Consider the Bernoulli regression model, P(yi = 1) = Pi = 14 pxiB' yi 6 (0, 1}, i = 1, ..., n, with / a one dimensional unknown parameter. The log-likelihood function is given by L(B) = BExiyi - Clog (1 + e" iB ) . (1) (10 pts) By finding dL/dB and d'L/d32, show that the Newton-Raphson algorithm for finding the maximum likelihood estimator B is given by B (t+1) _ B(t) + Linxi(yi - pit) ) where exiB(t) Pi 1 + exiB(t) . (2) (10 pts) If n = 10 and W is the n x n diagonal matrix with ith element pi(1 - pi), where pi is pi estimated at 3, and assuming that approximately B = N (B, (X'WX)-1), where X is the n x 1 vector of predictor variables (xi = i/10), find the approximate variance of B when its observed value is B = -0.34. (3) (10 pts) What is the value of the test statistic for testing the hypothesis B = 0. (4) (10 pts) What is the outcome of the test if the level of significance is chosen to be 0.1. (5) (10 pts) Write down an expresson for the deviance of the model and what is the approx- imate distribution of it if the model is correct