A quantitative factor \(x\) with four equally-spaced levels \(0,1,2,3\) may be coded using either the indicator basis \(e_{0}, e_{1}, e_{2}, e_{3}\) (such that \(e_{r}(i)=I\left(x_{i}=right)\) ) or the polynomial basis \(x^{0}, x^{1}, x^{2}, x^{3}\) (with \(x^{0}=1, x^{1}=x\) ). Show that, if every level occurs with equal frequency in the design, the polynomials \(1, z=2 x-3,\left(z^{2}-ight.\) 5) \(/ 4,\left(5 z^{3}-41 zight) / 12\) are orthogonal with respect to the standard inner product in \(\mathbb{R}^{n}\). Show that the components of the \(4 \times 4\) transformation matrix that expresses this polynomial basis in terms of the indicator basis are all integers.