Testing simple hypotheses, covariance and least squares
Decide how many heads make a fair-coin hypothesis implausible, compute covariance and the correlation coefficient for five points, and find the line with the smallest sum of squared errors.
- Germany: grade 12
- Russia: grade 11
Guide
This follows the Russian advanced-level grade 11 course: testing the simplest hypotheses, covariance and correlation coefficient, linear regression and least squares.
Hypothesis test: assume H0 ('the coin is fair'), find the probability of a result at least this extreme if H0 is true. If it is below the significance level (usually 0.05), the result is too rare for H0 and we reject it. This is a decision, not a proof.
Covariance cov(X, Y) = Σ(x − x̄)(y − ȳ) ÷ n. The correlation coefficient r = cov ÷ (σx σy) lies between −1 and 1. The least-squares line y = mx + c has m = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)² and c = ȳ − m x̄. Common mistakes: not dividing the covariance by n, taking cov for r, forgetting the intercept.