STA6349: Beta-Binomial

Consider a dining survey conducted by a restaurant owner in New York. The owner is also interested in knowing about the proportion \(\pi\) of students prefer eating out on Friday. He believes that 40% of the time, \(\pi < 0.7\) and 80% of the time, \(\pi < 0.9\).

1. What do the following prior distributions look like? (i.e., please graph them) Do either (or both) represent the owner’s beliefs well?

Beta(2, 2):

Beta(20, 20):

a. Do these priors have the same mean? Provide appropriate statistical justification in your response.

b. What can you see about the strength of the priors?

2. Suppose a survey yields 4 successes out of 10 responses. Find the posterior distributions with the following priors:

Beta(2, 2):

Beta(20, 20):

Comparison:

3. The restaurant owner finds information from the previous analysis, where his belief about the proportion of students’ favorite dining day being Friday was represented by a Beta(15, 10) distribution, resulted from a prior of Beta(3, 2) and survey results \(y=12\) and \(n=20\). The owner now wants to conduct another dining survey with the same question: Is Friday your favorite dining day?

a. The second survey gives a result of 8 yes out of 20 responses. Use the owner’s current beliefs and this information to update the restaurant owner’s belief about the proportion proportion, \(\pi\).

Prior: Beta(15, 10)

Posterior:

b. Let’s look at analyzing the combined data. Start with the initial prior of Beta(3, 2) and find the posterior distribution for \(y=12+8=20\) out of \(n=20+20=40\) responses.

Prior: Beta(3, 2)

Posterior:

c. Are the two posterior distribution the same in parts (a) and (b)? Why or why not?

d. Suppose the two survey results are reversed. That is, the first survey gives 8 yes and second survey gives 12 yes. Do you still observe the same posterior as in part (b)? Why or why not?

e. Suppose the two survey results are slightly different: the first survey gives 15 yes and the second survey gives 5 yes. What is the posterior distribution in this case?

Prior: Beta(3, 2)

Posterior:

4. Should we combine the two survey results together? Why or why not?

5. Describe a scenario where you believe it would be inappropriate to combine the survey results.