only part II of the attached pdf, the text document helps with the question. I need help with all parts of that question a-f.
Faculty of Science, Applied Science & Engineering Department of Mathematics and Statistics ASSIGNMENT 5 STAT 4203: INTRODUCTION TO MULTIVIARTE STATISTICAL ANALYSIS WINTER • 2023 Due Date: Wednesday, March 1st @ 12:30 pm Total Marks: 48 Part I: Questions requiring detailed solutions Instructions: A paper copy with solutions to these questions are due at 12:30 pm, in class, on March 1st, 2023. Show all of your work. Part I: Question 1 Marks: 9 Let 1 2 3 4, , andX X X X be independent and identically distributed random vectors with 2 1 0 = μ and 1 0 2 0 5 4 2 4 3 = Σ . a. Find the mean vector and covariance matrix for the linear combination 1 2 3 4 1 1 1 1 4 4 4 4 + + +X X X X . (3) b. Find the mean vector and covariance matrix for the linear combination 1 2 3 4− + −X X X X . (3) c. Obtain the covariance between the two linear combinations of random vectors in part a. and b. (3) Part I: Question 2 Marks: 19 Let 1 2 60, ,X X X be a random sample of size 60 from a bivariate normal distribution having mean μ and covariance matrix Σ . Completely specify each of the following: a. The distribution of X . (3) 2 b. The distribution of ( ) ( )11 1 −− −X μ Σ X μ . (2) c. The distribution of ( ) ( )160 −− −X μ Σ X μ . (2) d. The approximate distribution of ( ) ( )160 −− −X μ S X μ . (2) e. The distribution of 59S . (1) f. The distribution of (59 ) A S A , where 2 1 1 1 = A . (Note: You need NOT simplify/expand your solution.) (2) g. The distribution of 60X . (3) h. The distribution of 1 25 10+X X . (4) Part II: Questions requiring the use of R The R code solutions to these questions are to be submitted as a .txt document on the due date through D2L. When you are ready to submit your assignment in D2L: 1. Go to Assessments -> Assignments (at top of page). 2. Click “Assignment #5 R Code” 3. Go to “Add a file”. Upload your assignment. Multiple files will not be accepted. Note: Your R script will be pasted into R and run. In order to earn full marks, the output must be correct and your R script must be properly commented. Part II : Question 1 Marks: 20 a. Import the file AP.dat (provided in D2L) into R. Call your data AP and add the column names Wind, Solar Radiation, CO, NO, NO_2, O_3 and HC to columns 1 through 7 respectively. This data comes from Table 1.5 on p. 39 in your textbook. The description is as follows: “The data in Table 1.5 are 42 measurements on air-pollution variables recorded at 12:00 noon in the Los Angeles area on different days.” b. Construct QQ plots for each of the 7 variables and, for each, test for univariate normality using shapiro.test()as we did in class. Based on the plots, do you think we can assume multivariate normality? Explain. (5) c. Repeat part b. but plot the log transformed data (use log()) and calculate the p -values for the transformed data. What do you notice? (5) 3 d. Examine the pairs 5log( )X and 6log( )X for bivariate normality by calculating and displaying the squared statistical distances ( ) ( )1 , 1,2, ,42,j j j− − − =x x S x x and then creating an appropriate QQ plot to assess the distribution of the squared distances. (5) e. Refer to part c. and determine the proportion of observations 5[log( )jX , 6log( )]jX that fall within the approximate 50% probability contour of a bivariate normal distribution. (3) f. Based on your findings in part b.-d., do you think we can reasonably assume that the random vector with components 5log( )X and 6log( )X is bivariate normally distributed? Explain. (2) II 3S Chapter 1 Aspects of Multivariate Analysis .1.2. A morning newspaper lists the following used-car prices for a foreign compact with age XI measured in years and selling price X2 measured in thousands of dollars: 1 2 3 3 4 5 6 8 9 11 18.95 19.00 17.95 15.54 14.00 12.95 8.94 7.49 6.00 3.99 (a) Construct a scatter plot of the data and marginal dot diagrams. (b) Infer the sign of the sampkcovariance sl2 from the scatter plot. ( c) Compute the sample means X I and X2 and the sample variartces SI I and S22' Com- pute the sample covariance SI2 and the sample correlation coefficient '12' Interpret these quantities. (d) Display the sample mean array i, the sample variance-covariance array Sn, and the sample correlation array R using (I-8). 1.3. The following are five measurements on the variables Xl' X2, and X3: XI 9 2 6 5 8 X2 12 8 6 4 10 X3 3 4 0 2 Find the arrays i, Sn, and R. 1.4. The world's 10 largest companies yield the following data: The World's 10 Largest Companiesl Company Citigroup General Electric American Int! Group Bank of America HSBCGroup ExxonMobil Royal Dutch/Shell BP INGGroup Toyota Motor Xl = sales (billions) 108.28 152.36 95.04 65.45 62.97 263.99 265.19 285.06 92.01 165.68 X2 = profits (billions) 17.05 16.59 10.91 14.14 9.52 25.33 18.54 15.73 8.10 11.13 X3 = assets (billions) 1,484.10 750.33 766.42 1,110.46 1,031.29 195.26 193.83 191.11 1,175.16 211.15 IFrom www.Forbes.compartiallybasedonForbesTheForbesGlobaI2000, April 18,2005. (a) Plot the scatter diagram and marginal dot diagrams for variables Xl and X2' Com- ment on the appearance of the diagrams. (b) Compute Xl> X2, su, S22, S12, and '12' Interpret '12' 1.5. Use the data in Exercise 1.4. (a) Plot the scatter diagrams and dot diagrams for (X2, X3) and (x], X3)' Comment on thepattems. (b) Compute the i, Sn, and R arrays for (XI' X2, X3). Exercises 39 1.6. The data in Table 1.5 are 42 measurements on air-pollution variables recorded at 12:00 noon in the Los Angeles area on different days. (See also the air-pollution data on the web at www.prenhall.com/statistics. ) (a) Plot the marginal dot diagrams for all the variables. (b) Construct the i, Sn, and R arrays, and interpret the entries in R. Table 1.5 Air-Pollution Data Solar Wind (Xl) radiation (X2) CO (X3) NO (X4) N02 (xs) 0 3 (X6) HC(X7) 8 98 7 2 12 8 2 7 107 4 3 9 5 3 7 103 4 3 5 6 3 10 88 5 2 8 15 4 6 91 4 2 8 10 3 8 90 5 2 12 12 4 9 84 7 4 12 15 5 5 72 6 4 21 14 4 7 82 5 1 11 11 3 8 64 5 2 13 9 4 6 71 5 4 10 3 3 6 91 4 2 12 7 , 3 7 72 7 4 18 10 3 10 70 4 2 11 7 3 10 72 4 1 8 10 3 9 77 4 1 9 10 3 8 76 4 1 7 7 3 8 71 5 3 16 4 4 9 67 4 2 13 2 3 9 69 3 3 9 5 3 10 62 5 3 14 4 4 9 88 4 2 7 6 3 8 80 4 2 13 11 4 5 30 3 3 5 2 3 6 83 5 1 10 23 4 8 84 3 2 7 6 3 6 78 4 2 11 11 3 8 79 2 1 7 10 3 6 62 4 3 9 8 3 10 37 3 1 7 2 3 8 71 4 1 10 7 3 7 52 4 1 12 8 4 5 48 6 5 8 4 3 6 75 4 1 10 24 3 10 35 4 1 6 9 2 8 85 4 1 9 10 2 5 86 3 1 6 12 2 5 86 7 2 13 18 2 7 79 7 - 4 9 25 3 7 79 5 2 8 6 2 6 68 6 2 11 14 3 8 40 4 3 6 5 2 Source: Data courtesy of Professor O. C. Tiao.