Chapter 11.12, Problem 76E

The results that follow were obtained from an analysis of data obtained in a study to assess the relationship between percent increase in yield (Y) and base saturation (x₁, pounds/acre), phosphate saturation (x₂, BEC%), and soil pH (x₃). Fifteen responses were analyzed in the study. The least-squares equation and other useful information follow.

$\hat{y}=38.83-0.0092x_1-0.92x_2+11.56x_3,S_{yy}=10965.46,\text{SSE}=1107.01,$

${10}^4{\left(X^{\prime }X\right)}^{-1}=\left[\begin{array}{cccc}151401.8& 2.6& 100.5& -28082.9\\ 2.6& 1.0& 0.0& 0.4\\ 100.5& 0.0& 8.1& 5.2\\ -28082.9& 0.4& 5.2& 6038.2\end{array}\right]$

1. a Is there sufficient evidence that, with all independent variables in the model. β₂ < 0? Test at the α = .05 level of significance.
2. b Give a 95% confidence interval for the mean percent increase in yield if x₁ = 914, x₂ = 65 and x₃ = 6.