57. "To construct a large-sample confidence interval for a proportions, it is not necessary to substitute for the unknown value of x in the formula for the standard error of . A less approximate method for constructing a 95% confidence interval finds the endpoints by determining the 7 values that are 1.96 standard errors from the sample proportion. That is, one solves for or in the equation |-|=1.96 (1-x) n One can solve this by trial and error, using the endpoints of the usual confidence interval as initial guesses. Or one can square both sides of the equation and solve the resulting quadratic equation.

a) Use this method for the data in Problem 5.16, and compare the result to that obtained with the usual method.

b) Explain what happens with the usual method when the sample proportion equals 0 or 1. (The two methods give similar results for very large samples, but otherwise can be quite different if the sample proportion is near 0 or 1.)