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## Problem 3. The problem of comparing the means of two independent normal sampleswith equal variances

Problem 3. The problem of comparing the means of two independent normal sampleswith equal variances (see Lecture 13, Slide 4) can be formulated as a regression problemas follows: Denote n1 i.i.d. samples from a N (n1,a2) population by Y1,Y2, . . . , Yn1 and n2i.i.d. observations from a N (#2, 02) normal population by Ym+1,Ym+2, . . . , Ym+n2. De?ne adummy variable :13,- = 1, for 2′ = 1,2,…,n1 and :51; = 0, for i = n1 + 1,711 + 2,. . .,n1 +n2.Thus, if :21- = 1, then Y,; comes from the ?rst population, and if m,- = 0, then Y} comes fromthe second population. (a) (2 points) Show that this corresponds to a regression model Y;- = ?g +?1\$i +Â£51-3 with160=M2 and51=u1â,u2. A A (b) (2 points) Show that the least squares estimates ?g = 1172 and BI = ?l â 372. (c) (2 points) Show that the MSE for the regression is the same as the pooled varianceestimate 82 of 02, with n1 + n2 â 2 degrees of freedom. ((1) (2 points) Show that the regression tâtest of 51 = 0 is the same as the equal variancetâtest of #1 = #2. (e) (2 points) Show that the ANOVA F âtest for #1 2 #2 is the square of the equalvariance tâtest of M1 2 n2.Problem 5. Suppose X1, X2, . . . ,Xn are i.i.d. observations from a uniform distribution onthe interval [6′ â 1/2, 6 + 1/2]. (a) (5 points) Write down the likelihood function of 0. (b) (5 points) Show that any 6 between Xmax â 1/2 and ijn+ 1/2 maximizes the likeli-hood, and therefore, can be taken as the MLE, where Xmx = max{X1, X2, . . . ,Xn}and Xmin = min{X1,X2, . . . ,Xn}.Problem 6. Suppose X1, . . . ,Xn are i.i.d. from a continuous distribution with probabilitydensity function IV few) = exp(9 – 393;), a: 0, where 6 E (â00, 00).(a) (2 points) Find the MLE of 0.(b) (3 points) Denote the MLE of 6 from a random sample of size n by 9â. Find theasymptotic distribution of Â«509″, â 6).(c) (2 points) Give an approximate 95% con?dence interval of 6 when n 2 30.(d) (3 points) Propose a test statistic for testing Ho : 9 = 0 vs. H1 : 9 % 0 when n 2 30,and specify the rejection region for a = 0.01.