This lecture emphasizes conceptual understanding of the t-test, its logic, and how it fits into the broader structure of statistical reasoning.
The t-test is one of the most widely used statistical tools.
It compares two means and asks: Is the difference between them real, or could it be due to chance?
The t-test is closely related to the z-test.
When the population standard deviation is unknown and the sample size is small, we use t instead of z.
Types of t-Tests
- One-sample t-test: compares a sample mean to a known or hypothesized population mean.
- Independent-samples t-test: compares means from two separate groups.
- Paired-samples t-test: compares two scores from the same group (before vs. after).
Symbolic Formulas
One-sample t-test
$$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$$
Independent-samples t-test
$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}}$$
Paired-samples t-test
$$t = \frac{\bar{D}}{s_D / \sqrt{n}}$$
Degrees of Freedom
- One-sample: $$df = n - 1$$
- Independent-samples: $$df = n_1 + n_2 - 2$$
- Paired-samples: $$df = n - 1$$
Example (Independent t-Test)
Two groups of students try different study methods:
- Group A: \(n = 10\), mean = 80, SD = 10
- Group B: \(n = 10\), mean = 90, SD = 10
$$t = \frac{80 - 90}{\sqrt{\tfrac{10^2}{10} + \tfrac{10^2}{10}}} = \frac{-10}{\sqrt{10 + 10}} = \frac{-10}{\sqrt{20}} = \frac{-10}{4.47} = -2.24$$
Degrees of freedom = 18.
Compare this t-value to the critical value in the t-table at \(df = 18\).
Example (Paired t-Test)
Students take a test before and after tutoring.
Differences (After − Before): 4, 6, 5, 3, 2.
Mean difference:
$$\bar{D} = \frac{4 + 6 + 5 + 3 + 2}{5} = 4$$
Standard deviation of differences:
$$s_D = 1.58$$
$$t = \frac{4}{1.58 / \sqrt{5}} = \frac{4}{0.71} = 5.63$$
Degrees of freedom = 4.
This large t-value indicates strong evidence of improvement.
Definition
- Independent t-test: compares two separate groups.
- Paired t-test: compares the same group measured twice.
- Degrees of freedom (df): number of independent pieces of information.
Visuals
Figure L5.1 — Independent t-Test. Bar graph of two groups (A and B) with means and SEM error bars.
Figure L5.2 — Paired t-Test. Line plot showing before vs. after scores for each student.
Figure L5.3 — t vs. z Distribution. Overlay of the normal (z) curve and t curves with df = 5 and 20.
Why This Matters
The t-test is the workhorse of statistics.
It forms the foundation for many other methods (ANOVA, regression, mixed models).
Understanding t means understanding how we compare signal (mean difference) to noise (variability).
Practice self-test quiz
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