When the sum of the \(x\) values is equal to zero, the calculation of the coefficients of the regression line of \(Y\) on \(x\) is greatly simplified; in fact, their estimates are given by

\[\widehat{\alpha}=\frac{\sum y}{n} \text { and } \quad \widehat{\beta}=\frac{\sum x y}{\sum x^{2}}\]

This simplification can also be attained when the values of \(x\) are equally spaced; that is, when they are in arithmetic progression. We then code the data by substituting for \(x\) the values \(\ldots,-2,-1,0,1,2, \ldots\), when \(n\) is odd, or the values \(\ldots,-3,-1,1,3, \ldots\), when \(n\) is even. The preceding formulas are then used in connection with the coded data.

(a) Because of high lead residue, a faucet manufacturer cannot sell his product for home use unless each item is labeled as being hazardous to health. A consulting engineer suggests adding an acid bath at the end of the production line. An experiment is conducted with the bath having 0.2,0.4,0.6,0.8,1.0,1.2 and 1.4 percent acid solutions. Suppose the corresponding values of lead residue are 4.6, 4.0, 3.3, 3.6, 3.0, 2.4, and \(1.6 \mathrm{ppm}\).

Fit a least squares line and give the point prediction of the lead residue when using a 1.3 percent solution.

(b) Encouraged by the responses in Part (a), one further test was conducted with a 1.6 percent solution. Suppose the resulting lead residue is 1.1 ppm. Fit a least squares line using all eight runs. Again predict the lead residue for a 1.3 percent solution.