Psychological Science & Statistics – Chapter 9

The One-Sample t-Test and Confidence Intervals

In Chapter 8 you learned the logic of hypothesis testing using a simulation- based one-sample t-test. You saw how to:

state a null hypothesis,
compute a sample t-statistic,
build a null distribution of t-values under \(H_0\), and
approximate a p-value by checking how extreme \(t_\text{obs}\) is.

In this chapter we take the next step: the analytic (formula-based) one-sample t-test, and the closely related 95% confidence interval (CI) for a population mean.

This chapter connects the simulation-based intuition from Chapter 8 to the classical t-test formulas used throughout psychological science.

When to Use a One-Sample t-Test

Use a one-sample t-test when you want to compare a sample mean to a known or hypothesized population mean:

Does the population of students represented by this class have a mean stress score of 20?

Mathematically, the hypotheses are:

\[\begin{split}H_0: \mu = \mu_0 \\ H_1: \mu \ne \mu_0\end{split}\]

Why We Use t (Instead of z)

When the population standard deviation \(\sigma\) is unknown—as is almost always the case in psychology—we use the sample standard deviation \(s\). Substituting \(s\) introduces extra uncertainty, leading to a t distribution instead of a normal distribution.

The estimated standard error is:

\[SE = \frac{s}{\sqrt{n}}\]

The t-statistic is:

\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\]

Confidence Intervals

A 95% confidence interval around a mean provides a range of plausible population values:

\[\bar{x} \pm t^* \frac{s}{\sqrt{n}}\]

where \(t^*\) is the critical value from the t distribution with \(n-1\) degrees of freedom.

Interpretation:

If we repeated the study many times, 95% of the resulting CIs would contain the true population mean.

Connecting Confidence Intervals and Hypothesis Tests

A powerful insight:

If the 95% CI **does not include* \(\mu_0\), the two-sided t-test will reject \(H_0\) at \(\alpha = 0.05\).*

If the CI **includes* \(\mu_0\), the t-test will fail to reject \(H_0\).*

PyStatsV1 Lab: A One-Sample t-Test With Confidence Intervals

In this lab, you will:

Load a synthetic population of stress scores (same population used in Ch. 7–8).
Draw a random sample of size \(n\).
Compute:
- sample mean \(\bar{x}\),
- sample SD \(s\),
- standard error \(SE\),
- t-statistic,
- degrees of freedom \(df = n-1\),
- p-value (analytic),
- 95% confidence interval.
Compare:
- the analytic t-test,
- the 95% CI,
- and (optionally) the simulation results from Chapter 8.

All code for this chapter lives in:

scripts/psych_ch9_one_sample_ci.py

Running the Lab Script

From the project root:

python -m scripts.psych_ch9_one_sample_ci

If you have a Makefile target:

make psych-ch09

Expected Console Output

Your numbers will vary due to randomness, but output will look similar to:

Loaded synthetic population with 50000 individuals
Population mean stress_score = 19.98
Population SD   stress_score = 9.94

Drawn sample size n = 25
Sample mean = 22.10
Sample SD   = 10.44
SE          = 2.09
t statistic = 1.01
df          = 24

Analytic two-sided p-value = 0.323
95% CI = [17.80, 26.40]

Interpreting Your Output

Focus on:

the t-statistic: How many standard errors your mean is from the null;
the p-value: Is the result “rare” under \(H_0\)?
the CI: Does the interval contain the hypothesized value \(\mu_0\)?

Your Turn: Practice

Change the null value \(\mu_0\) and observe how the t-statistic changes.
Change the sample size \(n\) and see how the CI narrows or widens.
Run the analysis multiple times to see sampling variability.

Summary

In this chapter you learned:

the formula-based one-sample t-test,
how to compute a 95% confidence interval,
the connection between CIs and hypothesis tests.

In the next chapter (Chapter 10) we extend this logic to comparing two independent groups using the independent-samples t-test.