3. One question in the textbook gives a data set (in Exercise 6.24 page 347) on mea- surements on Egyptian skulls for three different time periods. In this question we will use the data only for the first two periods. I also changed some values in the data set to make the results more appropriate for this question. The data I created are given in the data file 2sample_data.txt (posted). You can read this if you like to get an idea about this data set but we will use the data in the data file 2sample_data.txt (posted) to answer the questions that I have given below. Note that we only use two populations (i.e. the two periods 1 and 2 only) and the questions we are answering here are also not the questions in Exercise 6.24 page 347. (a) (2 points) Calculate the observed sample means for the two populations. (b) (3 points) Calculate the calculate the observed sample covariance matrices for the two populations and the pooled covariance matrix. (c) (6 points) Use Box's M test to test for equality of the two population covariance matrices. Use a = 0.05. (d) (7 points) In this part, regardless of your answer to part (c) above, ASSUME that the covariance matrices are equal. Assuming equal covariance matrices, test for the equality of mean vectors. Use a = 0.05. If you rejected Ho : Hi M2 = 0, find the linear combination most responsible for the rejection. (e) (4 points) Calculate 95% simultaneous and Bonferroni simultaneous confidence intervals for the components of p1 - M2. x1 x2 x3 x4 Period 131 138 89 49 1 125 131 92 48 1 131 132 99 50 1 119 132 96 44 1 136 143 100 54 1 138 137 89 56 1 139 130 108 48 1 125 136 93 48 1 131 134 102 51 1 134 134 99 51 1 129 138 95 50 1 134 121 95 53 1 126 129 109 51 1 132 136 100 50 1 141 140 100 51 1 131 134 97 54 1 135 137 103 50 1 132 133 93 53 1 139 136 96 50 1 132 131 101 49 1 126 133 102 51 1 135 135 103 47 1 134 124 93 53 1 128 134 103 50 1 130 130 104 49 1 138 135 100 55 1 128 132 93 53 1 127 129 106 48 1 131 136 114 54 1 124 138 101 46 1 126 137 100 47 2 135 133 96 47 2 140 133 97 44 2 150 128 103 50 2 128 123 94 44 2 137 135 97 51 2 134 144 99 53 2 135 129 101 47 2 133 133 95 49 2 135 124 93 45 2 135 135 102 52 2 133 138 97 50 2 133 135 98 55 2 140 133 97 48 2 132 135 103 52 2 133 127 97 44 2 140 128 106 52 2 125 130 100 50 2 132 128 104 46 2 136 129 92 53 2 139 135 105 48 2 128 130 99 47 2 137 135 96 51 2 131 125 90 49 2 136 138 100 48 2 133 133 89 52 2 134 129 103 49 2 132 131 92 51 2 137 131 97 53 2 132 127 100 50 2 3. One question in the textbook gives a data set (in Exercise 6.24 page 347) on mea- surements on Egyptian skulls for three different time periods. In this question we will use the data only for the first two periods. I also changed some values in the data set to make the results more appropriate for this question. The data I created are given in the data file 2sample_data.txt (posted). You can read this if you like to get an idea about this data set but we will use the data in the data file 2sample_data.txt (posted) to answer the questions that I have given below. Note that we only use two populations (i.e. the two periods 1 and 2 only) and the questions we are answering here are also not the questions in Exercise 6.24 page 347. (a) (2 points) Calculate the observed sample means for the two populations. (b) (3 points) Calculate the calculate the observed sample covariance matrices for the two populations and the pooled covariance matrix. (c) (6 points) Use Box's M test to test for equality of the two population covariance matrices. Use a = 0.05. (d) (7 points) In this part, regardless of your answer to part (c) above, ASSUME that the covariance matrices are equal. Assuming equal covariance matrices, test for the equality of mean vectors. Use a = 0.05. If you rejected Ho : Hi M2 = 0, find the linear combination most responsible for the rejection. (e) (4 points) Calculate 95% simultaneous and Bonferroni simultaneous confidence intervals for the components of p1 - M2. x1 x2 x3 x4 Period 131 138 89 49 1 125 131 92 48 1 131 132 99 50 1 119 132 96 44 1 136 143 100 54 1 138 137 89 56 1 139 130 108 48 1 125 136 93 48 1 131 134 102 51 1 134 134 99 51 1 129 138 95 50 1 134 121 95 53 1 126 129 109 51 1 132 136 100 50 1 141 140 100 51 1 131 134 97 54 1 135 137 103 50 1 132 133 93 53 1 139 136 96 50 1 132 131 101 49 1 126 133 102 51 1 135 135 103 47 1 134 124 93 53 1 128 134 103 50 1 130 130 104 49 1 138 135 100 55 1 128 132 93 53 1 127 129 106 48 1 131 136 114 54 1 124 138 101 46 1 126 137 100 47 2 135 133 96 47 2 140 133 97 44 2 150 128 103 50 2 128 123 94 44 2 137 135 97 51 2 134 144 99 53 2 135 129 101 47 2 133 133 95 49 2 135 124 93 45 2 135 135 102 52 2 133 138 97 50 2 133 135 98 55 2 140 133 97 48 2 132 135 103 52 2 133 127 97 44 2 140 128 106 52 2 125 130 100 50 2 132 128 104 46 2 136 129 92 53 2 139 135 105 48 2 128 130 99 47 2 137 135 96 51 2 131 125 90 49 2 136 138 100 48 2 133 133 89 52 2 134 129 103 49 2 132 131 92 51 2 137 131 97 53 2 132 127 100 50 2