et X₁, X2,..., Xn be a random sample from a distribution with pdf x-1, 0 0. Show the likelihood has statistic II X₁. Use this to determine the UMP test for Ho: 0 = 0' : 0 <0', for fixed 0'>0. Guniformly most powerful test
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- In a hypothesis test with hypotheses Ho: μ ≤ 54 and H1: μ > 54, a random sample of 24 elements selected from the population produced a mean of 58.6 and a standard deviation of 13.4. The test is to be made at the 10% significance level. Assume the population is normally distributed. What is the critical value of t ?Consider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θSuppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]
- Suppose that a group of researchers is planning to test a new weight loss supplement. They have selected a random sample of 45 people who are trying to lose weight and plan to measure the amount of weight lost after one month of using the supplement. Assume that the researchers know from prior experiments that the standard deviation of weight lost in one month, , is 1.7 lb. To show that the supplement is effective, they plan to use a one-sample z‑test of Ho : u= 0lb against H1 : u > 0lb , where is the mean amount of weight lost in one month. They have also determined that, for a test with a significance level of 0.05, the power of the test is 0.9347 if the mean amount of weight lost is actually 0.8 lb. What is the probability that the researchers will reject their null hypothesis if the mean amount of weight lost is 0.8 lb or more? Give your answer as a percentage, precise to two decimal places..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=.05Suppose X1, . . . , Xn be a random sample from the Beta(θ, 1) distribution. Find the P-value for the LRT test of the hypotheses H0 : θ ≥ 1 vs H1 : θ < 1
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- To test H0: μ = 50 versus H1: μ < 50, a random sample of size n = 24 is obtained from a population that is known to be normally distributed with σ = 12a) If the sample is determined to be x = 47.1, compute the test statistic.b) If the researcher decides to test this hypothesis at α = 0.05 level of significance, determine the P-value.Let the following simple random sample following:1. Binomial pmf. (11, ¾);2. Uniform pmf;3. Uniform pdf (0, a);4. Exponential pdf with (µ) .Find the corresponding pmf/pdf of Y1 , Y4 , Y7 and F(Yi) where25. The State of California claims the population average of the amount of ice cream each Californian eats in the month of September is 6.85 pints with population standard deviation of 1.35 pints. An SRS of 500 Californians resulted in a sample average of 6.75 pints eaten per person in the month of September . At alpha=0.05, is there evidence to support the State of California's claim that Californians eat an average of 6.85 pints of ice cream in the month of September? Write a conclusion using the context of the problem.