Hypothesis Testing

Hypothesis testing lets you make statistically grounded decisions from sample data — is a difference real, or just noise? R has built-in functions for every classic test.

1. The Core Idea — Null and Alternative Hypotheses

Every test compares a null hypothesis (H₀) — "no effect / no difference" — against an alternative hypothesis (H₁). The p-value tells you how surprising your observed data would be if H₀ were actually true.

2. T-Tests — Comparing Means

One-Sample T-Test

scores <- c(72, 85, 90, 68, 75, 88, 95, 70)
t.test(scores, mu = 80)   # is the mean significantly different from 80?

Two-Sample T-Test (Independent Groups)

group_a <- c(85, 90, 78, 92, 88)
group_b <- c(70, 75, 68, 80, 72)

t.test(group_a, group_b)
# or with a data frame:
t.test(salary ~ gender, data = df)

Paired T-Test

before <- c(80, 85, 78, 90, 82)
after  <- c(88, 91, 85, 94, 89)
t.test(before, after, paired = TRUE)

3. Reading t.test() Output

4. ANOVA — Comparing 3+ Group Means

anova_model <- aov(salary ~ department, data = df)
summary(anova_model)

# Post-hoc test if ANOVA is significant
TukeyHSD(anova_model)

5. Chi-Square Test — Categorical Relationships

# Test of independence between two categorical variables
contingency_table <- table(df$department, df$gender)
chisq.test(contingency_table)

# Goodness-of-fit test
observed <- c(50, 30, 20)
expected_probs <- c(0.5, 0.3, 0.2)
chisq.test(observed, p = expected_probs)

6. Choosing the Right Test

7. Checking Test Assumptions

shapiro.test(scores)          # test for normality
var.test(group_a, group_b)   # test for equal variances (before t.test)

8. Production-Ready Checklist

• ✅ Know your hypotheses before testing — H0 and H1, clearly stated.
• ✅ Choose the correct test — based on data type and number of groups.
• ✅ Check assumptions — normality, equal variance — before trusting results.
• ✅ Interpret p-values carefully — statistical significance ≠ practical importance.