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Q28Section A (Section A)28 of 60

P-value of two-sided t-test

Let x1,x2,x3x_1, x_2, x_3 and x4x_4 be observed values of a random sample from an N(θ,σ2)N(\theta, \sigma^2) distribution, where θ∈R\theta\in\mathbb{R} and σ>0\sigma > 0 are unknown parameters. Suppose that xˉ=14∑i=14xi=3.6\bar{x}=\frac{1}{4}\sum_{i=1}^4 x_i = 3.6 and 13∑i=14(xi−xˉ)2=20.25\frac{1}{3}\sum_{i=1}^4 (x_i - \bar{x})^2 = 20.25. For testing the null hypothesis H0:θ=0H_0: \theta=0 against H1:θ≠0H_1: \theta\neq 0, the pp-value of the likelihood ratio test equals

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