Homework answers / question archive / STAT8121 Assignment 2 For your answers, you need to (1) state clearly the assumptions required for your answer; (2) attach the R codes you used to answer the question; (3) only a summary of R outputs can be used in your answers and you may attach the original R outputs in an Appendix Question 1 A physical anthropologist performed a mineral analysis of 9 ancient Peruvian hairs

STAT8121 Assignment 2 For your answers, you need to (1) state clearly the assumptions required for your answer; (2) attach the R codes you used to answer the question; (3) only a summary of R outputs can be used in your answers and you may attach the original R outputs in an Appendix Question 1 A physical anthropologist performed a mineral analysis of 9 ancient Peruvian hairs

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STAT8121 Assignment 2

For your answers, you need to (1) state clearly the assumptions required for your answer; (2) attach the R codes you used to answer the question; (3) only a summary of R outputs can be used in your answers and you may attach the original R outputs in an Appendix

Question 1 A physical anthropologist performed a mineral analysis of 9 ancient Peruvian hairs. The results for the chromium (x1) and strontium (x2) are stored in the data file “PeruHair.dat” (available on unit webpage). These observations comprise a random sample from a particular population. It is known that low levels (? 0.100 p.p.m.) of chromium suggest the presents of diabetes, while strontium is an indication of animal protein intake. (a) Construct and plot the 95% confidence ellipse for the population mean vector ? = (?1, ?2) T . (b) Obtain 90% Bonferroni simultaneous confidence intervals for ?1 and ?2. (c) Obtain 90% T2 simultaneous confidence intervals for ?1 and ?2. (d) Is it plausible that the mean strontium level in the population is 20.0? Explain your answer with reference to the previous parts

Question 2 In a study of “love and marriage” a sample of husbands and wives were asked to answer the following questions. (1) What is the level of passionate love you feel for your partner? (2) What is the level of passionate love that your partner feels for you? (3) What is the level of companionate that you feel for your partner? (4) What is the level of companionate that your partner feels for you? The responses were recorded on a 5-point scale: 1 = none 2 = very little 3 = some 4 = very much 5 = extremely Thirty husbands and thirty wives gave the responses and the data are included in the file “Love.dat” (available on unit webpage). In this data set the first 4 columns are the responses of husband rating wife, and the last 4 columns are the responses of wife rating husband.

(a) Plot the mean vectors for husbands and wives as sample profiles. (b) Perform hypothesis testing on the following questions. (Use ?=5%). 1. Is the husband rating wife profile parallel to the wife rating husband profile? 2. If yes to 1 then are these two profiles coincident? 3. If yes to both 1 & 2 then are these two profiles at the same level? What conclusions can be drawn from (b)?

Question 3. Suppose two independent multivariate random samples are collected, and they are denoted by 11 12 1n1 x and x, ,..., x 21 22 2n2 x , where n x, ,..., x 1 and n2 are the sample sizes and each xij is a p?1 vector. The first sample is collected from a population with mean ?1 and covariance matrix ?1 whilst the second sample from a population with mean ?2 and covariance matrix ?2. We assume these two samples are large, that is both n1 and n2 are large, however the populations distributions for these two samples are unknown. (a) Clearly explain how to test the hypothesis of H0: C ?1 = C ?2, where the C matrix is known and is r?p. You MUST provide the reasons for your test procedure. (b) Explain how to construct 100(1-?)% Bonferroni simultaneous confidence intervals for C (?1 - ?2).