Hypothesis Testing Problem — Full Solution
Problem Setup
We test the hypothesis:
H0: p = 0.6
Sample size: n = 15
Decision rule:
- Do NOT reject H0 if 6 ≤ X ≤ 12
- Reject H0 if X ≤ 5 or X ≥ 13
We compute:
(a) α when p = 0.6
(b) β when p = 0.5
(c) β when p = 0.7
Binomial Formula:
P(X=k) = C(15, k) p^k (1-p)^(15-k)
PART (a): α when p = 0.6
α = P(X ≤ 5) + P(X ≥ 13)
P(X ≤ 5) = 0.13379
P(X ≥ 13) = 0.02695
α = 0.16074
PART (b): β when p = 0.5
β = P(6 ≤ X ≤ 12) = P(X ≤ 12) - P(X ≤ 5)
P(X ≤ 5) = 0.1509
P(X ≤ 12) = 0.99631
β = 0.84541
PART (c): β when p = 0.7
β = P(X ≤ 12) - P(X ≤ 5)
P(X ≤ 5) = 0.0037
P(X ≤ 12) = 0.8732
β = 0.8695
Final Answers:
α = 0.1607
β (p = 0.5) = 0.8454
β (p = 0.7) = 0.8695