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📚 Mostly Harmless Statistics
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1 Introduction to Data2 Organizing Data3 Descriptive Statistics4 Probability5 Discrete Probability Distributions6 Continuous Probability Distributions7 Confidence Intervals for One Population8 Hypothesis Tests for One Population9 Hypothesis Tests and Confidence Intervals for Two Populations10 Chi-Square Tests11 Analysis of Variance12 Correlation and Regression13 Nonparametric Tests

Chapter 9: Hypothesis Tests and Confidence Intervals for Two Populations

Dependent samples or matched pairs, occur when the subjects are paired up, or matched in some way. Most often, this model is characterized by selection of a random sample where each member is observed under two different conditions, before/after some experiment, or subjects that are similar (matched) to each other are studied under two different conditions.

There are 3 types of hypothesis tests for comparing two dependent population means µ1 and µ2, where, µD is the expected difference of the matched pairs.

Table of the three hypothesis setups for a paired-means t-test: two-tailed with H0: μD = 0 and H1: μD ≠ 0, right-tailed with H1: μD > 0, and left-tailed with H1: μD < 0, each above a normal curve shaded in both tails, the right tail, or the left tail respectively.
Figure 9-1

Note: If each pair were equal to one another then the mean of the differences would be zero. We could also use this model to test with a magnitude of a difference, but we rarely cover that scenario, therefore we are usually test against the difference of zero.

The t-test for dependent samples is a statistical test for comparing the means from two dependent populations (or the difference between the means from two populations). The t-test is used when the differences are normally distributed. The samples also must be dependent.

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University), hosted on LibreTexts (stats.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.