Case 1 — Independent t-test (Two Groups)
Scenario: A teacher wants to compare math test scores between students taught with traditional lectures and those taught with interactive software.
Question: Are the two teaching methods different in average test score?
Design/Test: Independent-samples t-test.
Worked Example:
- Group A (Lecture): mean = 78, SD = 10, n = 20
- Group B (Software): mean = 85, SD = 12, n = 20
Formula:
$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}}$$
In words:
$$t = \frac{\text{mean}_1 - \text{mean}_2}{\sqrt{\tfrac{\text{variance}_1}{n_1} + \tfrac{\text{variance}_2}{n_2}}}$$
Plugging in values:
$$t = \frac{78 - 85}{\sqrt{\tfrac{100}{20} + \tfrac{144}{20}}} = \frac{-7}{\sqrt{5 + 7.2}} = \frac{-7}{\sqrt{12.2}} = \frac{-7}{3.49} = -2.01$$
Degrees of freedom = 38.
Case 2 — Paired t-test (Before and After)
Scenario: Students take a memory test before and after a week of practice.
Question: Did memory scores improve after training?
Design/Test: Paired-samples t-test.
Worked Example:
Differences (After – Before): 2, 4, 3, 5, 6
- Mean difference:
$$\bar{D} = \frac{2+4+3+5+6}{5} = 4$$ - Standard deviation of differences: $$s_D = 1.58$$
Formula:
$$t = \frac{\bar{D}}{s_D / \sqrt{n}}$$
Plugging in values:
$$t = \frac{4}{1.58/\sqrt{5}} = \frac{4}{0.71} = 5.63$$
Degrees of freedom = 4.
Case 3 — One-way ANOVA (Three Groups)
Scenario: A psychologist tests three methods of stress reduction: meditation, exercise, and music.
Question: Do the methods differ in average stress score?
Design/Test: One-way ANOVA.
Worked Example (summary):
- Group means: Meditation = 65, Exercise = 70, Music = 80
- $$SS_{\text{between}} = 300, , df_{\text{between}} = 2, , MS_{\text{between}} = 150$$
- $$SS_{\text{within}} = 200, , df_{\text{within}} = 12, , MS_{\text{within}} = 16.7$$
Formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$
Plugging in values:
$$F = \frac{150}{16.7} = 9.0$$
df = (2, 12).
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