# Sampling Distributions for the Difference Between Sample Proportions

> AP Statistics · AP Stats 2024-2026
> Source: https://www.owlsprep.com/study/ap-statistics-u12-sampling-distributions-for-the-difference/

This module covers properties of sampling distributions for the difference of two independent sample proportions, including required conditions, parameter calculations, and normal approximation for probability estimation.

**Prerequisites:** [Single sample proportion sampling distributions](https://www.owlsprep.com/study/ap-statistics-u12-single-sample-proportion-sampling-distributions/); [Normal distribution probability calculations](https://www.owlsprep.com/study/ap-statistics-unit-5-normal-probability/)

## Learning objectives

- Define the sampling distribution of $\hat{p}_1 - \hat{p}_2$ for two independent populations
- Verify the three required conditions for normality of the difference distribution
- Calculate the exact mean and standard deviation of the difference in sample proportions
- Estimate probabilities for values of $\hat{p}_1 - \hat{p}_2$ using normal approximation

## Core Definition and Key Parameters

When you take independent random samples from two separate populations, the difference between their sample proportions will vary across repeated sampling. This set of all possible differences forms a unique sampling distribution with predictable, fixed parameters.

**Sampling Distribution of $\hat{p}_1 - \hat{p}_2$** — The full set of values for the difference between sample proportions from two independent groups, calculated across all possible random samples of size $n_1$ and $n_2$

*Notation:* $\hat{p}_1 - \hat{p}_2$

*Example:* Comparing the proportion of 18-24 year old voters who support a policy to the proportion of 25-40 year old voters who support the same policy

$$\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$$

The mean of the difference distribution is exactly equal to the difference between the two population proportions, no approximation required. The standard deviation uses the property that variances of independent variables add, even for differences.

$$\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$$

**Worked example:** Population 1 has true proportion $p_1 = 0.6$, sample size $n_1 = 100$. Population 2 has true proportion $p_2 = 0.45$, sample size $n_2 = 80$. Calculate the mean and standard deviation of the sampling distribution of $\hat{p}_1 - \hat{p}_2$.

1. Step 1: Calculate the mean of the distribution directly using the population proportions
2. $$\mu_{\hat{p}_1 - \hat{p}_2} = 0.6 - 0.45 = 0.15$$
3. Step 2: Compute the individual variances for each sample proportion first
4. $$\text{Var}(\hat{p}_1) = \frac{0.6(0.4)}{100} = 0.0024, \quad \text{Var}(\hat{p}_2) = \frac{0.45(0.55)}{80} = 0.00309$$
5. Step 3: Sum the variances and take the square root to get the combined standard deviation
6. $$\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{0.0024 + 0.00309} \approx 0.074$$

## Conditions for Normal Approximation

To treat the sampling distribution of $\hat{p}_1 - \hat{p}_2$ as approximately normal, you must verify three non-negotiable conditions for AP exam full credit.

- **Random**: Both samples are selected independently via simple random sampling from their respective populations
- **Independent (10% Condition)**: Each sample size is no larger than 10% of its parent population when sampling without replacement
- **Normal (Large Counts)**: For both groups, the expected number of successes and failures is at least 10

> **Full Credit Requirement**
>
> If any single condition fails, the normal approximation is invalid, and you cannot use z-scores to calculate probabilities for the difference distribution.

**Worked example:** Check if the sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal given $n_1=50$, $p_1=0.2$, $n_2=40$, $p_2=0.3$.

1. Step 1: Confirm both samples are independent random samples as stated in the problem context
2. Step 2: Verify 10% condition: Both populations are at least 500 and 400 people respectively, which is standard for real survey contexts
3. Step 3: Check all four large count products
4. $$n_1p_1 = 10, n_1(1-p_1) = 40, n_2p_2 =12, n_2(1-p_2)=28$$
5. All values are ≥10, so the normal approximation is fully valid.

**Exam command terms**

AP exam FRQs use very specific language for this topic, and you will lose points if you skip required steps:

- **Show that the sampling distribution is approximately normal** — You must explicitly state and verify all three conditions, no partial credit for skipping any check *(You cannot just write 'normal' without proving large counts are satisfied)*

## Probability Calculations for $\hat{p}_1 - \hat{p}_2$

Once you have confirmed the distribution is approximately normal, you can use standard z-score calculations to find the probability of observing a difference as extreme or more extreme than a given value.

$$z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}}$$

> **Easy Exam Memory Hook**
>
> RIN: Random, Independent, Normal. Recite this before every two-proportion problem to make sure you never miss a condition check.

**Worked example:** Using the earlier distribution with mean 0.15 and standard deviation 0.074, calculate the probability that $\hat{p}_1 - \hat{p}_2 > 0.2$.

1. Step 1: Plug the observed difference, mean, and standard deviation into the z-score formula
2. $$z = \frac{0.2 - 0.15}{0.074} \approx 0.676$$
3. Step 2: Use a standard normal table or calculator to find the area to the right of z=0.676
4. Step 3: Final probability ≈ 0.249, meaning roughly 25% of all possible sample differences will exceed 0.2

## Common pitfalls

- **Wrong:** Adding standard deviations directly instead of adding variances then taking the square root
  - Why it fails: Variances of independent variables add, but standard deviations do not, so this will produce an overestimated combined SD
  - Correct: Calculate variance for each sample proportion separately, sum the two variances, then take the square root to get the combined standard deviation
- **Wrong:** Using a pooled proportion to calculate the SD of the sampling distribution
  - Why it fails: Pooling is only required for two-proportion hypothesis tests, not for describing the theoretical sampling distribution
  - Correct: Use the individual known population proportions $p_1$ and $p_2$ in the SD formula for sampling distribution calculations
- **Wrong:** Checking large counts only for the combined total sample instead of each individual group
  - Why it fails: Normality requires both individual single-proportion sampling distributions to be approximately normal, not just the combined difference
  - Correct: Verify all four product terms $n_1p_1$, $n_1(1-p_1)$, $n_2p_2$, $n_2(1-p_2)$ are ≥10
- **Wrong:** Forgetting to confirm the 10% independence condition when sampling without replacement
  - Why it fails: If sample size exceeds 10% of the parent population, observations are not independent, and the SD formula will be inaccurate
  - Correct: Explicitly state that both samples are less than 10% of their respective populations for full exam credit
- **Wrong:** Using the combined total sample size $n_1 + n_2$ in the SD denominator
  - Why it fails: The two samples are fully independent, so their sample sizes cannot be merged for variance calculations
  - Correct: Use $n_1$ only for group 1 variance, $n_2$ only for group 2 variance

## Cheatsheet

| Parameter | Formula | Required Condition |
| --- | --- | --- |
| Mean of $\hat{p}_1 - \hat{p}_2$ | $p_1 - p_2$ | Independent random samples |
| Standard Deviation of $\hat{p}_1 - \hat{p}_2$ | $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$ | 10% condition satisfied for both samples |
| Z-score for Probability | $z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sigma_{\hat{p}_1 - \hat{p}_2}}$ | Large counts condition satisfied for both samples |

## What's next

Mastering sampling distributions for the difference in sample proportions is the critical foundational step before moving to formal inference for two proportions. You will apply these exact properties to construct confidence intervals for the difference between two population proportions, which lets you estimate how large a gap exists between two groups with a specified level of confidence. You will also use the same core framework to run two-proportion z hypothesis tests, to evaluate if there is statistically significant evidence of a difference between two population proportions. This topic is heavily weighted on the AP Stats exam, appearing in roughly 15% of all FRQ points across recent administrations, so fluency with condition checks and parameter calculations will directly boost your score on multi-part inference questions.

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