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3 Learning About a Proportion 3.1 Introduction Suppose data y is observed from a sampling distribution f ( y | θ ) that depends on an unknown parameter θ . We assume that one has beliefs about θ before sampling that are expressed through a prior density g ( θ | y ). Once a value of y is observed, then one’s updated beliefs about the parameter θ are reflected in the posterior density, the conditional density of θ given y : g ( θ | y ) = f ( y | θ ) g ( θ ) f ( y ) , where f ( y ) is the marginal density of y f ( y ) = f ( y | θ ) g ( θ ) dθ. In the computation of the posterior density, note that the only terms in- volving the unknown parameter θ are the likelihood function L ( θ ) = f ( y | θ ) and the prior density g ( θ ). Bayes’ rule says that the posterior density is pro- portional to the product of the likelihood and the prior, or g ( θ | y ) L ( θ ) g ( θ ) . In a Bayesian analysis, both the posterior density and the marginal density play important roles. The posterior density contains all information about the parameter contained in both the prior density and the data. One performs different types of inference by computing relevant summaries of the posterior density. The marginal density f ( y ) reflects the distribution of the data y before observing any data. This density is called the predictive density since f ( y ) is used to make predictions about future data values.

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2 3 Learning About a Proportion 3.2 An Example on Learning About a Proportion In this chapter, we discuss the basic elements of a Bayesian analysis through the problem of learning about a population proportion p . We take a random sample from the population of size n and observe y successes – for a given value of p , the probability of y is given by the binomial formula f ( y | p ) = n y p y (1 - p ) n - y . As an example, suppose that coordinator of developmental math courses at a particular university is concerned about the proportion of students in these courses who have math anxiety, where “math anxiety” is defined by obtaining a particular score on an anxiety rating instrument. A sample of 30 students takes the instrument and 10 have math anxiety. What can be said about the proportion of all developmental math course students who have math anxiety? The standard estimate of p is the proportion of successes in the sample ˆ p = y/n and the traditional Wald “large-sample” confidence interval for p is given by ˆ p - z α/ 2 ˆ p (1 - ˆ p ) n , ˆ p + z α/ 2 ˆ p (1 - ˆ p ) n , where z α is the 1 - α quantile of the standard normal distribution. For large samples, this interval will cover the unknown proportion in re- peated sampling with probability 1 - α . However this interval estimate has questionable value for samples with very few observed successes or failures. Suppose that no students in our sample have math anxiety. Then y = 0, ˆ p = 0 / 30 = 0 and the confidence interval will be degenerate at zero. (Simi- larly, if all the students have math anxiety, then ˆ p = 30 / 30 = 1 and the confi- dence interval will be degenerate at one.) Since one certainly believes that the
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