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- If X1 and X2 constitute a random sample of size n = 2from an exponential population, find the efficiency of 2Y1relative to X, where Y1 is the first order statistic and 2Y1and X are both unbiased estimators of the parameterFor an exponential random variable (X) having θ = 4 and pdf given by: f(x) = (1/θ)e^(−x/θ ) where x ≥ 0, compute the following: a) E(X). b) Var(X). c) P(X > 3)..A sample of 9 measurements, randomly selected from a normally distributed population, resulted in x= 2.6, and s= 0.9 Conduct a hypothesis test to verify the claim that the population mean is greater than 2.5 . Use a=.05
- given normally distributed sample x=12 and s=3, use the Empirical Rule determine the upper and lower bounds to contain approximately 95% of your dataLet (X1,...,Xn) be a random sample of random variables with EX21 < ∞. Consider estimating µ = EX1 under the squared error loss. Show that (i) any estimator of the form a ¯ X+bis inadmissible, where ¯ X is the sample mean, a and b are constants, and a>1;Consider a random sample X1, … , Xn from the pdff (x; u) = .5(1 + (THETA)x) -1 <= x <= 1where -1 <= theta <= 1 (this distribution arises in particlephysics). Show that theta = 3X is an unbiased estimator oftheta. [Hint: First determine mu = E(X) = E(X).]
- Determine the appropriate critical value(s) for each of the following tests concerning the population mean: a. upper-tailed test: α=0.005; n=25; σ=5.0 b. lower-tailed test: α=0.025; n=28; s=9.0 c. two-tailed test: α=0.02; n=41; s=4.4 d. two-tailed test: α=0.05J 1 Consider the hypothesis statement to the right using a=0.01 and the data to the right from two independent samples. A) calcuate the appropriate test statistic and interpret the results b) calculate the p value and interpret the results