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Problem 1. Goodness-of- fit test We want to test if the samples X1,.... X100 are from an exponential distribution, i.e., (x(x) = re for some A>0. We observe 35 samples in the interval (0,1), 20 samples in (1,2], 25 samples in (2, 4] and 20 samples in (4,+00). We also know =2. (a) 9 points] Do an appropriate goodness-of fit test with the significance level =0.05. Problem 2. Two sample test Suppose X1,..... X, are i.i.d. N(Hx,0%) and are i.i.d. N(Hv,02). and X1's are inde- pendent of Yi's. Assume o} and of are unknown. We test Ho against H1 ex (a) [6 points] If we observe n = 9.m 6,5 = 1.3, 1 y 2.5. 8 10.8, s 6.3. make your decision for the test assuming o} to? given the significance levela 0.1. (b) (6 points] If s² = s3 the method of two-sample test assuming equal variances equivalent to the two-sample test assuming unequal variances? Justify your answer by comparing the test statistics and decision rules Problem 3. Test for two variances Suppose X1 ...X. are i.i.d. N(Hx,0-) and are i.i.d. N(HY of). and X;'s are inde- pendent of Yi's. Assume x and py are unknown. We test Ho:ox =o; against H1 of #02 (a) [5 points] If we observe 72 8. m = 9.82 and s2 4.2. make your decision for the test given the significance level 0.01. (b) [6 points] Start with two different F distributions, derive two formulae of the 100(1 -a) confidence interval for the variance ratio 03/03 Your two answers are both in terms of Sx s3 and F critical values.

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