Question 14.5: At the beginning of this section we noted that the concentra...
At the beginning of this section we noted that the concentration of vanadium can be determined spectrophotometrically by making the solution acidic with \mathrm{H}_{2} \mathrm{SO}_{4} and reacting with \mathrm{H}_{2} \mathrm{O}_{2} to form a reddish brown compound with the general formula (\mathrm{VO})_{2}\left(\mathrm{SO}_{4}\right)_{3}. Palasota and Deming { }^{7} studied the effect on the absorbance of the relative amounts of \mathrm{H}_{2} \mathrm{SO}_{4} and \mathrm{H}_{2} \mathrm{O}_{2}, reporting the following results for a 2^{2} factorial design.
\begin{array}{ccc} \mathrm{H}_{2} \mathrm{SO}_{4} & \mathrm{H}_{2} \mathrm{O}_{2} & \text{Response }\\ \hline+1 & +1 & 0.330 \\ +1 & -1 & 0.359 \\ -1 & +1 & 0.293 \\ -1 & -1 & 0.420 \end{array}
Four replicate measurements were made at the center of the factorial design, giving responses of 0.334,0.336,0.346, and 0.323 . Determine if a first-order empirical model is appropriate for this system. Use a 90 \% confidence interval when accounting for the effect of random error.
We begin by determining the confidence interval for the response at the center of the factorial design. The mean response is 0.335 , with a standard deviation of 0.0094 . The 90 \% confidence interval, therefore, is
\mu=\bar{X} \pm \frac{t s}{\sqrt{n}}=0.335 \pm \frac{(2.35)(0.0094)}{\sqrt{4}}=0.335 \pm 0.011
The average response, \bar{R}, from the factorial design is
\bar{R}=\frac{1}{n} \sum R_{i}=\frac{1}{4}(0.330+0.359+0.293+0.420)=0.350
Because \bar{R} exceeds the confidence interval’s upper limit of 0.346 , there is reason to believe that a 2^{2} factorial design and a first-order empirical model are inappropriate for this system. A complete empirical model for this system is presented in problem 10 in the end-of-chapter problem set.