Applet Exercise Activate the applet Binomial Revision and scroll down to the section

Chapter 16, Problem 16.13

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Applet Exercise Activate the applet Binomial Revision and scroll down to the section labeled Credible Interval. Change the value of the Bernoulli proportion to 0.45 and the parameters of the beta prior to = 3 and = 5 and press Enter on your computer. a What is the data-free credible interval for p based on the beta (3, 5) prior? b Use the applet Beta Probabilities and Quantiles (accessible at the www.thomsonedu.com/ statistics/wackerly) to calculate the prior probability that p is larger than the upper endpoint of the interval that you obtained in part (a). Also calculate the probability that p is smaller than the lower endpoint of the interval that you obtained in part (a). c Based on your answers to part (b), what is the prior probability that p is in the interval that you obtained in part (a)? Do you agree that the interval obtained in part (a) is a 95% credible interval for p based on the beta (3, 5) prior? d Click the button Next Trial once. Is the posterior based on the sample of size 1 different than the prior? How does the posterior differ from the prior? e What is a 95% credible interval based on the prior and the result of your sample of size 1? Is it longer or shorter than the interval obtained (with no data) in part (a)? f Click the button Next Trial once again. Compare the length of this interval (based on the results of a sample of size 2) to the intervals obtained in parts (a) and (e). g Use the applet Beta Probabilities and Quantiles to calculate the posterior probability that p is larger than the upper endpoint of the interval that you obtained in part (f). Does the value of this posterior probability surprise you? h Click the button Next Trial several times. Describe how the posterior is changed by additional data. What do you observe about the lengths of the credible intervals obtained using posteriors based on larger sample sizes?

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