An article published in Engineering Structures and Technologies (Sep. 2012) presented the results of a study on the use of processed straw as thermal insulation for homes. Chopped straw specimens were prepared and tested for thermal conductivity at a temperature of 10º Celcius. Two variables were measured for each of 25 straw specimens: y = thermal conductivity (watts per meter-Kelvin) and x = density (kilograms per cubic meter). The data are provided in the  accompanying table. Consider the quadratic model,

E(y) = β_o + β₁ + β₂x²

a. Construct Y and X matrices for the data.

b. Find X'X and X'Y.

c. Use the technique outlined in Appendix A.4 to find (X'X)^-1.

d. Find the β̂ matrix and the least-squares prediction equation.

e. Find SSE and s².

f. Test H_o: β₂ = 0 vs. H_a: β₂ ≠ 0 using α = .05.

g. Conduct a test of overall model adequacy using α = .05.

h. Find and interpret R_a².

i. Find and interpret a 95% confidence interval for E(y) when x = 75.

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STRAW specimen Thermal Density (X) conductivity (y) 1 0.052 49 2 0.045 50 3 0.055 51 4 0.042 56 5 0.048 57 6. 0.049 62 7 0.046 64 8. 0.047 65 0.051 66 10 0.047 68 11 0.049 78 12 0.048 79 13 0.048 82 14 0.052 83 15 0.051 84 16 0.053 98 17 0.054 100 18 0.055 100 19 0.057 101 20 0.055 103 21 0.074 115 22 0.075 116 23 0.077 118 24 0.076 119 25 0.074 120