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2. Student's T Test Bookmark this page Deriving the Student's T Test from Likelihood Ratio 2 possible points (qualifiable) Let X1, ..., Xn " X ~N ((1, ;). Consider the null and alternative hypotheses Ho : #1 = 5 H1 : M # 5. Assume that / is unknown, but of is. The test statistic for the likelihood ratio test associated to the above hypothesis can be expressed in terms of n. Xn . and of. What is T,? (Enter barX_n for X,, and sigma_142 for of.) T! = STANDARD NOTATION If owere unknown and we used the estimator of = _. (Xi - X. )" in both log-likelihoods, what would be the distribution of VT,? ta-1 tn O tn- 1l None of the above. STANDARD NOTATION(C) 3 points possible (graded) Xi ~ Exp (X) , A>0, which means that each X1 has density fx () = Ne , =>0. (Enter barX_n for X", the sample average.) Maximum likelihood estimator * = (If the Fisher information is not defined, enter DNE.) Fisher information I (X) = Use Fisher Information to find the asymptotic variance V ()) of the MLE A. V ( ) ) = STANDARD NOTATION(d) 7 points possible (graded) Xi ~ N ( H, 0 ' ) , HER, 0' > 0, which means that each X1 has density 1 exp V2702 202 Hint: Keep in mind that we consider of as the parameter, not 7. You may want to write 7 = " in your computation. (Enter barX_n for the sample average X, and bar(X_n^2) for the sample average of second moments XH .) Maximum likelihood estimator / = (Enter barX_n for the sample average X, and bar(X_n^2) for the sample average of second moments X, .) Maximum likelihood estimator of = Hint: One of the formulas for Fisher information will lead to a much shorter computation. (If the Fisher information is not defined, enter DNE for all boxes below.) [I (14, ?)1,1 = . [I (14, 2)12 = [I (16, 02 )12,1 = Using the Fisher Information you obtain above, what is the asymptotic variance V (') of the MLE o? ?Compare this with your result from Homework 5 Problem 3. V (o' ) =Likelihood Ratio Test for Shifted Exponential I 2 point; possible (graded! While we cannot formally take the log of zeror it makes sense to dene the loglikelihood of a shifted exponential to be F! a\" (a, a) = (n In 1 A 2 (X.- :2 mm, + (oo} Imam. .=1 Keep in mind that the likelihood is zero when min, (Xi) #2- What is the test statistic T'" for the two-sample student's T test associated to Ho and Hi ?Express your answer in terms of n, m, 61 , 62 , Xn, and Y m. (Enter bark n for Xn, barY_m for Y m, hat(sigma_142) for of, and hat(sigma_212) for ?.) Th = STANDARD NOTATION Submit You have used 0 of 4 attempts Save Applying the Welch-Satterthwaite Formula 2 points possible (graded) Suppose we observe Xn = 6.2, Ym = 6,01 = 0.1, and 02- = 0.2 with n = 50 and m = 50. Using the Welch-Satterthwaite formula, what is the approximate number of degrees of freedom for the test statistic In? What is the p-value for this test? (You may consult a table of values or use software for the student's T distribution.) Submit You have used 0 of 3 attempts Cave