22S:138, Bayesian StatisticsFall 2009, Homework 3Due: Mon. 9/211. Refer to your answers to homework 2. Were your analyses robust to thedifferent prior specifications that you use d? Explain.2. (problem from Johnson and Albert) In 1986, the St. Louis Post Dispatchwas interested in measuring public support for the c onstruction of an indoorstadium. The newspaper conducted a survey in which they interviewed 301registered voters. In ths sample, 135 voters opposed the construction of a newstadium. Let p denote the proportion of all registered voters in the St. Louisvoting district opposed to the stadium.Suppose a city councilman wants to increase the city sales tax to pay for theconstruction of the new stadium. She claims that under half of the registeredvoters are opposed to s tadium construction. She would like to use the samplesurvey data of the newspaper to test the two hypotheses:H0: p ≥ .5Ha: p < .5(a) A frequentist method of testing these hypotheses is based on the p-value.The p-value in this case is the probability of observing the sample re-sult obtained, or something more extreme, if indeed exactly half of theregistered voters in St. Louis were opposed to construction; that is,p − value = P r(y ≤ 135|p = .5)where y is a binomial random variable with sample size n = 301 andsuccess probability p = 0.5. Compute the p-value for this example (useof an R function will make this easy). If this probability is small, thenone concludes that there is significant evidence in support of hypothesisHa: p < .5.(b) Now consider a Bayesian approach to testing these hypotheses. Supposethat a uniform prior is assigned to p. Find the posterior distribution ofp and use it to compute the posterior probabilities of H0and H1.3. Referring to the previous problem, again suppose that a uniform prior is placedon the prop ortion p, and that from a random sample of 301 voters, 135 opposeconstruction of a new stadium. Also suppose that the newspaper plans ontaking a new survey of 20 voters. Let y∗denote the number in this newsample who oppose construction.(a) Find the posterior predictive probability that y∗= 8. (Use one of the Rfunctions that I wrote and that were introduced in lab.)1(b) Find the 90% prediction interval for y∗. Do this by finding the predictiveprobabilities for each of the possible values of y∗and ordering them fromlargest probability to smallest. Then keep adding the most probablevalues y∗into the probability set until the total probability exceeds .90for the first time.4. (REQUIRED for stats and biostats grad students; OPTIONAL for undergradsand students in other majors.) Referring to the previous problem, supposemore generally that the newspape r plans on taking a new survey of n∗voters.Find the formula for the posterior predictive probability of y∗successes in asample of size n∗(still conditioning on the uniform prior and the results ofthe first survey). [Hint: You will need to compute the integral of the kernelof a beta density. Look at the expression for the normalized beta density, andrecall that it has to integrate to 1. The predictive probability that you needcan be expressed as a ratio involving some
View Full Document
Unlocking...