A sampling distribution for a sample proportion shows how the sample proportion p̂ varies across all possible samples of size n. Its center is the population proportion p, its standard deviation is √(p(1-p)/n), and it is approximately normal when np ≥ 10 and n(1-p) ≥ 10. Knowing the center, spread, and shape lets you calculate probabilities about p̂ in context.

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Why This Matters for the AP Statistics Exam

This topic is the bridge between probability and the inference you will do in later units. Once you can describe the center, spread, and shape of the distribution of p̂, you can find the probability of getting a sample result as extreme as the one observed. That same reasoning powers confidence intervals and significance tests for proportions in Unit 6.

On the AP Statistics exam, you should be ready to:

Calculate the mean and standard deviation of p̂ from a given p and n.
Check whether the sampling distribution of p̂ is approximately normal.
Find probabilities about p̂ using a z-score and the normal model.
State what a probability or parameter means using proportion or percent units in context.

Key Takeaways

The sample proportion p̂ is an unbiased estimator of p, so the mean of its sampling distribution is μ_p̂ = p.
The standard deviation of p̂ is σ_p̂ = √(p(1-p)/n). Larger n makes the spread smaller, since SE is proportional to 1/√n.
The Large Counts condition, np ≥ 10 and n(1-p) ≥ 10, tells you when p̂ is approximately normal.
Use the 10% condition when sampling without replacement, so n is less than 10% of the population, keeping the standard deviation formula valid.
Convert a value of p̂ to a z-score with z = (p̂ − p)/σ_p̂, then use the normal model to find probabilities.
Always interpret results with the right units (proportion or percent) and tie them to the specific population.

Formulas

The mean and standard deviation of the sampling distribution of p̂ are:

Mean: μ_p̂ = p
Standard deviation: σ_p̂ = √(p(1-p)/n)

These formulas can be found on page 2 of the AP Statistics formula sheet.

The standard deviation formula assumes independent samples, which happens when you sample with replacement. If you sample without replacement, the true standard deviation is a bit smaller. As long as your sample is less than 10% of the population, that difference is small enough to ignore. This is the 10% condition.

Source: AP Statistics Formula Sheet

Large Counts Condition

Before you treat the sampling distribution of p̂ as normal, you have to check that the sample is large enough. The Large Counts condition handles this.

The condition is np ≥ 10 and n(1-p) ≥ 10, where n is the sample size and p is the population proportion. In plain terms, you need at least 10 expected successes (np) and at least 10 expected failures (n(1-p)). When both are met, the sampling distribution of p̂ is approximately normal, so you can use normal-based methods like z-scores, and later, confidence intervals and significance tests.

When p is close to 0 or 1, the distribution of p̂ tends to be skewed. The Large Counts condition is what protects you from treating a skewed distribution as if it were normal.

For the shape of distributions of sample means, you check the Central Limit Theorem. For proportions, you always check the Large Counts condition instead.

How to Use This on the AP Statistics Exam

Problem Solving

Work these steps in order so your reasoning is easy to follow:

Identify p and n from the problem.
Find the center: μ_p̂ = p.
Find the spread: σ_p̂ = √(p(1-p)/n).
Check Large Counts: confirm np ≥ 10 and n(1-p) ≥ 10 before using the normal model.
Convert to a z-score: z = (p̂ − p)/σ_p̂.
Use the normal model to find the probability, then interpret it in context.

Free Response

Showing each piece clearly is important for clear exam work. Name the distribution of p̂, state the center and spread with their formulas, check the conditions, and write your final probability as a statement about the proportion in context, not just a bare number.

Common Trap

A z-score for a proportion uses the standard deviation of p̂, not the standard deviation of a single observation. Always divide by σ_p̂ = √(p(1-p)/n), not by √(p(1-p)).

Practice Problem

Suppose that you are conducting a survey to estimate the proportion of people in your town who support a new public transportation system. You decide to use a simple random sample of 1000 people, and you ask them whether or not they support the new system. After collecting the data, you find that 600 people out of the 1000 respondents support the system.

a) Calculate the sample proportion of respondents who support the new system.

b) Explain what the sampling distribution for the sample proportion represents and why it is useful in this situation.

c) Suppose that the true population proportion of people who support the new system is actually 0.6. Describe the shape, center, and spread of the sampling distribution for the sample proportion in this case.

d) Explain why the sampling distribution for the sample proportion can be modeled as approximately normal in this situation.

e) Calculate a 95% confidence interval for the population proportion of people who support the new system based on the sample data. (Optional for now, but feel free to answer if you already checked out the section on confidence intervals!)

f) Discuss one potential source of bias that could affect the results of this study, and explain how it could influence the estimate of the population proportion.

Answers

a) The sample proportion of respondents who support the new system is 600/1000 = 0.6.

b) The sampling distribution for the sample proportion represents the distribution of possible values for the sample proportion if the study were repeated many times with samples of the same size. It is useful here because it lets us make inferences about the population proportion based on the sample data.

c) If the true population proportion is 0.6, the sampling distribution of p̂ is approximately normal with center μ_p̂ = 0.6 and standard deviation σ_p̂ = √((0.6)(0.4)/1000) ≈ 0.0155. The shape is approximately normal because the Large Counts condition is met.

d) The sampling distribution of p̂ can be modeled as approximately normal because the Large Counts condition holds: np = 1000(0.6) = 600 ≥ 10 and n(1-p) = 1000(0.4) = 400 ≥ 10. Both expected counts are well above 10.

e) A 95% confidence interval for the population proportion can be calculated as 0.6 ± (1.96 × √((0.6)(1-0.6)/1000)). This gives a confidence interval of about (0.570, 0.630).

f) One potential source of bias is nonresponse bias, which occurs when certain groups are more or less likely to respond to the survey. If people who support the new system are more likely to respond, the sample could be biased toward higher support and overestimate the population proportion. If people who do not support the system are more likely to respond, the sample could be biased toward lower support and underestimate the population proportion.

Common Misconceptions

Mean and standard deviation are not the same thing. The center of the sampling distribution is p, while the spread is √(p(1-p)/n). Mixing these up is a common error.
The Large Counts condition uses expected successes and failures, np and n(1-p), not the raw sample size alone. A big n does not automatically guarantee normality if p is extreme.
p and p̂ are different. p is the fixed population proportion (a parameter), and p̂ is the proportion from your sample (a statistic) that varies from sample to sample.
The 10% condition and the Large Counts condition do different jobs. The 10% condition keeps the standard deviation formula valid when sampling without replacement; Large Counts checks whether the shape is approximately normal.
A larger sample does not make the center move. Increasing n shrinks the standard deviation but leaves μ_p̂ = p unchanged.

Vocabulary

The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.

Term	Definition
approximately normal	A distribution that closely follows the shape of a normal distribution, allowing for the use of normal probability methods.
categorical variable	A variable that takes on values that are category names or group labels rather than numerical values.
independent samples	Two or more separate groups of data where the values in one group do not influence or depend on the values in another group.
mean of the sampling distribution	The expected value of a sample statistic; for sample proportions, μp̂ = p.
parameter	A numerical summary that describes a characteristic of an entire population.
population proportion	The true proportion or percentage of a characteristic in an entire population, typically denoted as p.
probability	The likelihood or chance that a particular outcome or event will occur, expressed as a value between 0 and 1.
sample proportion	The proportion of individuals in a sample that have a particular characteristic, denoted as p-hat (p̂).
sample size condition	The requirement that np ≥ 10 and n(1-p) ≥ 10 must be satisfied for a sampling distribution of a sample proportion to be approximately normal.
sampling distribution	The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population.
sampling with replacement	A sampling method in which an item selected from a population can be selected again in subsequent draws.
sampling without replacement	A sampling method in which an item selected from a population cannot be selected again in subsequent draws.
standard deviation of the sampling distribution	The measure of variability in a sampling distribution; for sample proportions, σp̂ = √(p(1-p)/n).

Frequently Asked Questions

What is the sampling distribution of a sample proportion?

The sampling distribution of a sample proportion describes the values of p-hat from all possible samples of the same size from a population. Its mean is p, and its standard deviation is sqrt(p(1-p)/n) when the independence conditions are met.

What is the formula for the mean of p-hat?

The mean of the sampling distribution of p-hat is the population proportion p. This means p-hat is an unbiased estimator of p.

What is the standard deviation formula for a sample proportion?

The standard deviation of p-hat is sqrt(p(1-p)/n), where p is the population proportion and n is the sample size. If sampling without replacement, the 10% condition lets you use this formula as a close approximation.

What is the Large Counts condition for sample proportions?

The Large Counts condition is np >= 10 and n(1-p) >= 10. It checks whether the sampling distribution of p-hat can be treated as approximately normal.

When do you use Large Counts instead of the Central Limit Theorem?

Use the Large Counts condition for sample proportions because the variable is categorical. Use Central Limit Theorem reasoning for sample means involving quantitative variables.

How is Topic 5.5 tested on the AP Statistics exam?

AP Statistics questions can ask you to find the mean and standard deviation of p-hat, check Large Counts and the 10% condition, calculate a z-score, find a probability with the normal model, and interpret results in context.