion of measure supporters in samples of size 75) represents.

1. First, identify the population: all 1500 registered town voters, with true population proportion p = 0.62.
2. The sampling distribution describes every unique $\hat{p}$ value you could get if you selected every possible SRS of size 75 from this 1500-person population.
3. It also quantifies the relative frequency (probability) of each possible $\hat{p}$ value occurring across all those samples.

**Check your understanding**

Confirm you can tell the three distribution types apart

1. Which distribution describes the values of a categorical variable for every individual member of the full population?

   - Sampling distribution of $\hat{p}$
   - Population distribution
   - Single sample distribution

   *Why:* The population distribution includes raw individual outcomes, not aggregated sample statistics.

## Mean and Standard Deviation of the Sampling Distribution

The sampling distribution of $\hat{p}$ has two fixed, easily calculated parameters when the sample is selected randomly. The mean of the distribution is exactly equal to the true population proportion p, meaning $\hat{p}$ is an unbiased estimator with no systematic tendency to over or underestimate p.

$$\mu_{\hat{p}} = p$$

$$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$

> **info**
>
> The spread of the sampling distribution shrinks as sample size n increases, so larger samples produce $\hat{p}$ values that cluster much closer to the true population p.

**Derivation:** Prove that $\hat{p}$ is an unbiased estimator of p

*Starting from:* X is a binomial random variable counting successes in n independent trials, with E[X] = np

1. Define the sample proportion as $\hat{p} = \frac{X}{n}$
2. Apply linearity of expectation: $E[\hat{p}] = E\left[\frac{X}{n}\right] = \frac{1}{n}E[X]$
3. Substitute the known binomial expectation: $\frac{1}{n}(np) = p$

*Conclusion:* The expected value of $\hat{p}$ is exactly equal to the population proportion p, confirming no estimation bias.

**Worked example:** A factory produces lightbulbs, 3% of which are defective. If you select an SRS of 400 bulbs, calculate the mean and standard deviation of the sampling distribution of $\hat{p}$ (proportion of defective bulbs in the sample).

1. Identify given values: population proportion p = 0.03, sample size n = 400
2. Calculate the mean: $\mu_{\hat{p}} = p = 0.03$
3. Calculate standard deviation: $\sigma_{\hat{p}} = \sqrt{\frac{0.03(0.97)}{400}} \approx 0.0085$

## Conditions for Normal Approximation

To use the normal distribution to approximate the shape of the sampling distribution of $\hat{p}$, you must verify three non-negotiable conditions, all of which are explicitly graded on AP exam FRQs.

**Large Counts Condition** — The requirement that the expected number of successes and failures in the sample are both at least 10, to ensure the discrete binomial distribution is well-approximated by a continuous normal curve.

*Notation:* $np \geq 10$ and $n(1-p) \geq 10$

**Exam command terms**

AP graders award 1 full point for complete condition verification, and deduct points if you skip any step

- **Random** — Sample must be a properly randomized SRS or experimental group to eliminate selection bias *(Check: 'The problem states we selected an SRS of 80 customers, so this condition is met.')*

- **10% Condition** — Sample size n cannot exceed 10% of the total finite population when sampling without replacement *(Check: 'n=80, population size is 1000, 80 < 0.1*1000 = 100, so condition is satisfied.')*

- **Large Counts** — Expected successes and failures are both ≥10 to justify normality *(Check: np=80*0.25=20 ≥10, n(1-p)=60 ≥10, so large counts holds.)*

**Worked example:** Determine if the sampling distribution of $\hat{p}$ is approximately normal for a sample of size 25 drawn from a population where p=0.3.

1. Check Random: Assume the sample is a valid SRS as specified in the problem
2. Check 10% Condition: Assume population size is at least 250, so n=25 is less than 10% of the population
3. Check Large Counts: np = 25*0.3 = 7.5 < 10, n(1-p) = 17.5 ≥ 10
4. Conclusion: The large counts condition fails, so the sampling distribution is not approximately normal.

## Calculating Probabilities for Sample Proportions

Once all three conditions are verified, you can treat the sampling distribution of $\hat{p}$ as exactly normal, and use standard normal CDF calculations to find the probability of observing any range of $\hat{p}$ values.

$$z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}$$

> **R-L-N Exam Mnemonic**
>
> R-L-N: Remember the three conditions in order: Random, 10% Limit, Large Counts (Normal) to never miss grading points on AP FRQs.

**Worked example:** Public health data shows 18% of US adults smoke. We take an SRS of 300 adults. Find the probability that the sample proportion of smokers is less than 14%.

1. Verify conditions: Random (SRS), 10% (300 < 10% of all US adults), Large Counts: np=54 ≥10, n(1-p)=246 ≥10. All conditions met.
2. Calculate parameters: $\mu_{\hat{p}} = 0.18$, $\sigma_{\hat{p}} = \sqrt{\frac{0.18*0.82}{300}} \approx 0.0222$
3. Calculate z-score: $z = \frac{0.14 - 0.18}{0.0222} \approx -1.80$
4. Find P(Z < -1.80) = 0.0359. The probability of observing a sample proportion below 14% is ~3.6%.

## Common pitfalls

- **Wrong:** Using the sample $\hat{p}$ instead of the true population p to calculate $\sigma_{\hat{p}}$
  - Why it fails: The theoretical standard deviation of the sampling distribution is defined using the known population proportion, not an estimate from one sample.
  - Correct: Always use the given population p value when calculating the exact spread of the sampling distribution.
- **Wrong:** Skipping condition checks before applying normal approximation
  - Why it fails: AP graders deduct 1-2 full points for proportion inference problems if you do not explicitly verify all three conditions.
  - Correct: State and confirm Random, 10%, and Large Counts conditions for every problem that uses normal approximation for $\hat{p}$.
- **Wrong:** Using n instead of $\sqrt{n}$ in the denominator for $\sigma_{\hat{p}}$
  - Why it fails: This gives the standard deviation of the binomial count X, not the sample proportion $\hat{p}$, leading to drastically incorrect spread values.
  - Correct: Use the formula $\sqrt{p(1-p)/n}$ to get the correct standard deviation for the proportion distribution.
- **Wrong:** Checking the 10% condition when sampling with replacement
  - Why it fails: Observations are fully independent when sampling with replacement, so the 10% rule is unnecessary and irrelevant.
  - Correct: Only verify the 10% condition when sampling without replacement from a finite population.
- **Wrong:** Requiring observed sample successes and failures to be ≥10 instead of expected np and n(1-p)
  - Why it fails: The large counts condition uses expected values calculated from the population p, not observed counts from your specific sample.
  - Correct: Calculate np and n(1-p) using the given population proportion p, not your sample $\hat{p}$.

## Cheatsheet

| Parameter | Formula | Required Condition |
| --- | --- | --- |
| Mean of $\hat{p}$ | $\mu_{\hat{p}} = p$ | Sample is randomly selected |
| Standard Deviation of $\hat{p}$ | $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$ | n ≤ 10% of total population |
| Normal Approximation | $\hat{p} \sim N(\mu_{\hat{p}}, \sigma_{\hat{p}})$ | $np \geq 10$, $n(1-p) \geq 10$ |
| Z-score for $\hat{p}$ | $z = \frac{\hat{p} - p}{\sqrt{p(1-p)/n}}$ | All 3 conditions verified |

## What's next

Mastering sampling distributions for sample proportions is the non-negotiable foundation for all subsequent categorical inference content, which makes up roughly 15% of your total AP Statistics exam score. You will reuse the exact conditions and parameter formulas you learned here to construct confidence intervals that estimate unknown population proportions, and run formal hypothesis tests to evaluate claims about population proportions. This topic also directly connects to sampling distributions for sample means, which you will apply for quantitative inference later in the course. Explicit condition verification is one of the most commonly missed grading points on AP FRQs, so prioritize practicing that step for every problem. Move on to the next topics to build your full proportion inference toolkit.

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