The relative standard error of the negative of the natural logarithm of p is equal to the square root of the posterior variance of p divided by the product of p and the negative of the natural logarithm of p. The relative standard error of the negative of the natural logarithm of 1 minus p is equal to the square root of the posterior variance of 1 minus p divided by the product of 1 minus p and the negative of the natural logarithm of 1 minus p.

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This figure is titled "Small Area Estimate versus Effective Sample Size When the Relative Standard Error Equals 10 Percent." It is a graph of a function within a coordinate plane; the horizontal axis shows the proportion estimated (as a percentage), and the vertical axis shows the effective sample size. A horizontal line through the graph indicates an effective sample size of 208. At EFN = 208, the estimated proportion is either 5.23 or 50 percent or 94.77 percent. The graph is symmetric about a vertical line drawn at the estimated proportion that equals 50 percent. It decreases from an infinitely large required effective sample size when the estimated proportion is close to zero and approaches a local minimum of 154 when the estimated proportion is 20 percent. The graph increases for estimated proportions greater than 20 percent until a required effective sample size of 208 is reached for an estimated proportion of 50 percent. The graph decreases for estimated proportions greater than 50 percent and approaches a local minimum of 154 for the required effective sample size when the estimated proportion is 80 percent. The graph increases for estimated proportions greater than 80 percent and reaches an infinitely large required effective sample size when the estimated proportion is close to 100 percent.

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