In 1986, the space shuttle Challenger exploded during take-off. The explosion was the result of an O-ring failure, a splitting of a ring of rubber that seals the parts of the ship together. The accident was believed to have been caused by the unusually cold weather at the time of launch (around 31 Fahrenheit degrees, or 0 Celcious degrees). Data on previous space shuttle at lunches and O-ring failures is given in the dataset challenger available in Moodle. The first column corresponds to the failure indicator y; (1 indicates failure,) and the second column to the corresponding temperature x in Fahrenheit degrees (1 i 24). (a) Generate 20,000 iterations of a Metropolis-within-Gibbs algorithm to estimate the intercept and the slope for the Challenger logistic regression (use the code provided in class). Plot the histograms, the convergence of the means and the autocorrelations of the intercept parameter and the slope parameter. Com- ment. (b) Compute the expected probability of failure at 65, 50 and 35 Fahrenheit de- grees and make a histogram for each case.