The correlation coefficient measures linear association. Suppose that a random variable X takes on values evenly (uniformly distributed) over the range from 5 to 5, and that Z = X². i) Show that the correlation (Pearson) of X and Z is zero, even though there is clearly a systematic relationship. Explain. ii) What would be the expected value of the Spearman's rank correlation, and why? Using a statistical package, generate a random sample of 1000 values of X uniformly distributed on [5, 5], compute Z = X² for each value of X, add e, randomly generated standard normal errors to get the 1000 values of Y = Z + e, and check the sample correlation between X and Y . iii) What is the population correlation between X and Y . iv) Do you get the correlation (sample) zero? In either case, justify your answer.