The Cheating Coin

The statistician is more likely correct. A z-score of 2.0 gives a one-sided p-value of approximately 2.28%, below the standard 5% significance level. The data statistically support the conclusion that the coin is biased toward heads.

Frame this as a one-sided z-test. Under H0: coin is fair (p=0.5). Under H1: coin is biased toward heads (p > 0.5). Compute the z-score for the observation and look up the tail probability. Step 1: Under H0: number of heads X ~ Binomial(100, 0.5). Mean = np = 50. Variance = np(1-p) = 25. Standard deviation = 5. Step 2: Observed: x = 60 heads. Z-score = (60 − 50) / 5 = 10 / 5 = 2.0. Step 3: P-value = P(Z ≥ 2.0) where Z is the standard normal distribution ≈ 0.0228, or about 2.28%. Step 4: Decision rule: if p-value < 0.05 (standard significance threshold), reject the null hypothesis. 2.28% < 5% → reject H0. Step 5: Conclusion: The result is statistically significant. The data provide meaningful evidence that the coin is biased toward heads. Step 6: Note: this does not prove bias with certainty. About 1 in 44 fair coins would produce this result by chance. But it is unlikely enough to act on — the statistician is more likely correct. Key Insight: Statistical significance does not mean certainty — it means the observation is unlikely enough under the null hypothesis that we are comfortable rejecting it. The z-score of 2.0 sits at the border of 'moderately strong' evidence, and a one-sided p-value of 2.28% comfortably clears the standard 5% threshold.