Mathfolis

Sampling Distribution for a Difference of Proportions

Unit 5 · Sampling Distributions

What AP Stats asks here

When comparing proportions from two independent samples, the sampling distribution centers at p1−p2p_1 - p_2 and inherits variability from both samples. Variances add — even though we are subtracting — because Var(−Y) = Var(Y). The recurring trap: a student who subtracts the variances or the SEs gets a smaller-than-real estimate of uncertainty.

Sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2

Center
μp^1−p^2=p1−p2\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2
Standard error
σp^1−p^2=p1(1−p1)n1+p2(1−p2)n2\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}

Conditions

Independence
samples drawn independently\text{samples drawn independently}
Large counts (both)
nipi≥10 and ni(1−pi)≥10n_i p_i \ge 10 \text{ and } n_i (1 - p_i) \ge 10
10% condition (both)
ni<0.10Nin_i < 0.10 N_i
AP Tip: Estimating a difference is inherently noisier than estimating a single proportion — to match precision, each sample needs roughly twice the size of a one-sample design.
Caution: Variances add for differences of independent variables. If two SEs were allowed to subtract, two equal-SE samples would imply zero combined uncertainty — clearly absurd.
Type 1

Describe the sampling distribution

Verify the conditions and compute p1(1−p1)/n1+p2(1−p2)/n2\sqrt{p_1(1-p_1)/n_1 + p_2(1-p_2)/n_2} for the SE.

Example 1
True proportions p1=0.50p_1 = 0.50, p2=0.40p_2 = 0.40; sample sizes n1=400n_1 = 400, n2=500n_2 = 500. Which best describes the sampling distribution of p^1−p^2\hat{p}_1 - \hat{p}_2? (A) N(0.10,0.0332)N(0.10, 0.0332) (B) N(0.10,0.0008)N(0.10, 0.0008) (C) N(0.10,0.001)N(0.10, 0.001) (D) N(0.45,0.05)N(0.45, 0.05)

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Type 2

Variance still adds on subtraction

Var(−Y) = Var(Y), so Var(X − Y) = Var(X) + Var(Y) for independent X and Y.

Example 2
A student writes 'SE of p^1−p^2\hat{p}_1 - \hat{p}_2 equals SE(p^1\hat{p}_1) − SE(p^2\hat{p}_2) because we are subtracting.' Which critique is correct? (A) The student is correct. (B) Variances still add on subtraction because Var(−Y) = Var(Y); combine variances then square-root, never subtract SEs. (C) The student is correct only when the SEs are equal. (D) The formula does not apply because the variables are independent.

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Type 3

Compute probabilities

Standardize p^1−p^2\hat{p}_1 - \hat{p}_2 with the combined SE and use the z-table.

Example 3
For p^1−p^2∼N(0.10,0.0332)\hat{p}_1 - \hat{p}_2 \sim N(0.10, 0.0332), what is P(p^1−p^2>0.15)P(\hat{p}_1 - \hat{p}_2 > 0.15)? (A) ≈ 0.066 (B) ≈ 0.50 (C) ≈ 0.95 (D) ≈ 0.10

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