Aluminum scraps that are recycled into alloys are classified into three categories: soft drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements (e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: x1 = proportion of aluminum scraps from cans,
Table for Exercise 12.14

x2 = proportion of aluminum scraps from pots/pans, and
x3 = proportion of aluminum scraps from crank chambers. The first-order model, E(y) = β0 + b1x1 + b2x2 + b3x3, was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the table above.
a. Is the overall model statistically useful (at a = .05) for predicting the percentage of silver in the alloy? If so, give a practical interpretation of R2.
b. Is the overall model statistically useful (at a = .05) for predicting the percentage of iron in the alloy? If so, give a practical interpretation of R2.
c. Based on the parameter estimates, sketch the relationship between percentage of silver (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at a = .05.
d. Based on the parameter estimates, sketch the relationship between percentage of iron (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at a = .05?

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Element (y) Fp-value R° .075 Bo .0020 (.001) -5.958 B2 -.0000123 (.035) .084 B3 -.0000357 (.009) .054 -.0000141 (.015) .079 Silver .049 Iron <.001 .783 (<.001) (<.001) (<.001) (<.001)