Test if one value comes from N(μ,σ²)

Q: Test if one value comes from N(μ,σ²)

This question evaluates understanding of hypothesis testing and statistical inference, specifically the formulation and interpretation of a z-statistic and p-value, and the implications of testing with a single observed value including required assumptions.

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Assume the population is N(μ, σ²) with μ and σ known. You observe a single value x = 1.37. Formulate a two-sided hypothesis test H0: X ~ N(μ, σ²) vs H1: X is not drawn from that distribution. Specify the z statistic, compute the tail probability P(|Z| ≥ |(x−μ)/σ|), and report the p-value. State all assumptions that make such a test meaningful with n = 1, discuss its power and Type I error control, and explain how the procedure changes if μ or σ are unknown.

Test if one value comes from N(μ,σ²)

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