confidence interval involves the inversion of a hypothesis test. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. The collection of values, , for which the normal approximation is valid can be represented as where is the quantile of a standard normal distribution. Since the test in the middle of the inequality is a Wald test, the normal approximation interval is sometimes called the Wald interval or Wald method, after Abraham Wald, but it was first described by Pierre-Simon Laplace in 1812. Standard error of a proportion estimation when using weighted data Let there be a simple random sample where each is i.i.d from a Bernoulli(p) distribution and weight is the weight for each observation. Standardize the (positive) weights so they sum to 1. The weighted sample proportion is: . Since the are independent and each one has variance , the sampling variance of the proportion therefore is: . The standard error of is the square root of this quantity. Because we do not know , we have to estimate it. Although there are many possible estimators, a conventional one is to use , the sample mean, and plug this into the formula. That gives: For unweighted data, , giving . The SE becomes , leading to the familiar formulas, showing that the calculation for weighted data is a direct generalization of them. Wilson score interval The Wilson score interval is an improvement over the normal approximation interval in multiple respects. It was developed by Edwin Bidwell Wilson (1927). Unlike the symmetric normal approximation interval (above), the Wilson score interval is asymmetric. It does not suffer from problems of overshoot and zero-width intervals that afflict the normal interval, and it may be safely employed with small samples and skewed observations. The observed coverage probability is consistently closer to the nominal value, . Like the normal interval, the interval can be computed directly from a formula. Wilson started with the normal approximation to the binomial: with the analytic formula for the sample standard deviation given by Combining the two, and squaring out the radical, gives an equation that is quadratic in : Transforming the relation into a standard-form quadratic equation for , treating and as known values from the sample (see prior section), and using the value of that corresponds to the desired