Use the following steps to show that \(r\) is bounded by -1 and 1 (These steps are due to Koch, 1990).

a. Let \(a\) and \(c\) be generic constants. Verify \[\begin{aligned}0 & \leq \frac{1}{n-1} \sum_{i=1}^{n}\left(a \frac{x_{i}-\bar{x}}{s_{x}}-c \frac{y_{i}-\bar{y}}{s_{y}}\right)^{2} \\& =a^{2}+c^{2}-2 a c r .\end{aligned}\]

b. Use the results in part (a) to show \(2 a c(r-1) \leq(a-c)^{2}\).

c. By taking \(a=c\), use the result in part(b) to show \(r \leq 1\).

d. By taking \(a=-c\), use the results in part (b) to show \(r \geq-1\).

e. Under what conditions is \(r=-1\) ? Under what conditions is \(r=1\) ?