Analysis of proportions: a survey was done of bicycle and other vehicular traffic in the neighborhood of the campus of the University of California, Berkeley, in the spring of 1993. Sixty city blocks were selected at random; each block was observed for one hour, and the numbers of bicycles and other vehicles traveling along that block were recorded. The sampling was stratified into six types of city blocks: busy, fairly busy, and residential streets, with and without bike routes, with ten blocks measured in each stratum. Table 3.3 displays the number of bicycles and other vehicles recorded in the study. For this problem, restrict your attention to the first four rows of the table: the data on residential streets.

 

(a) Let y₁,…, y₁₀ and z₁,…, z₈ be the observed proportion of traffic that was on bicycles in the residential streets with bike lanes and with no bike lanes, respectively (so y₁ =16/ (16+58) and z₁=12/(12+113), for example). Set up a model so that the y_i’s are iid given parameters θ y and the zi ’s are iid given parameters θ_z.

(b) Set up a prior distribution that is independent in θ_y and θ_z.

(c) Determine the posterior distribution for the parameters in your model and draw 1000 simulations from the posterior distribution. (Hint: θ_y and θ_z are independent in the posterior distribution, so they can be simulated independently.)

(d) Let μ_y =E(yi |θ_y) be the mean of the distribution of the y_i’s; μ_y will be a function of θ y . Similarly, define μz . Using your posterior simulations from (c), plot a histogram of the posterior simulations of μ_y −μ_z, the expected difference in proportions in bicycle traffic on residential streets with and without bike lanes.