Clarification on tests for independence and homogeneity?

Question

1. If the null hypothesis for a test of independence is true, what distribution does the test statistic have? Would it still follow a chi-square distribution, or a normal distribution?
2. In a goodness-of-fit test, it is possible to select the expected frequencies before the observed frequencies are known (i.e., before the sample is taken). Would this ever be possible in a test of independence?
3. Is a test of homogeneity equivalent to a 2 sample Z-test for proportions? A test of homogeneity is non-parametric and expands on a Z-test.

Thanks!

Answer 1

1. If $X,\;Y$ are bivariate Normal with correlation, $\rho ,$ then obtain the sample correlation coefficient, $R,$ and form the statistic $$ T = \frac{{R\sqrt {n - 2} }}{{\sqrt {1 - {R^2}} }}$$ A level $\alpha $ test is to reject ${H_0}:\rho = 0$ when $|T| \ge {t_{\frac{\alpha }{2},n - 2}}$