Difference of Proportions

Start with two distinct populations. Population 1 has a true proportion $p_1$ of individuals with some characteristic, and Population 2 has a true proportion $p_2$. For example, $p_1$ might be the proportion of left-handed students at School A, and $p_2$ the proportion at School B. These are fixed but usually unknown parameters.

Take a simple random sample of size $n_1$ from Population 1 and an independent simple random sample of size $n_2$ from Population 2. Compute the sample proportions $\hat{p}_1 = x_1 / n_1$ and $\hat{p}_2 = x_2 / n_2$, where $x_1$ and $x_2$ are the counts of successes in each sample.

Because $E[\hat{p}_1] = p_1$ and $E[\hat{p}_2] = p_2$, the expected value of the difference is $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$. The statistic is unbiased: on average across all possible samples, the difference of sample proportions equals the true difference of population proportions.

Since the two samples are independent, the variances add. The variance of $\hat{p}_1$ is $p_1(1-p_1)/n_1$ and the variance of $\hat{p}_2$ is $p_2(1-p_2)/n_2$. Therefore, the standard deviation of the sampling distribution (called the standard error) is $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}$.

The sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal provided: (1) Both samples are randomly selected and independent of each other. (2) The success-failure condition holds for each sample: $n_1 p_1 \geq 10$, $n_1(1-p_1) \geq 10$, $n_2 p_2 \geq 10$, and $n_2(1-p_2) \geq 10$. (3) Each sample is less than 10% of its population (the 10% condition) to ensure approximate independence. When these conditions hold, we can apply the Central Limit Theorem.

Once we know the distribution is approximately $N(p_1 - p_2,\; \sigma_{\hat{p}_1 - \hat{p}_2}^2)$, we standardize using a $z$-score: $z = \dfrac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}}$. This $z$-score follows the standard normal distribution $N(0,1)$, enabling us to look up probabilities in a standard normal table or use technology.

• \sigma_{\hat{p}_1 - \hat{p}_2} — Standard error of the difference (dimensionless): Measures the typical spread of the statistic $\hat{p}_1 - \hat{p}_2$ across all possible pairs of samples.
• p_1 — Population proportion 1 (dimensionless (0 to 1)): The true proportion of successes in Population 1; a fixed parameter.
• p_2 — Population proportion 2 (dimensionless (0 to 1)): The true proportion of successes in Population 2; a fixed parameter.
• n_1 — Sample size from Population 1 (count): Number of individuals randomly sampled from Population 1.
• n_2 — Sample size from Population 2 (count): Number of individuals randomly sampled from Population 2.

Problem. A survey finds that 55% of adults in City A support a new transit policy ($p_1 = 0.55$) and 48% in City B support it ($p_2 = 0.48$). Independent random samples of $n_1 = 200$ from City A and $n_2 = 150$ from City B are taken. (a) Describe the sampling distribution of $\hat{p}_1 - \hat{p}_2$. (b) What is the probability that the sample from City A shows a proportion at least 0.10 higher than the sample from City B, i.e., $P(\hat{p}_1 - \hat{p}_2 \geq 0.10)$?

Answer. The sampling distribution of the difference is approximately Normal with mean 0.07 and standard error ≈ 0.0539. The probability that the City A sample proportion exceeds the City B sample proportion by at least 0.10 is approximately 28.9%.

• Sampling Distribution of a Proportion — The difference of proportions builds directly on how a single sample proportion $\hat{p}$ is distributed; mastering one makes the other straightforward.
• Central Limit Theorem — The CLT is the theoretical reason why the sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal for large sample sizes.
• Confidence Interval for Difference of Proportions — The sampling distribution derived here is exactly what is used to construct and interpret confidence intervals for $p_1 - p_2$.
• Two-Proportion Z-Test — Hypothesis testing for whether $p_1 = p_2$ relies on the sampling distribution of $\hat{p}_1 - \hat{p}_2$ under the null hypothesis.