An Introduction to Probability and Statistical Inference, 2nd Edition

Exercises

4.1 A coin, with probability θ of falling heads, is tossed independently 100 times and 60 heads are observed. At level of significance α = 0.1:

(i) Use the LR test in order to test the hypothesis H₀: θ = 1/2 (against the alternative H_A: θ ≠ 1/2).

(ii) Employ the appropriate approximation (see relation (10) in Chapter 8) to determine the cutoff point.

4.2 Let X₁, X₂, X₃ be independent r.v.'s distributed as B(1, θ), θ ∈ Ω = (0, 1), and let t = x₁ + x₂ + x₃, where the x_i's are the observed values of the X_i's.

(i) Derive the LR test λ for testing the hypothesis H₀: θ = 0.25 (against the alternative H_A: θ ≠ 0.25) at level of significance α = 0.02.

(ii) Calculate the distribution of λ(T) and carry out the test, whereT = X₁ + X₂ + X₃