Question

Question Let Xij and X2, denote the responses to treatments 1 and 2, respectively, with j-1,2, -. ,n on the n paired comparison experiments. Assume that the n observed differences d, -Xij -X2j can be regarded as a random sample from a multivariate normal distribution with mean vector δ and variance-covariance Σd. The descriptive statistical results from an experiment of a sample of n = 30 as follows 13-4 2 14 4-2 dSd4 13 -2and S14 14 2 2 2 17 2 -2 10 I. Test at the 5% significance level the hypothesis that there is no treatment difference. 2. Find the 95% T2 simultaneous confidence interval for the individual population mean differences δι'δ2, and δ3 Hint : Use dit V (n-pe Fa (p, n-p) 4/n; i = 1, 2, . . . , p 3. Find the 95% Bonferroni simultaneous confidence interval for the individual population mean differences δ,82, and δ3 Hint : Used, ±ta/2p(n-1)、/sa/m i= 1.2, ,p 4. Use parts 2 and 3 to test the hypothesis Ho : δί-0, for i-1, 2, and 3 using 0.05. Hint: Check whether the each interval contains zero or not

Answer 1

1, Here. Ho : δ=0 14 4 -2 4 14 2 Hence value of T2 -30*(1,0,-1) 06.4815 162-2 2 ow.TN (n -1)p Hence value of test statistic -((30 -3)/(30- 1)3) 6.4815 2.0115 p - value- P(F> 2.0115FF3,27)0.1360 > 0.05 Hence we fail to reject Ho at 5% level of significance 2. Fo·05.327 2.9604. 、1 (n-1)pF 95% C. I. for δι ー_ Folp, n-p) 3.0885 (n - 1)p (-10.1358, 12.1358) 95% C.1. for 2 (n - 1)p =(-11.1358, 11.1358)

95% C. I. for δ3 (n - 1)p (n - 1) (-10.7667, 8.7667) 3. t0.05/6,29 = 2.5409 95% C.I. for б (d to.05/6,2 s,/n, di + to.o5/6,29 1 s2,/n) 95% C.1. for δ2 -0.6726, 2.6726) d-t05/6.2y i,/n, d+ toos/6./m) (1.6726, 1.6726) d as/,,dt/,a/m) (-2.4670, 0.4670) 95% C.1 . for 3 (4) We fail to reject Ho : δί-0, i = 1, 2, 3 since all the confidence intervals in (2), (3) contain zero