Extending the life of an aluminum smelter pot. Refer to The American Ceramic Soc

Question

Extending the life of an aluminum smelter pot. Refer to The American Ceramic Society Bulletin (Feb. 2005) study of aluminum smelter pots, Exercise 10.9 (p. 497). Recall that the life length of a smelter pot depends on the porosity of the brick lining. The apparent porosity of each of six brick specimens, as well as the mean pore diameter of each brick, is reproduced in the accompanying table. Use matrix algebra and the method of least-squares to fit the straight-line model E(y) = beta_0 + beta_1x to the six data points. Source: Bonadia, P., et al. "Aluminosilicate refractories for aluminum cell linings." The American Ceramic Society Bulletin, Vol. 84, No. 2, Feb. 2005 (Table II). a. Construct Y and X matrices for the data. B. Find X'X and X'Y. c. Verify that (X'X)^-1 = [1.50384 -.12940 -.12940 012525] d. Find the beta matrix, then give the least-squares prediction equation. e. Find SSE and s^2.

Explanation / Answer

Solution:

a)

X = [1 12; 1 9.7 ; 1 7.3 ; 1 5.3 ; 1 10.9 ; 1 16.8]

Y = [18.8;18.3;16.3;6.9;17.1;20.4]

X =

1.0000 12.0000
1.0000 9.7000
1.0000 7.3000
1.0000 5.3000
1.0000 10.9000
1.0000 16.8000

Y =

18.8000
18.3000
16.3000
6.9000
17.1000
20.4000

b)

X'*X

X'*X

ans =

6.0000 62.0000
62.0000 720.5200

X'*Y

ans =

1.0e+03 *

0.0978
1.0878

c)

inv(X'*X)

ans =

1.5038 -0.1294
-0.1294 0.0125

d)

b =  inv(X'*X) *X'*Y

ans =

6.3126
0.9665

hence b0 = 6.3126 ,b1 = 0.9665

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