You have a friend who claims to have an unbiased 3-sided dice. You dispute this claim, so you will attempt the resolve t

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You Have A Friend Who Claims To Have An Unbiased 3 Sided Dice You Dispute This Claim So You Will Attempt The Resolve T 1 (103.69 KiB) Viewed 27 times

You have a friend who claims to have an unbiased 3-sided dice. You dispute this claim, so you will attempt the resolve the dispute with statistics. Using the same probability model and data as in Question 1, this amounts to per- forming a hypothesis test H. : P1 = P2 = 1/3 against H: Pi + P2. Answer the following: (a) Explain why it is possible to use the likelihood ratio test to perform this hypothesis test. (b) Show that the likelihood ratio test statistic is given by: m m LR = -2 mn log P2 un wes (?) - 1s (9) 1 Зр log dia – log () **)£ Σ 12 р where p =1-P1-P2. (c) Assume we have collected data from m = 20 experiments of n = 10 rolls of our friend's dice and calculate L-111,1 = 63 and 2-101,2 = 86. Under the asymptotic assumption of the distribution of the likelihood ratio test statistic, compute the p-value for the hypothesis test. Inter- pret the p-value in the context of the hypothesis test. Note: you will need a method to calculate the cumulative distribution function for the appropriate xà distribution, you can use the pchisq in R for this. (d) The debate between you are your friend continues and you decide a second opinion with information criteria is needed. Write the expressions for AIC and BIC for both hypotheses. (e) Evaluate the penalty terms using the values in part (c). Are these penal- ties large enough to change your conclusions compared with the likeli- hood ratio test?