The Z-Test › Summary

This chapter covered the main aspects of the z-test:

When to Use Z-Tests

• Known population standard deviation (rare for means)
• Testing proportions (most common application)
• Very large samples where t ≈ z

The Core Mathematics

• Test statistic: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$
• Under $H_0$, z follows a standard normal distribution
• The standard error $\sigma/\sqrt{n}$ quantifies how much sample means vary

Test Procedures

• One-sample z-test: Test whether a mean differs from a specific value
• Two-sample z-test: Test whether two means differ from each other
• Proportion z-test: Test hypotheses about population proportions
• Two-proportion z-test: Compare proportions between groups (A/B testing)

Key Points

• Always verify normal approximation conditions for proportion tests
• Confidence intervals provide the same information as hypothesis tests
• The z-test is foundational for understanding more complex tests