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 H0θ = 1/2 (against the alternative HAθ ≠ 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 xi's are the observed values of the Xi's.

(i) Derive the LR test λ for testing the hypothesis H0θ = 0.25 (against the alternative HAθ ≠ 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

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