Chapter 13.4, Problem 52E

Utilization of sucrose as a carbon source for the production of chemicals is uneconomical. Beet molasses is a readily available and low-priced substitute. The article “Optimization of the Production of b-Carotene from Molasses by Blakeslea Trispora” β. of Chem. Tech. and Biotech., 2002: 933–943) carried out a multiple regression analysis to relate the dependent variable y = amount of β-carotene (g/dm³) to the three predictors amount of lineolic acid, amount of kerosene, and amount of antioxidant (all g/dm³).

Obs	Linoleic	Kerosene	Antiox	Betacaro
1	30.00	30.00	10.00	0.7000
2	30.00	30.00	10.00	0.6300
3	30.00	30.00	18.41	0.0130
4	40.00	40.00	5.00	0.0490
5	30.00	30.00	10.00	0.7000
6	13.18	30.00	10.00	0.1000
7	20.00	40.00	5.00	0.0400
8	20.00	40.00	15.00	0.0065
9	40.00	20.00	5.00	0.2020
10	30.00	30.00	10.00	0.6300
11	30.00	30.00	1.59	0.0400
12	40.00	20.00	15.00	0.1320
13	40.00	40.00	15.00	0.1500
14	30.00	30.00	10.00	0.7000
15	30.00	46.82	10.00	0.3460
16	30.00	30.00	10.00	0.6300
17	30.00	13.18	10.00	0.3970
18	20.00	20.00	5.00	0.2690
19	20.00	20.00	15.00	0.0054
20	46.82	30.00	10.00	0.0640

a.    Fitting the complete second-order model in the three predictors resulted in R² = .987 and adjusted R² = .974, whereas fitting the first-order model gave R² = .016. What would you conclude about the two models?

b.    For x₁ = x₂ = 30, x₃ = 10, a statistical software package reported that $\hat{Y}$ = .66573, $s_{\hat{Y}}$ =.01785, based on the complete second-order model. Predict the amount of β-carotene that would result from a single experimental run with the designated values of the independent variables, and do so in a way that conveys information about precision and reliability.