# comparing normal populations

## Summary

A common task in statistics is to determine whether two normally distributed populations have the same mean. The appropriate test can depend on factors such as the sample size and whether the populations are paired or independent.

## Context

This concept has the prerequisites:

- Gaussian distribution
- statistical hypothesis testing (Comparing populations involves hypothesis testing.)
- Student-t distribution (The t test uses the student-t distribution.)
- expectation and variance (The tests require computing variances.)

## Goals

- Know what the z-test and t-test are, and how to compute the p-values.
- When would you use one rather than the other?

- What are paired z-tests and t-tests? Why might they be preferable to ones assuming independent populations?

- What is the difference between a one-sided and a two-sided test, and why would you use one over the other?

- Be aware that the assumption that the populations are normally distributed is a strong one, and often not justified.

## Core resources (we're sorry, we haven't finished tracking down resources for this concept yet)

## Supplemental resources (the following are optional, but you may find them useful)

## -Paid-

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

- Section 9.5, "The t test," pages 576-585
- Section 9.6, "Comparing the means of two normal distributions," pages 587-596

→ All of Statistics

A very concise introductory statistics textbook.

- Section 10.1, "The Wald test," pages 152-156
- Section 10.10.2, "The t-test," page 170

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

- Section 11.2, "Comparing two independent samples," up through 11.2.1, "Methods based on the normal distribution," pages 421-435
- Section 11.3, "Comparing paired samples," up through 11.3.1, "Methods based on the normal distribution," pages 444-447

## See also

-No Additional Notes-