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Stat 407 Quiz 2 Name 1. (2pts) Which of the following samples is most likely to correspond to this correlation matrix " # 1 0.5 ? 0.5 1 The middle plot because correlation between the two variables is 0.5. 2. (2pts) From this formula for a bivariate normal density tell me what the mean is and the variance of the distribution is. 1 1 x1 − 10 2 x1 − 20 2 f (x1 , x2 ) = √ × exp − ( √ ) +( √ ) 2 2π 2 × 3 2 3 The mean is " 10 10 # and the variance is " 2 0 0 3 # 3. (2pts) Based on the following SAS output, would you describe the variable LLENGTH as coming from a normal distribution? The UNIVARIATE Procedure Variable: LLength Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation Stem 51 50 49 48 47 46 45 24 4.90065934 0.16249809 -0.53298 577.002416 3.31584132 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Leaf 8 0344679 013359 11999 # 1 7 6 5 3359 4 8 1 ----+----+----+----+ Multiply Stem.Leaf by 10**-1 1 24 117.615824 0.02640563 -0.5642116 0.60732945 0.03316978 Boxplot | +-----+ *--+--* +-----+ | | | The distribution is slightly skewed, and bimodal, but it is hard to really tell whether it is a sample from a normal because the sample is so small. 4. (3pts) Calculate the T 2 = "n(X̄#− µ0 )0 S−1 n−1 (X̄ − µ0 ) statistic from the following sample values given n = 3 9 and a hypothesized mean 5 X̄ = " 8 6 # −1 S = " 1/3 1/9 1/9 4/27 # Would you reject the null hypothesis? T 2 = 0.78(= 7/9). With a value so small we are unlikely to reject the null hypothesis. 5. (1pt) Which of the following is NOT an assumption underlying the contruction of the test statistic when testing hypotheses about the multivariate mean (such as in the previous question)? (a) Ho is true. (b) The sample is taken by simple random sampling methods. (c) The variance-covariance matrices are equal. (d) The population takes a multivariate normal shape. The answer is (c). There is only one variance-covariance matrix so there is not an assumption of equal covariances. (a), (b) and (d) are all assumptions underlying the construciton of the test statistic. 2