Statistics for Business: Decision Making and Analysis, 3rd Edition

... confidence interval that includes zero bounds the magnitude of the difference but does not identify which population has the larger parameter.

Confidence interval for p₁ − p₂.

The 100(1 − α)% confidence interval for p₁ − p₂ is

$${\widehat{p}}_1-{\widehat{p}}_2-z_{\alpha /2} \text{se}\left({\widehat{p}}_1-{\widehat{p}}_2\right) \text{to} {\widehat{p}}_1-{\widehat{p}}_2+z_{\alpha /2} \text{se}\left({\widehat{p}}_1-{\widehat{p}}_2\right)$$

where

$$\text{se}\left({\widehat{p}}_1-{\widehat{p}}_2\right)=\sqrt{\frac{{\widehat{p}}_1\left(1-{\widehat{p}}_1\right)}{n_1}+\frac{{\widehat{p}}_2\left(1-{\widehat{p}}_2\right)}{n_2}}$$

Checklist

(same as for the two-sample z-test for proportions)

• ✓ No obvious lurking variable.

• ✓ SRS condition.

• ✓ Sample size condition.

What Do You Think?

1. An advertiser compared the brand loyalty of those owning domestic cars and those owning foreign cars. It sampled 200 owners of Ford, General Motors, and Chrysler vehicles and 150 owners of European brands Audi, BMW, and Mercedes. Among owners of domestic brands, 146 said ...