The sampling distribution of a sample proportion

Warm-up

Three quick questions from earlier lessons.

Discover

In a city 40% of adults cycle. In random samples of 100, around which value will p̂ be centered?

0.40
40

Key idea

Mean, spread and shape of p̂.

For random samples of size

n

from a population with proportion

p

:

\mu_{\hat{p}} = p

and

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

(if

n

is at most 10% of the population). The shape is approximately normal when

np \ge 10

and

n(1 - p) \ge 10

.

Worked example

$p = 0.4$ , $n = 100$ .

\mu_{\hat{p}} = 0.4

Center.

\sigma_{\hat{p}} = \sqrt{0.0024} \approx 0.049

√(0.4 × 0.6/100).

Approximately normal

40 ≥ 10 and 60 ≥ 10.

Your turn, guided

Your turn: $p = 0.5$ , $n = 100$ . Find $\sigma_{\hat{p}}$ .

0.05

√(0.25/100).

Practice ★

The population proportion is p = 8/10. Find the standard deviation of p̂ for random samples of size 83.
The population proportion is p = 2/10. Find the standard deviation of p̂ for random samples of size 227.

The population proportion is p = 5/10. What is the mean of the sampling distribution of p̂ for samples of size 351?

Watch out!

Practice ★★

The population proportion is p = 1/8. What is the smallest sample size n for which np ≥ 10 and n(1 − p) ≥ 10?
The population proportion is p = 1/2. What is the smallest sample size n for which np ≥ 10 and n(1 − p) ≥ 10?

Boss problem. In a population, p = 8/10. Random samples of 100 are taken.

Exit ticket

Three questions to check you have got it.

The population proportion is p = 8/10. Find the standard deviation of p̂ for random samples of size 83.
The population proportion is p = 2/10. Find the standard deviation of p̂ for random samples of size 227.
The population proportion is p = 4/10. Find the standard deviation of p̂ for random samples of size 394.

Summary card

Sampling distribution of p̂

$\mu = p$ , $\sigma = \sqrt{p(1 - p)/n}$ ; normal if $np, n(1 - p) \ge 10$ .

Watch out: proportion, not count.

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