If and are large, then the beta distribution can be approximated by the normal distribution

If α and β are large, then the beta distribution can be approximated by the normal distribution using the beta mean and variance given. This is useful in case beta distribution software is unavailable. Use the approximation to compute the credible interval in Example 15.8.

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Example 15.8 The posterior distribution of the parameter 0 = the pro- portion of all U.S. adults that incorrectly believe antibiotics kill viruses was a Beta(490, 455) dis- tribution. The .025 and .975 quantiles of this beta distribution are 1.025 = .4866 and 1.975.5503. So, after observing the results of the NSF survey, there is a 95% posterior probability that is between 4866 and .5503. For comparison, the one-proportion z interval based on y/n = 488/939 = 5197 is 5197(1-5197) 939 Due to the large sample size, the two intervals are quite similar. It must be emphasized that, even if the confidence interval is nearly the same as the credible interval for a parameter, they have different interpretations. To interpret the Bayesian credible interval, we say that there is a 95% probability that the parameter 0 is in the interval. .5197 +1.96 = (.4877, .5517)