This took me a second to land on the right framing.
Start by framing the null hypothesis that outcomes are purely random (luck), then outline the chi-square goodness-of-fit test using observed vs. expected win distributions. Walk through the calculation of the test statistic, degrees of freedom, and p-value, and finish by discussing limitations and practical interpretations.
Pro tip: Emphasize that rejecting the null doesn't prove skill—it only suggests the game isn't purely luck; you need further analysis (e.g., consistency across players) to infer skill. Also, mention that with large n, even tiny deviations can be significant, so effect size matters.
Null hypothesis (H0): The game is purely luck, meaning each player's win probability is equal (e.g., 0.5 for two-player games) and outcomes are independent. Alternative (H1): The game involves skill, so win probabilities differ across players.
Aggregate win/loss records for each player over many rounds. For a chi-square goodness-of-fit test, you need observed counts of wins (and losses) per player, and compute expected counts under H0 (e.g., total wins / n for each player).
Calculate χ² = Σ (O_i - E_i)² / E_i, where O_i is the observed number of wins for player i and E_i is the expected number of wins under H0. Ensure expected counts are sufficiently large (e.g., ≥5) for validity.
Degrees of freedom (df) = number of categories - 1 = n - 1 (if comparing n players). Compare the test statistic to a chi-square distribution with df to obtain the p-value. If p < α (e.g., 0.05), reject H0.
If H0 is rejected, conclude the game is not purely luck, but note that this doesn't prove skill—it could be due to other factors (e.g., player advantages). Limitations: assumes independence, large sample required, sensitive to sample size, and doesn't measure effect size or skill magnitude.
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