T-Tests

What You'll Learn

• One-sample t-test
• Two-sample t-test
• Paired t-test
• When to use each

What is a T-Test?

Purpose: Test if means are significantly different

Three types:

1. One-sample: Compare sample to known value
2. Two-sample: Compare two groups
3. Paired: Compare before/after

One-Sample T-Test

Question: Is sample mean different from known value?

Example: Average height is 170cm. Is your sample different?

Hypotheses:

• H0: μ = 170 (null)
• H1: μ ≠ 170 (alternative)

Excel: =T.TEST(range, 170, 2, 1) Python: scipy.stats.ttest_1samp(data, 170)

Two-Sample T-Test

Question: Are two group means different?

Example: Do men and women have different average salaries?

Types:

• Independent: Different groups
• Equal variance assumed
• Unequal variance (Welch's)

Excel: =T.TEST(group1, group2, 2, 2) Python: scipy.stats.ttest_ind(group1, group2)

Paired T-Test

Question: Is there difference before vs after?

Example: Weight before vs after diet (same people)

Key: Same subjects measured twice!

Excel: =T.TEST(before, after, 2, 1) Python: scipy.stats.ttest_rel(before, after)

P-Value

What it means: Probability of seeing this result if null hypothesis is true

Decision rule:

• p < 0.05: Reject null (significant!)
• p ≥ 0.05: Fail to reject null

Example: p = 0.03 → Reject null (groups ARE different)

Assumptions

T-tests assume:

• Normal distribution (or large sample)
• Independent observations
• For two-sample: Equal variances (or use Welch's)

Check before testing!

Effect Size

Statistical significance ≠ Practical significance

Cohen's d: d = (Mean1 - Mean2) / Pooled SD

Interpretation:

• d = 0.2: Small
• d = 0.5: Medium
• d = 0.8: Large

Practice Exercise

Group A: 80, 85, 90, 95, 100 Group B: 70, 75, 80, 85, 90

Conduct two-sample t-test. Are they different?

Next Steps

Learn about Chi-Square Tests!

Tip: Always check assumptions before running t-test!