Difference of Two Proportions

Comparing two groups

• Often we compare two proportions — say, support in group $1$ vs group $2$.
• The statistic is the difference of two sample proportions 两个样本比例之差 $\hat{p}_1 - \hat{p}_2$.
• Its sampling distribution is centered at the true difference: $\mu = p_1 - p_2$.
• It lets us judge whether two groups genuinely differ.

Combining the spreads

• The standard deviation combines both samples' variability:
• $$\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$$
• Note the $+$: variances add (echoing the "combining variables" rule) even for a difference.
• Each group contributes its own $\frac{p(1-p)}{n}$ term.

The distribution of p̂₁ − p̂₂: centres subtract and variances add

Check both samples

• Apply the large counts condition to each group: $n_1p_1, n_1(1-p_1), n_2p_2, n_2(1-p_2)$ all $\ge 10$.
• Require independence: the two samples are independent of each other.
• Also the 10% condition for each if sampling without replacement.
• Only then is the normal model for $\hat{p}_1 - \hat{p}_2$ valid.

Using the normal model

• With conditions met, $\hat{p}_1 - \hat{p}_2$ is approximately normal.
• Standardize with a $z$-score: (observed difference $-$ true difference) $\div$ the combined SD.
• Then read probabilities from the standard normal, as before.
• This is the foundation for two-proportion tests and intervals (Unit 6).

Write from a difference of 0.25

• One sample proportion is 0.40.
• The other is 0.15. The lesson difference uses 0.1 and 0.07.