Hypothesis Testing

HYPOTHESIS TESTING STEPS:
  1. State H₀ (null hypothesis — "no effect" / "no difference")
  2. State H₁ (alternative hypothesis — what you want to show)
  3. Choose significance level α (usually 0.05 = 5%)
  4. Collect data
  5. Compute test statistic
  6. Compute p-value
  7. Decision:
     • p < α → Reject H₀ (evidence for H₁)
     • p ≥ α → Fail to reject H₀ (insufficient evidence)

p-value interpretation:
  p-value = P(observing this data or more extreme | H₀ is true)
  Small p-value → data is unlikely under H₀ → reject H₀

Common thresholds:
  p < 0.001 → Very strong evidence against H₀ (***)
  p < 0.01  → Strong evidence (***)
  p < 0.05  → Significant evidence (*)
  p ≥ 0.05  → Not significant (ns)

# H₀: μ = 70000 (population mean salary = $70,000)
# H₁: μ ≠ 70000 (two-tailed)
set.seed(42)
sample_salaries <- rnorm(30, mean=75000, sd=12000)

result <- t.test(sample_salaries, mu=70000, alternative="two.sided")
print(result)
cat("\n")
cat(sprintf("t-statistic: %.3f\n", result$statistic))
cat(sprintf("p-value:     %.4f\n", result$p.value))
cat(sprintf("95%% CI:      [%.0f, %.0f]\n", result$conf.int[1], result$conf.int[2]))
cat(sprintf("Decision:    %s H₀\n", ifelse(result$p.value < 0.05, "REJECT", "FAIL TO REJECT")))

# One-tailed: H₁: μ > 70000
t.test(sample_salaries, mu=70000, alternative="greater")

# Compare two groups
set.seed(42)
group_A <- rnorm(25, mean=78000, sd=15000)  # Department A
group_B <- rnorm(25, mean=72000, sd=14000)  # Department B

# H₀: μ_A = μ_B (no salary difference between departments)
result <- t.test(group_A, group_B, var.equal=FALSE)  # Welch's t-test
cat(sprintf("Dept A mean: $%.0f\n", mean(group_A)))
cat(sprintf("Dept B mean: $%.0f\n", mean(group_B)))
cat(sprintf("p-value: %.4f → %s\n", result$p.value,
             ifelse(result$p.value < 0.05, "Significant difference!", "No significant difference")))

# Effect size (Cohen's d)
cohen_d <- (mean(group_A) - mean(group_B)) /
           sqrt((var(group_A) + var(group_B)) / 2)
cat(sprintf("Cohen&#039;s d: %.3f (%s effect)\n", cohen_d,
             ifelse(abs(cohen_d) > 0.8, "Large",
             ifelse(abs(cohen_d) > 0.5, "Medium", "Small"))))

# Paired t-test (before/after measurements)
before <- c(72, 68, 75, 80, 65, 70, 78)
after  <- c(78, 74, 80, 85, 72, 76, 83)
paired_result <- t.test(before, after, paired=TRUE, alternative="less")
cat("\nPaired t-test (training effectiveness):\n")
cat(sprintf("Mean improvement: %.1f points\n", mean(after - before)))
cat(sprintf("p-value: %.4f\n", paired_result$p.value))

# ─── CHI-SQUARED TEST (categorical variables) ─────────
# H₀: No relationship between gender and department preference
observed <- matrix(c(45, 30, 25, 35, 40, 30), nrow=2,
                   dimnames=list(c("Male","Female"), c("Tech","Sales","HR")))
chi_result <- chisq.test(observed)
cat(sprintf("Chi-squared: %.3f, df=%d, p=%.4f\n",
             chi_result$statistic, chi_result$parameter, chi_result$p.value))

# ─── ONE-WAY ANOVA (compare 3+ group means) ──────────
set.seed(42)
dept_salaries <- data.frame(
  salary = c(rnorm(20, 85000, 10000), rnorm(20, 72000, 12000), rnorm(20, 78000, 11000)),
  dept   = rep(c("IT", "HR", "Finance"), each=20)
)
anova_result <- aov(salary ~ dept, data=dept_salaries)
summary(anova_result)
# If significant: which groups differ?
TukeyHSD(anova_result)  # Post-hoc pairwise comparisons

# ─── SURVEY ANALYSIS TOOL ────────────────────────────
set.seed(123)
n <- 100
survey <- data.frame(
  gender        = sample(c("Male","Female"), n, replace=TRUE, prob=c(0.55,0.45)),
  dept          = sample(c("IT","HR","Finance","Marketing"), n, replace=TRUE),
  job_satisfact = round(runif(n, 1, 10), 1),
  before_train  = round(runif(n, 60, 80)),
  after_train   = round(runif(n, 65, 92))
)

cat("=== SURVEY ANALYSIS REPORT ===\n\n")

# 1: Is satisfaction different by department?
cat("1. ANOVA: Satisfaction by Department\n")
anova_sat <- aov(job_satisfact ~ dept, data=survey)
p_anova <- summary(anova_sat)[[1]][["Pr(>F)"]][1]
cat(sprintf("   F-test p-value: %.4f — %s\n\n", p_anova,
             ifelse(p_anova < 0.05, "Significant differences!", "No significant differences")))

# 2: Does training improve performance?
cat("2. Paired t-test: Before vs After Training\n")
t_res <- t.test(survey$before_train, survey$after_train, paired=TRUE, alternative="less")
cat(sprintf("   Mean improvement: %.2f points\n", mean(survey$after_train - survey$before_train)))
cat(sprintf("   p-value: %.4f — %s\n\n", t_res$p.value,
             ifelse(t_res$p.value < 0.05, "Training effective!", "Training not significant")))

# 3: Gender vs Department independence
cat("3. Chi-squared: Gender vs Department\n")
chi_res <- chisq.test(table(survey$gender, survey$dept))
cat(sprintf("   Chi² = %.3f, p = %.4f — %s\n", chi_res$statistic, chi_res$p.value,
             ifelse(chi_res$p.value < 0.05, "Association exists!", "Independent")))