Hypothesis Testing: One-Tailed and Two-Tailed Tests

1. 1 A coin is flipped 20 times and gives 15 heads. 'Is the coin biased, or is this just bad luck?' We need a rigorous method. 'How unlikely would this have to be before we conclude the coin is biased?'
2. 2 H₀ (null hypothesis): the coin is fair, p = 0.5. H₁ (alternative hypothesis): the coin is biased in favour of heads, p > 0.5 (one-tailed). Significance level α = 0.05 (5%). 'We will reject H₀ only if our result has probability < 0.05 under H₀.'
3. 3 Under H₀, X ~ B(20, 0.5). Find c such that P(X ≥ c) < 0.05. Using tables or calculator: P(X ≥ 15) = 0.0207 < 0.05. P(X ≥ 14) = 0.0577 > 0.05. Critical region: X ≥ 15. Since our result (15) is in the critical region, we reject H₀.
4. 4 Model the required language: 'There is sufficient evidence at the 5% significance level to reject H₀. We conclude that the coin is biased in favour of heads.' Students practise writing a full conclusion for a different scenario.
5. 5 H₁: the coin is biased (either direction), p ≠ 0.5. Split the significance level: 0.025 in each tail. Find lower critical value: P(X ≤ c₁) < 0.025, and upper: P(X ≥ c₂) < 0.025. Critical region: X ≤ 4 or X ≥ 16 (for B(20, 0.5)).
6. 6 A die is suspected of being biased against 6. In 30 rolls, a 6 appears 2 times. Test at the 5% level. H₀: p = 1/6, H₁: p < 1/6. X ~ B(30, 1/6). P(X ≤ 2) = ? Students calculate and conclude.
7. 7 What does 'fail to reject H₀' mean? (Not the same as proving H₀ is true.) Why 5%? Could we use 1%? What changes? What is a Type I error? (Rejecting H₀ when it is actually true.)

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