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Bayesian interval estimation 11.58. Measurements of the diameters of a random sample of 200 ball bearings made by a certain machine during one week showed a mean of 0.824 inch and a standard deviation of 0.042 inch. The diameters are normally distributed. Find a (a) 90%, (b) 95%, and (c) 98% Bayesian HPD credibility interval for the mean diameter u of all ball bearings made by the machine. Assume that the prior distribution of u is normal with mean 0.8 inch and standard deviation 0.05.
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The posterior mean and standard deviation are respectively 0.824 and 0.0030. (a) The 90% HPD interval is given by 0.824 (b) The 95% HPD interval is given by 0.824 (c) The 98% HPD interval is given by 0.824 (1.645 (1.96 (2.33 0.003) or [0.819, 0.829]. 0.003) or [0.818, 0.830]. 0.003) or [0.817, 0.831].
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11.59. A sample poll of 100 voters chosen at random from all voters in a given district indicated that 55% of them were in favor of a particular candidate. Suppose that, prior to the poll, we believe that the true proportion u of voters in that district favoring that candidate has Jeffreys prior (see Problem 11.23) given by 1 p(u) , 0 u 1. Find 95% and 99% equal tail area Bayesian credibility intervals for !u(1 u) the proportion u of all voters in favor of this candidate.
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We have n 100 and x 55. From Problem 11.23, the posterior density of u is beta with parameters a 55.5 and b 45.5. This density has the following percentiles: x0.005 0.423, x0.025 0.452, x0.975 0.645, x0.995 0.673. This gives us the 95% Bayesian equal tail credibility interval [0.452, 0.645] and the 99% Bayesian equal tail credibility interval [0.423, 0.673]. (It is instructive to compare these with the traditional intervals we obtained in Problem 6.13.)
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CHAPTER 11 Bayesian Methods
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11.60. In the previous problem, assume that u has a uniform prior distribution on [0, 1] and find (a) 95% and (b) 99% equal tail area credibility intervals for u.
The posterior distribution of u is beta with parameters 56 and 46 (see Theorem 11-1). (a) We need the percentiles x0.025 and x0.975 of the preceding beta distribution. These are respectively 0.452 and 0.644. The 95% interval is [0.452, 0.644]. (b) We need the percentiles x0.005 and x0.995 of the preceding beta distribution. These are respectively 0.422 and 0.644. The 99% interval is [0.422, 0.672].
11.60. In 40 tosses of a coin, 24 heads were obtained. Find a 90% and 99.73% credibility interval for the proportion of heads u that would be obtained in an unlimited number of tosses of the coin. Use a uniform prior for u.
By Theorem 11-1, the posterior density of u is beta with a 25 and b 17. This density has the following percentiles: x0.00135 0.367, x0.05 0.469, x0.95 0.716, x0.99865 0.800. The 90% and 99.73% Bayesian equal tail area credibility intervals are, respectively, [0.469, 0.716] and [0.367, 0.800]. (The traditional confidence intervals are given in Problem 6.15.)
11.62. A sample of 100 measurements of the diameter of a sphere gave a mean x 4.38 inch. Based on prior # experience, we know that the diameter is normally distributed with unknown mean u and variance 0.36. (a) Find 95% and 99% equal tail area credibility intervals for the actual diameter u assuming a normal prior density with mean 4.5 inches and variance 0.4. (b) With what degree of credibility could we say that the true diameter is 4.38 0.01
(a) From Theorem 11-3, we see that the posterior mean and variance for u are 4.381 and 0.004. The 95% credibility interval is [4.381 (1.96 0.063), 4.381 (1.96 0.063)] [4.26, 4.50]. Similarly, the 90% credibility interval is [4.381 (1.645 0.063), 4.381 (1.645 0.063)] [4.28, 4.48]. (b) We need the area under the posterior density from 4.37 to 4.39. This equals the area under the standard normal density between (4.37 4.381)>0.063 0.17 and (4.39 4.381)>0.063 0.14. This equals 0.1232, so the required degree of credibility is roughly 12%.
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