Business Context

PayWave, a fintech company, launched a new fraud-screening rule and tested whether it increases the false-positive rate above the historical benchmark. A product manager sees a small p-value and says, “So there’s only a 2% chance the null hypothesis is true.”

Problem Statement

Use the test results below to explain quantitatively why a p-value is not the probability that the null hypothesis is true. Show both the frequentist interpretation of the p-value and a separate Bayesian calculation for the probability that the null is true after seeing the data.

Given Data

PayWave tested whether the false-positive rate is higher than the historical rate of 5.0%.

Assume a one-sided hypothesis test: the team only cares whether the false-positive rate increased.

Requirements

1. State the null and alternative hypotheses.
2. Compute the z-statistic and one-sided p-value.
3. Explain in words what the p-value means.
4. Compute $P(H_0 \mid D)$ using Bayes' theorem with the provided priors and likelihood under $H_1$.
5. Compare the p-value to $P(H_0 \mid D)$ and explain why they are different quantities.
6. State the correct business interpretation for the fraud-rule decision.

Assumptions

• The 400 reviewed transactions are independent.
• The normal approximation for the sample proportion is acceptable.
• The Bayesian prior and $P(D \mid H_1)$ are provided for illustration; they are not derived from the z-test itself.