Given the below data, where we have samples of three-dimensional vectors Xi = (Xi,Yi,Zi) with mean = (X ,Y ,Z ) for each of 25 lizards. Assume that the assumption X1,...,X25 iid N3(,) is reasonable.

(a) Determine the sample mean X and sample covariance matrix S based on the full data set. (b) What is the distribution of T²(X,)? (c) Determine the value of T²(X,0) for 0 = (10, 70, 140). (d) What is the p-value of the test H0 : = (10, 70, 140). What can you conclude? (e) Determine the sample covariance matrix S2,3 for the variables Y (SVL) and Z (HLS). (f) Draw a scatterplot of the data and add 95% and 99% confidence ellipsoids for (Y ,Z ). Is the point (70, 140) in any of these ellipsoids? To draw the ellipsoids you may use the following R code:

source("bivCI.R") biv = data with last two columns of lizard data plot(biv, col = "red", pch = 16, cex.lab = 1.5) lines(bivCI(s = var(biv), xbar = colMeans(biv), n = dim(biv)[1],alpha = .01, m = 1000),type = "l", col = "blue") lines(bivCI(s = var(biv), xbar = colMeans(biv), n = dim(biv)[1],alpha = .05, m = 1000),type = "l", col = "red", lwd = 1) # Add ''+'' sign lines(colMeans(biv)[1], colMeans(biv)[2], pch = 3, cex = .8,type = "p", lwd = 1)

(g) Determine a 95% confidence t-interval for Z Y and compare it to the 95% T2- confidence interval of Z Y . What is the actual coverage of the latter?

1 5.526 59.0 113.5 2 10.401 75.0 142.0 3 9.213 69.0 124.0 4 8.953 67.5 125.0 5 7.063 62.0 129.5 6 6.610 62.0 123.0 7 11.273 74.0 140.0 8 2.247 47.0 97.0 9 15.493 86.5 162.0 10 9.004 69.0 126.5 11 8.199 70.5 136.0 12 6.601 64.5 116.0 13 7.622 67.5 135.0 14 10.067 73.0 136.5 15 10.091 73.0 135.5 16 10.888 77.0 139.0 17 7.610 61.5 118.0 18 7.733 66.5 133.5 19 12.015 79.5 150.0 20 10.049 74.0 137.0 21 5.149 59.5 116.0 22 9.158 68.0 123.0 23 12.132 75.0 141.0 24 6.978 66.5 117.0 25 6.890 63.0 117.0