PROBLEM NO. 1 (20 points) 1. Analysis of variance tests use the __________. A. t distribution B. normal distribution C. F distribution D. exponential distribution 2. The statistical methods of analysis of variance assume __________. A. convenience samples B. judgment samples C. random samples D. quota samples 3. Determining the table value for the F distribution is different from finding values in the t distribution tables because the F table requires __________ values for degrees of freedom. A. one B. two C. three D. more than three 4. The number of degrees of freedom in a chi-square goodness of fit test is __________. A. the number of categories B. the number of categories minus 1 C. the number of categories minus the number of parameters estimated D. the number of categories minus the number of parameters estimated minus 1
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1. Let S^2 be the sample variance based on X_1, X_2, ..., X_n from a normal distribution. Then we have seen that (n - 1)S^2 / σ^2 follow a χ^2_{n-1} distribution. Consider the prior distribution of σ^2 as h(σ^2) = (1 / (Γ(α)β^α)) (1 / (σ^2)^{α+1}) e^{-1/(βσ^2)}, 0 < σ^2 < ∞, where α and β are positive constants. (a) Find the posterior distribution of σ^2. (b) Find the Bayes estimator of σ^2. 2. Let X_1, X_2 be a random sample from a pdf given by f(x; θ_1, θ_2) = (1 / π) (θ_2 / (θ_2^2 + (x - θ_1)^2)), -∞ < x < ∞, where -∞ < θ_1 < ∞, 0 < θ_2. Use the non-informative prior h(θ_1, θ_2) ∝ 1. (a) Find the posterior pdf of θ_1, θ_2. (b) Evaluate the posterior pdf if x_1 = 1, x_2 = 4 for θ_1 = 1, 2, 3, 4 and θ_2 = 0.5, 1.0, 1.5, 2.0. (c) From the 16 values in part (b), where does the maximum of the posterior pdf occur? Explain.
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