代写MH4501 Multivariate Analysis作业

日期：2024-02-28 08:56

MH4501 Multivariate Analysis, Semester 2 AY 2023-24

Nanyang Technological University, SPMS, MAS

Assignment 2

Due date: 6:30 PM, 7 March 2024

Remark. You are allowed to use R for the computations, but there is no need to paste your R code in your

assignment. What matters the most is to explain the reasoning and assumptions that you make.

Question 1

Consider n = 15 realisations x1, . . . , x15 from N3(μ,Σ) where μ and Σ are unknown. The sample mean is

xˉ =

(

0.494 ?1.785 ?1.369)?

and the sample covariance matrix is

S =

?? 5.8282 ?1.3361 1.7434?1.3361 7.5816 2.2085

1.7434 2.2085 1.9713

??

i) Test the hypothesis H0 : μ = (0,?1,?1)? at the significance level α = 0.01.

ii) Test the three hypotheses H0 : μ1 = 0, H0 : μ2 = ?1 and H0 : μ3 = ?1 and compare the results to i).

Question 2

Suppose a medical researcher hypothesises that a treatment consisting of the simultaneous administration

of two drugs is more effective than a treatment consisting of the administration of only one of the drugs. A

study is designed in which 20 subjects are randomly divided into 4 groups of 5 subjects each:

? Group 1: subjects are given a placebo.

? Group 2: subjects are given a combination of two drugs

? Group 3: subjects are given one of two drugs

? Group 4: subjects are given the other drug

The effectiveness of the drugs is measured by two response variables x1 and x2, reported in Table 1. Test

the hypothesis that the means of all the treatments are equal. Is the effectiveness of the drug given to the

third treatment group significantly different from the effectiveness of the drug given to to fourth treatment

group?

1

Table 1: Data for drug effectiveness study

Treatments

1 2 3 4

x1 x2 x1 x2 x1 x2 x1 x2

1 2 8 9 2 4 4 5

2 1 9 8 3 2 3 3

3 2 7 9 3 3 3 4

2 3 8 9 3 5 5 6

2 2 8 10 4 6 5 7

Means 2 2 8 9 3 4 4 5

Question 3

Two measurements were collected on each of 36 flea-beetles; 18 of the beetles were from a species called

Chaetocnema concinna and the other 18 were from another species called Chaetocnema heikertingeri. The

first variable consisted of the sum of widths (in micrometres) of the first joints of the first two tarsi (feet);

and the second variable consisted of the corresponding sum for the second joints. It is of interest to know

whether or not the population means of the two species are different.

The sample means are x1 = (181.50, 129.17)

? and x2 = (205.06, 120.44)? and the sample covariance

matrices are

S1 =

(

120.58 56.25

56.25 44.63

)

S2 =

(

203.94 73.42

73.42 47.14

)

Conduct a suitable hypothesis test. State your conclusion in words. What assumptions have you made in

constructing the test? Do any of these assumptions seem suspect with these data?