A study has been conducted to investigate the association between type II diabetes and incidence of chronic kidney disease (CKD). 100 people with type II diabetes and 100 people with no diabetes were recruited into the study and followed up for 10 year. The participants had normal kidney function at recruitment and the primary outcome was whether the patients develop chronic kidney disease during follow-up.

The dataset contains 200rows (one row per patient) and three columns-Subject: subject number (not to be used in analysis)-Diabetes coded with 1 if the patient had type 2 diabetes at recruitment and 0 if the patient had no diabetes at recruitment-CKD coded with1 if the participant has developed CKD during follow-up and zero otherwise.

(i)At the 5% level of significance test for an association between Diabetes and CKD incidence.

(ii)Calculate a 95% confidence interval for the difference in proportions of patients with CKD in the diabetic vs non-diabetic group.

(iii)Compare the results under points (i) and (ii) and formulate the conclusion.

2. A random walk is expressed as X1 = Z1, Xt = Xt-1 + Zt, t = 2,3, ..., where Zt ~ WN(uz, oz), that is, E(Zt) = uz, Var(Zt) = of, and Cov(Zt, Zs) = 0 for t * s. Determine which statements are true with respect to a random walk model; show calculations and provide complete explanations. I. If uz + 0, then the random walk is nonstationary in the mean. II. If oz = 0, then the random walk is nonstationary in the variance. III. If of > 0, then the random walk is nonstationary in the variance.A Markov chain with state space S = {0, +1, +2, 13, .. } and transition probabilities specified by Pi,i+1 = p, Pli-1 = 1 - p, where 0 Wk qkj - "j lj, j = 0, 1, 2, .. . are satisfied. (c) An example where we have a stationary solution, but the detailed balance equations have no solution, is given by the infinitesimal matrix 3 A = 2 Find the stationary distribution for this chain and show that the detailed balance equa- tions have no solution. Can you give verbal description of how this chain moves in the long run? How is this compared to the interpretation at the end of question (a)?\f