Unit 3 · Topic 3.2 · about 20 minutes

Sampling Distributions for Sample Proportions

Find the mean and standard deviation of the sampling distribution of a sample proportion, check its conditions, and use it to find and interpret probabilities in context.

Predict first

A basketball player makes 70% of her free throws in the long run. Which is more likely: that she makes at least 80% of her shots in a 10-shot practice, or that she makes at least 80% in a 100-shot practice?

The sampling distribution of a sample proportion

Take a random sample of size $n$ and compute $\hat{p}$ , the proportion of successes. Another random sample of the same size gives a different $\hat{p}$ . The sampling distribution of $\hat{p}$ is the distribution of the values of $\hat{p}$ from all possible samples of that size, the same idea you simulated in 2.12 Sampling Distributions and the Central Limit Theorem.

For a population with proportion $p$ , when the sampled values are independent, that distribution has a mean and a standard deviation you can write down without simulating anything:

μ_{\hat{p}} = p σ_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}}

The mean says the values of $\hat{p}$ center on $p$ , which is why $\hat{p}$ is an unbiased estimator (3.1 Estimators). The standard deviation is the typical distance between $\hat{p}$ and $p$ in repeated samples of size $n$ . Because $n$ sits under a square root, quadrupling the sample size only cuts that distance in half. Both formulas are on the formula sheet.

Three conditions

The formulas and the normal shape each depend on how the data were collected.

Randomization condition: the data come from a random sample. Without one, the values of $\hat{p}$ may not center on $p$ at all.
10% condition: when sampling without replacement, the population is at least 10 times the sample size, $n \leq 0.10 N$ . Each person you remove changes the chances for the next pick a little. When the sample is a small slice of the population, that change is too small to matter, the observations behave as if independent, and the formula for $σ_{\hat{p}}$ is accurate.
Normality condition: the expected number of successes and the expected number of failures are both at least 10, so $n p \geq 10$ and $n (1 - p) \geq 10$ . Then the sampling distribution of $\hat{p}$ is approximately normal.

The first two make the mean and standard deviation trustworthy. The third is the one that lets you find probabilities with a normal curve. When it fails, the distribution is skewed, as in the graph below, where only 4 successes are expected.

Sampling distribution of $\hat{p}$ when $n = 100$ and $p = 0.04$

With $n p = 4$ , the distribution is skewed right and piles up against 0. A normal curve with the same mean and standard deviation would put about 2% of its area below 0, on proportions that cannot happen.

Lab

Sampling Distribution Machine

Switch the lab to proportions and set $p = 0.1$ . Start with a small sample size and raise it until the simulated distribution of $\hat{p}$ looks roughly symmetric, then check $n p$ at that point. Next, quadruple the sample size and watch what happens to the spread.

Open the full Sampling Distribution Machine lab

Worked exampleBus riders in a large district

District records show that 36% of the 21,500 students in a large school district ride a school bus. A transportation survey goes to a random sample of 400 of these students. Let $\hat{p}$ be the proportion of the sample who ride a bus. Find the probability that more than 40% of the sample ride a bus.

Center and spread. $μ_{\hat{p}} = p = 0.36$ and $σ_{\hat{p}} = \sqrt{\frac{0.36 (0.64)}{400}} = 0.024$ . In random samples of 400 students from this district, the proportion who ride a bus typically differs from 0.36 by about 0.024.
Conditions. Randomization: the 400 students were randomly selected. 10%: 400 is less than 10% of 21,500, which is 2,150. Normality: $n p = 400 (0.36) = 144$ and $n (1 - p) = 400 (0.64) = 256$ are both at least 10, so the sampling distribution of $\hat{p}$ is approximately normal.
Standardize. $z = \frac{0.40 - 0.36}{0.024} \approx 1.67$ .
Find the area. $P (\hat{p} > 0.40) = P (z > 1.67) \approx 0.0475$ from the table. Technology, working from the unrounded $z$ , gives 0.0478.
Interpret. If many random samples of 400 students were taken from this district, more than 40% of the sampled students would ride a bus in only about 4.8% of those samples.

Answer.

About 0.048. A sample with more than 40% bus riders would be unusual for this district, though not impossible.

Sampling distribution of $\hat{p}$ for random samples of 400 students

Mean 0.36 and standard deviation 0.024. The shaded area, about 0.048, is the probability that more than 40% of a random sample of 400 ride a bus.

Check your understanding

Records show that 82% of the 4,600 flights a regional airline flew this year arrived on time. An auditor selects a random sample of 150 of this year's flights. Find the standard deviation of the sampling distribution of the sample proportion of flights that arrived on time. Round to four decimal places.

Your answer

An orchard's records show that 15% of its apples are bruised. Inspectors take random samples of 500 apples, and the standard deviation of the sampling distribution of the sample proportion of bruised apples is about 0.016. Which is the best interpretation of 0.016?

Random samples of size $n$ are taken from a large population with population proportion $p$ . For which pair of values is the sampling distribution of $\hat{p}$ NOT approximately normal?

A high school has 1,200 students, and 30% of them play a school sport. A reporter for the school paper randomly selects 150 students without replacement and records the proportion who play a sport. Which condition for using $σ_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}}$ is NOT met?

For the orchard where 15% of apples are bruised, the probability that a random sample of 500 apples has a sample proportion of 0.18 or more bruised is about 0.03. Which is the correct interpretation?

Practice

Practice until it is automatic

New numbers every time. Each one is checked the moment you answer, with the full working shown.

Sampling distribution of a sample proportion practice page

Course alignment, for teachers

AP Statistics topic 3.2, Unit 3: Inference for Categorical Data: Proportions.

Skill 3.D: Calculate means, standard deviations, and parameters for probability distributions.
Skill 4.D: Interpret statistical calculations and results to assess meaning or a claim.
Skill 4.E: Justify the use of a chosen statistical inference method by verifying conditions.