4.If X_1, X_2, , X_n are i.i.d. r.v.'s from binomial distribution B(m, θ), θ (0, 1), show that X/m is the MLE of θ. (7%)

5.The lifetime of an automobile battery is described by an r.v. \( X \) having normal distribution with unknown mean \( \mu \) hour and known s.d. \( \sigma=150 \) hour. Let \( X_1, X_2, \cdots, X_{25} \) be independent r.v.'s distributed as \( X \) and suppose that \( \bar{x}=1725 \) hour. Test the hypothesis \( H_0: \mu=1800 \) versus \( H_1: \mu<1800 \) at level of significance \( \alpha=0.01 \) and make a conclusion. (10%)

4.If \( X_1, X_2, \dots, X_n \) are i.i.d. r.v.'s from binomial distribution \( B(m, \theta) \), \( \theta \in (0, 1) \), show that \( \overline{X}/m \) is the MLE of \( \theta \). (7%)

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