Playing Dice l Ill point (graded) You and your friend play dice games for fun, but one dayr you suspect that the die your friend uses is not fair. You will gather data and use the tools of hypothesis testing in this problem to provide a plausible answer as to whether or not the die is fair. Your statistical model is ({1,2,3,4, 5, 6}, {P1, }peA5)' You roll the die 15 times, writing the sample as random variables 's'd . . . . X1, . . . ,X15 3v Pp- where 1)\" Is the true parameter. Your null and alternative hypotheSIs are, respectively, Eh :p' ==[1/6 1/6 1/6 1/6 1/6 1/6?2 in :1; =# u/s 1/6 1/6 1/6 1/6 1/6P2 Let denote the MLE for the true parameter I)\". You use the following test statistic to test the hypotheses: \"(el}: _ 1 6 TnnE T. 6 i=1 Tn converges to a random variable X m X: for some integer In. What is k? Playing Dice || 1.11 point (graded) We use the same statistical set-up as above. In particular, our test-statistic is Miaif _ 5 6 11"\"ET' 6 3:]. You use a test of the form %=un>. What value of C should be chosen so that '6') is a test of asymptotic level 5%? Give a numerical value with at least 2 decimals. Use this table to find the quantiles of a chi-squared distribution. l1.07 V Playing Dice Ill 1 point possible lgradedl We use the same statistical set-up as above. Recall that you use a test of the form 111571 :1(Tn > C)! where C is a constant chosen in the previous problem so that 11')\" has asymptotic level 5%. Suppose you observe that data set iLthaLi Do you reject or fail to reject the null hypothesis that the die is fair? (You are encouraged to use computational tools.) O Reject 0 Fall to reject