I am trying to solve exercise 5.13 in Gelman et. al. Bayesian Data Analysis (third edition). The exercise says:
Hierarchical binomial model: Exercise 3.8 described a survey of bicycle traffic in Berkeley, California, with data displayed in Table 3.3. For this problem, restrict your attention to the first two rows of the table: residential streets labeled as ‘bike routes,’ which we will use to illustrate this computational exercise.
The data are
y=[16 9 10 13 19 20 18 17 35 55] and
n=[ 74 99 58 70 122 77 104 129 308 119].
(a) Set up a hierarchical model for the observed number of bicycles on streets $ j =1, \cdots , 10$ that is binomial with unknown probability $\theta_j$ and sample size equal to the total number of vehicles (bicycles included) in that block. Assign a beta population distribution for the parameter $\theta_j$ and a non-informative hyperprior distribution. Write down the joint posterior distribution.
(b) Compute the marginal posterior density of the hyperparameters and draw simulations from the joint posterior distribution of the parameters and hyperparameters.
(c) Compare the posterior distributions of the parameters $\theta_j$ to the raw proportions in location j.
(d) Give a 95% CI for the average underlying proportion of traffic that is bicycles.
(e) A location on a new residential street with a bicycle route is sampled at random during which time 100 vehicles of all kinds go by. Give a 95% CI for the Number of those vehicles that are bicycles. Discuss how much you trust this interval in application.
I have managed to solve (a), (b) and (c). I would like to ask if you could help me with (d) and (e). For question (d) I thought to do the following: For each variable $\theta_j$ I have computed the 95% CI. So the mean of these lower(upper) bounds will be the lower(upper) bound for the wanted CI. Is this correct?
For question (e) I thought the following: Let [a,b] be the 95%CI from (d) question. The meaning of this CI is (correct me please if I am wrong) :[a,b] is the interval where with 95% probability the observed vehicle is bicycle. So, from the 100 new observed vehicles I predict that 95 of them will lie among a and b will be bicycle. But, I don't understant if I can trust this interval. Could you please give me a hint.
Please, I don't want you to solve the excercise for me, just give me hints to continue.