MATH4630 Assignment 2 Solved

Question 1: Posten (1962) performed an experiment with 2 factors: velocity (2 levels: V1 and V2) and lubricants (three types: L1, L2, and L3). The ultimate torque x1 and the ultimate strain x2 of homogeneous pieces of bar steel are measured at each treatment combinations. The data are given below:

V1 V2

x1 x2 x1 x2

1. L1  7.80 90.4 7.12 85.1 7.10 88.9 7.06 89.0 7.89 85.9 7.45 75.9 7.82 88.8 7.45 77.9
2. L2  9.00 82.5 8.19 66.0 8.43 92.4 8.25 74.5 7.65 82.4 7.45 83.1 7.70 87.4 7.45 86.4
3. L3  7.60 94.1 7.06 81.2 7.00 86.6 7.04 79.9 7.82 85.9 7.52 86.4 7.80 88.8 7.70 76.4

1. State clearly the model in terms of the overall mean, main e↵ects and interaction e↵ects. This should include all the necessary assumptions and constraints such that we can answer the rest of the questions.
2. Obtain the all the necessary sum of squares.
3. Is there any evidence that treatment e↵ect exists?
4. Regardless of the answer in part (c), test for interaction e↵ect and then test for main e↵ect.

Question 2: Using the data set given in Question 1 and ignoring the velocity factor.

1. Is there any evidence that lubricant e↵ect exists?
2. Obtain the 95% confidence ellipsoid for the mean di↵erence between the L1 and L3.
3. Is there evidence of heterogeneity in variance?

1

Question 3: To compare two types of coating for resistance to corrosion, 15 pieces of pipe were coated with each type of coating. Two pipes, one with each type of coating, were buried together and left for the same length of time at 14 loactions. Corrosion for the coating was measured by two variables:

x1 = x2 =

The data are:

maximum depth of pit in thousandths of an inch number of pits

Coating 1

Coating 2

Location x1

1. 1  73
2. 2  43
3. 3  47
4. 4  53
5. 5  58
6. 6  47
7. 7  52
8. 8  38
9. 9  61
10. 10  56
11. 11  56
12. 12  34
13. 13  55
14. 14  65
15. 15  75

x2 x1 x2 31 51 35 19 41 14 22 43 19 26 41 29 36 47 34 30 32 26 29 24 19 36 43 37 34 53 24 33 52 27 19 57 14 19 44 19 26 57 30 15 40 7 18 68 13

Do the two coatings di↵er significantly in their e↵ect on corrosion? Clearly state the needed assumptions for your analysis.

Question 4: Let 22e

xe1, . . . , xen be
with 1 and 2 be the diagonal entries. Derive the likelihood ratio statistic for testing H0 :12 =2 =2.

a sample from

N2(μ, ⌃), where ⌃ is a diagonal matrix

2