Sampling Distribution of p-hat Simulator (AP Statistics)

How to use the simulator

Two sliders set up the population and the sample:

Population proportion (p): 0.05 to 0.95 in steps of 0.01, starting at 0.50.
Sample size (n): 5 to 200 in steps of 1, starting at 30.

Moving either slider clears all samples, so each histogram belongs to one

p

and one

n

. The buttons:

Draw 1, Draw 100 and Draw 1000 take that many new random samples of size $n$ and add each sample's $\hat{p}$ to the histogram. They add to what is already there.
Reset clears the samples.
Normal overlay: OFF/ON draws a normal curve with mean $p$ and standard deviation $\sqrt{p(1-p)/n}$ over the histogram (once there are at least two samples).

The histogram runs from 0 to 1 in 40 bars, each 0.025 wide, with a dashed line at the true

p

. Above it, the Large Counts box shows

np

and

n(1-p)

to one decimal, with a ✓ when both are at least 10 and a ✗ otherwise. Below it, four boxes report Samples, Mean of p̂, SD of p̂ and Theory SE √(p(1−p)/n).

That last box uses the true

p

, so on the formula sheet it is the standard deviation

\sigma_{\hat{p}}

. The name "standard error" is usually kept for the estimate made with

\hat{p}

in place of

p

.

Small samples can look gappy, because

\hat{p}

can only take the values

0, 1/n, 2/n, \dots, 1

. Try

p = 0.10

,

n = 40

:

np = 4

, the box shows ✗, and the histogram is skewed right.

The formulas

For random samples of size

n

from a population with proportion

p

:

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

$\mu_{\hat{p}} = p$ means $\hat{p}$ is an unbiased estimator: its sampling distribution is centered on the parameter, whatever the sample size.
The standard deviation formula assumes independent observations. When sampling without replacement, check the 10% condition: $n \le 0.10N$ .
The sampling distribution is approximately normal when the Large Counts condition holds: $np \ge 10$ and $n(1-p) \ge 10$ .
$n$ is under a square root, so to halve the spread you need four times the sample size.

See the square-root effect directly: at

p = 0.50

, Theory SE is 0.0707 for

n = 50

and 0.0354 for

n = 200

.

Worked example

Problem: At a large high school, 30% of students bike to school. A random sample of 100 students is taken. Describe the sampling distribution of

\hat{p}

, and find the probability that at least 36% of the sample bike to school.

Step 1: Center.

\mu_{\hat{p}} = p = 0.30

.

Step 2: Spread. The school has well over

10(100) = 1{,}000

students, so the 10% condition holds and

\sigma_{\hat{p}} = \sqrt{\dfrac{0.30(0.70)}{100}} \approx 0.0458

.

Step 3: Shape.

np = 100(0.30) = 30

and

n(1-p) = 100(0.70) = 70

are both at least 10, so the sampling distribution is approximately normal.

Step 4: Probability.

z = \dfrac{0.36 - 0.30}{0.0458} \approx 1.31

, and

P(\hat{p} \ge 0.36) = P(Z \ge 1.31) \approx 0.095

. About 9.5% of random samples of 100 students would have 36% or more bikers.

Check it in the simulator: set p to 0.30 and n to 100. The Large Counts box shows ✓ with np = 30.0 and n(1−p) = 70.0, and Theory SE reads 0.0458. Click Draw 1000: the Mean of p̂ lands close to 0.300 and the SD of p̂ close to 0.046. Turn on the overlay to see the normal curve match the histogram.

Common mistakes on the AP exam

Mixing up three distributions. The population distribution is the individuals, a sample's distribution is the data you collected, and the sampling distribution is the $\hat{p}$ values from all possible samples. The histogram here is the third.
Checking Large Counts with $n \ge 30$ . That rule of thumb belongs to sample means. For proportions, check $np \ge 10$ and $n(1-p) \ge 10$ .
Swapping the conditions. The 10% condition justifies the standard deviation formula; Large Counts justifies the normal shape. Name the right one for each job.
Thinking a bigger sample shifts the center. Larger $n$ reduces variability only; the center stays at $p$ . Bias comes from how the sample is chosen, not its size.
Forgetting the square root. Doubling $n$ divides $\sigma_{\hat{p}}$ by $\sqrt{2}$ , not by 2.
Describing without all three features. "Describe the sampling distribution" means center, spread and shape, each with its justification.

When the AP exam uses this

The sampling distribution of

\hat{p}

is topic 3.2, and

\mu_{\hat{p}}

and

\sigma_{\hat{p}}

are on the AP Statistics formula sheet. It underpins confidence intervals and significance tests for proportions, so expect questions that ask you to describe it, check its conditions, or calculate the probability of a sample result.

Sampling Distribution of p-hat Simulator

How to use the simulator

The formulas

Worked example

Common mistakes on the AP exam

When the AP exam uses this

More AP Statistics simulators