Ethyl vanillin is used in food, cosmetics, and pharmaceuticals for its vanilla-like scent. The article “Determination and Cor Correlation of Ethyl Vanillin Solubility in Different Binary Solvents at Temperatures from 273.15 to 313.15 K” (J. Chem. Engr. Data 2017: 1788–1796) reported an experiment to determine y = ethyl vanillin solubility (mole fraction) as a function of x₁ = initial mole fraction of the chemical propan-2-one in the solvent mixture and x₂ = temperature (°K). The experiment was run at seven x₁ and nine x₂ values. The accompanying table shows the response, y, at each combination.

a. Create scatterplots of y versus x₁ and y versus x₂. Does it appear the predictors are linearly related to y, or would quadratic terms be appropriate? 

b. Would a scatterplot of x₁ versus x₂ indicate whether an interaction term might be suitable? Why or why not? 

c. Perform a regression using the complete second-order model. Based on a residual analysis, does it appear that the model assumptions are satisfied?
d. Test various hypotheses to determine which term(s) should be retained in the model.

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.4 5 6 7 8 9 1.0 = 273.15 10.6 13.4 16.8 18.5 19.5 19.8 278.15 12.8 16.2 19.2 21.0 21.4 21.5 283.15 13.7 18.8 20.9 23.1 23.4 23.5 23.8 26.4 26.7 26.9 29.6 29.8 32.1 32.4 36.2 41.3 45.7 288.15 17.3 20.7 293.15 20.5 25.2 26.9 29.1 298.15 23.7 27.2 29.8 31.2 303.15 27.4 31.6 33.4 308.15 34.3 36.9 38.7 313.15 38.5 42.1 43.4 35.0 36.2 40.7 41.0 45.3 45.5 19.9 21.6 23.7 27.1 30.0 33.0 36.9 41.6 45.9