Observations (Xi, Yi), i 1, ,n, are made from a bivariate normal population with parameters ( Icirc frac14 X, Icirc frac14 Y, Iuml 2X, Iuml 2Y, p), and the model Yi Icirc plusmn Icirc sup2 xi Icirc micro i is going to be fit (a) Argue that the hypothesis H0 Icirc sup2 0 is true if and only if the h...

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Observations (Xi, Yi), i = 1,...,n, are made from a bivariate normal population with parameters (Î¼X, Î¼Y,

Question:

Observations (Xi, Yi), i = 1,...,n, are made from a bivariate normal population with parameters (Î¼X, Î¼Y, Ïƒ2X, Ïƒ2Y, p), and the model Yi = Î± + Î²xi + Îµi is going to be fit.
(a) Argue that the hypothesis H0: Î² = 0 is true if and only if the hypothesis Ho: p = 0 is true.
(b) Show algebraically that

where r is the sample correlation coefficient, the MLE of p.
(c) Show how to test H0: p = 0, given only r2 and n, using Student's t with n - 2 degrees of freedom. (Fisher derived an approximate confidence interval for p, using a variance stabilizing transformation. See Exercise 11.4.)