• The p-value is the probability of obtaining a result equal to or more extreme than what was observed, assuming the null hypothesis is true.
• A small p-value indicates strong evidence against the null hypothesis.
• Common thresholds: 0.05, 0.01, 0.001.

• In clinical trials, p-values help test the hypothesis that vaccines significantly reduce infection rates.
• Misinterpretation of p-values leads to flawed public perception of effectiveness.
• High p-values ≠ vaccine doesn’t work — it may reflect small sample size or bad design.

\[ p = P(\text{observing data as extreme as the sample data} \mid H_0 \text{ is true}) \]

\[ p = \int_{x_{obs}}^\infty f(x \mid H_0) \, dx \]

set.seed(123)
placebo <- rbinom(1000, 1, 0.10)  # 10% infection rate
vaccine <- rbinom(1000, 1, 0.08)  # 8% infection rate

data <- data.frame(
  group = rep(c("Placebo", "Vaccine"), each = 1000),
  infected = c(placebo, vaccine)
)

test_result <- prop.test(x = c(sum(placebo), sum(vaccine)),
                         n = c(length(placebo), length(vaccine)),
                         alternative = "greater")
test_result$p.value

## [1] 0.1301831

• Null Hypothesis: No difference in infection rate
• Alternative: Placebo has higher infection rate
• p-value < 0.05 suggests vaccine helps