The Sampling Distribution of a Sample Proportion

You’ll be able to

State the mean and standard deviation of the sampling distribution of p-hat
Check the conditions for approximate Normality of p-hat
Compute probabilities involving a sample proportion

Center and spread of p-hat

For an SRS of size n from a population with true proportion p, the sample proportion p-hat has mean p (unbiased) and standard deviation sqrt(p(1−p)/n). As with means, the spread shrinks as n grows. This standard deviation drives the width of confidence intervals and the test statistic for proportions in later units.

The Large Counts condition

The sampling distribution of p-hat is approximately Normal only when both the expected number of successes and failures are large enough: the Large Counts condition requires np ≥ 10 and n(1−p) ≥ 10. If either count is below 10, the distribution is too skewed for the Normal approximation, and Normal-based probabilities are unreliable.

Sampling distribution of p-hat

μ_(p-hat) = p · σ_(p-hat) = sqrt( p(1−p) / n ) · Normal if np ≥ 10 and n(1−p) ≥ 10

Also requires the 10% condition (n ≤ 0.10·N) so trials are approximately independent. The standard deviation uses the true p (or p-hat when p is unknown).

Worked example

Suppose 60% of a large population supports a measure (p = 0.6). For an SRS of n = 100, verify Normality and find P(p-hat < 0.5).

1.Check Large Counts: np = 100(0.6) = 60 ≥ 10 and n(1−p) = 100(0.4) = 40 ≥ 10, so p-hat is approximately Normal.
2.Mean and standard deviation: μ_(p-hat) = 0.6; σ_(p-hat) = sqrt(0.6·0.4/100) = sqrt(0.0024) ≈ 0.049.
3.Standardize: z = (0.5 − 0.6)/0.049 ≈ −2.04.
4.P(p-hat < 0.5) = P(Z < −2.04) ≈ 0.021.

Answer: Large Counts holds (60 and 40 are both ≥ 10), so p-hat is approximately Normal with mean 0.6 and standard deviation ≈ 0.049. Then z ≈ −2.04 and P(p-hat < 0.5) ≈ 0.021.

Checkpoint

The true proportion is p = 0.5 and the sample size is n = 400. What is the standard deviation of the sampling distribution of p-hat?

Watch out

The Normal approximation for p-hat is only valid when np ≥ 10 and n(1−p) ≥ 10 (the Large Counts condition). With a small p or small n, one count can fall below 10 and the sampling distribution is skewed — do not use z-based methods there.

Checkpoint

A true proportion is p = 0.04, and a researcher takes an SRS of n = 150. Can the sampling distribution of p-hat be treated as approximately Normal?

Answer the 2 checkpoints as you read.