You can use wolfram alpha to do any calculus (though I don’t think you’ll need to), but otherwise are limited to your notes and the text book. Stop working 48 hours after starting. Due by the end of...

Math 152 - Midterm 2 You can use wolfram alpha to do any calculus (though I don't think you'll need to), but otherwise are limited to your notes and the text book. Stop working 48 hours after starting. Due by the end of Wednesday. 1. Consider the following random sample coming from a continuous pdf that is symmetric about: 1, 3, 4, 6, 10, 15. Consider a test forH : = based on a 0 0 p-value coming from a binomial distribution, compute a 95 percent condence interval for . 2. Consider testing H :p =:5 vs H :p>:5 using a decision rule of the form 0 a \reject H if Xk", where XBin(10;p). 0 a. It seems reasonable to only consider values of k > 5. For k = 6;:::; 10, calculate the  value corresponding to each of these tests. b. Notice that there does not exist a value k for which  = :05. This is often the case when our random variable is discrete. We can create a rule that will have  = :05 by using a randomized decision rule. For some values of X, we know to reject. For others, we know to accept H . For some values, we \ ip a 0 coin", the outcome of which tells us how to conclude. Give such a rule that has  =:05 (you need to tell me for what value you need to ip the coin and what the probability of heads for the coin needs to be). 3. Consider a shifted exponential distribution with  = 1 known. (x) f (xj) =e 1(x) X a. Derive the likelihood ratio test for H : = 2 vs H : =6 2: 0 a b. For n = 2, give the rejection region corresponding to an  =:05. c. Using the rejection region given in b, invert it to construct a condence interval for . 4. a. Consider trying to use a `goodness-of-t' test for assessing the normality 2 of a random sample (this test is the common name for our  approximation to the GLRT for multinomial data, recall free-throws/Hardy-Weinberg from the homework). We decide to group the data into 3 bins to make it look multinomial: 12 observations fell less than 5, 64 fell between between 5 and 10, and 24 fell above 10. The null hypothesis...