Chapter 11 Problem Set – Paired-Samples t Test

Where this problem set fits in the story

This problem set extends the psych_ch11_paired_t chapter on within-subjects designs and the paired-samples t test.

Chapter 11 introduces designs where each participant serves as their own control (for example, pre–post designs). Track C adds a set of worked, fully reproducible examples that show how to:

Simulate pre–post data for different research scenarios.
Run paired-samples t tests using the PyStatsV1 helpers.
Interpret the resulting means, t values, p values, and effect sizes.

Learning goals

By the end of this problem set, you should be able to:

Recognize when a paired-samples t test is appropriate.
Explain how the test is based on difference scores (post – pre).
Describe how sample size and effect size jointly determine power.
Use the PyStatsV1 solution code as a template for your own pre–post data.

How to run the worked solutions

From the project root, run:

make psych-ch11-problems

This wraps:

python -m scripts.psych_ch11_problem_set

and regenerates all synthetic datasets, the summary CSV, and the group means plot.

Conceptual warm-up

In a paired design, each participant is measured twice (or more), so we compare them to themselves.
This reduces error by controlling for stable individual differences.
The paired-samples t test is equivalent to running a one-sample t test on the difference scores (post – pre).
Effect sizes (Cohen’s d) are typically calculated using the variability of the difference scores.

Applied exercises

Each exercise in this problem set corresponds to a realistic research scenario:

Exercise 1 – Moderate improvement (n = 40)

A typical lab-based intervention with a medium-sized effect. The paired t test should be clearly significant.
Exercise 2 – Small / ambiguous effect (n = 30)

A small effect with modest sample size. The t test will often be non-significant, highlighting how hard it is to detect small effects without sufficient power.
Exercise 3 – Strong improvement (n = 25)

A large effect with a smaller sample. Despite the lower n, the effect is strong enough to be detected with high confidence.

PyStatsV1 Lab: Paired-samples t problem set in action

The solution script scripts.psych_ch11_problem_set shows how to:

Generate pre–post data for each scenario (exercise label, group means, effect size, etc.).
Run the paired t tests using scripts.psych_ch11_paired_t.run_paired_t().
Save one CSV per exercise plus a summary CSV with the key statistics side-by-side.
Produce a simple bar plot comparing pre and post means for each exercise.

Running the Chapter 11 problem set lab

After running:

make psych-ch11-problems

you should see the following outputs:

data/synthetic/psych_ch11_exercise1.csv – Moderate-effect pre–post study (n = 40).
data/synthetic/psych_ch11_exercise2.csv – Small-effect pre–post study (n = 30).
data/synthetic/psych_ch11_exercise3.csv – Strong-effect pre–post study (n = 25).
outputs/track_c/ch11_problem_set_results.csv – Summary table of the three paired t tests.
outputs/track_c/ch11_problem_set_means.png – Group means plot (pre vs post for each exercise).

Conceptual summary

Paired-samples t tests compare mean differences within participants rather than between independent groups.
They are often more powerful than independent-samples designs, because each person serves as their own control.
Power depends on the magnitude of the true effect, the variability of the difference scores, and the sample size.
PyStatsV1 solution scripts give you a reusable pre–post template: drop in your own dataset, rerun the analysis, and verify the results using transparent, version-controlled code.