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Stat 407 Quiz 3 - SOLUTION Name 1. For the following data: T reat X1 X2 6.6 6.3 1 10 7 6.6 6.3 1 11 9 6.6 6.3 2 5 6 = 6.6 6.3 + 2 4 2 6.6 6.3 2 5 3 6.6 6.3 3 6 8 6.6 6.3 3 5 9 3.9 1.7 −0.5 3.9 1.7 0.5 −1.9 −2.6 0.3 −1.9 −2.6 + −0.7 −1.9 −2.6 0.3 −1.1 2.2 0.5 −1.1 2.2 −0.5 −1.0 1.0 2.3 −1.7 −0.7 −0.5 0.5 (a) (1pt) Calculate the overall mean. X̄ = " 6.6 6.3 # (b) (1pt) Calculate the group means. X̄1 = " 10.5 8.0 # , X̄2 = " 4.7 3.7 # , X̄3 = " 5.5 8.5 # (c) (2pts) Break up each observation into its mean, treatment and residual component. (d) (2pts) Given the between sums of squares (B) and within sums of squares (W) matrices below calculate the Wilks’ Λ∗ = |W|/|W + B|. B= " 44.0 23.7 23.7 36.3 |W| = 14.0, |W + B| = 1500, Λ∗ = 0.0093. 1 # , W= " 1.67 2.17 2.17 11.2 # 2. (2pts) For the following plots, which do you think is most likely to have Wilks’ Λ close to 1. Why? The left most plot, because there is the least difference between means. 3. (2pts) In testing the hypothesis that the means of two groups are different, you get a p-value equal to 0.05. If the variance-covariance matrices for the two groups are as follows: S1 = " 8 2 2 10 # , S2 = " 4 2 2 5 # would you expect that the true significance of the test statistic is greater than or less than 0.05? Explain. Probably we would expect the true significance to be larger than 0.05, that is we truly have a less significant result. If one of the variances is larger then we would have more uncertainty about the true value of the this population mean, and the pooled variance underestimates this uncertainty. 2