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2. Let the prior probability of the null hypothesis (H_0) being true is 0.5. If the posterior odd ratio of testing a null hypothesis against an alternative hypothesis is 4. Then, the posterior probability of the null hypothesis being true is * (1 Point) O P (Ho ( x) = 0.5 O P (Ho | x) = 0.2 O P (Ho | x) = 0.8 P(Ho [ x) = 0.1 3. T * (2 Points) E ( H | x) = If the data are highly precise (large n), then the weight is large on the sample mean of data, For large n, Bayesian and classical inference are approximately identical for mu O If the prior is highly precise, then the weight is less on the mean of prior distribution If the prior is highly precise, then the weight is large on the mean of prior distribution 4. Assume a random sample of size three is drawn from the Raleigh distribution. Let the observations be 3, 5, 8. Using Jefferys prior, the Bayes estimator under quadratic loss function is * (2 Points) f ( x) - -e 2. 0>. x>0. 3.778 3.333 3.877