From Section 1.4.2 the midpoint π̴ of the score confidence interval for Ï€ is the sample proportion for an adjusted data set that adds z²_a/2/2 observations of each type to the sample. This motivates an adjusted Wald interval,

Show that the variance π̴(1 €“ π̴)/n* at the weighted average is at least as large as the weighted average of the variances that appears under the square root sign in the score interval. Thus, this interval contains the score interval.

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where n* = n + zap. ñ± Za/½ V#(1 - ñ)/n* ,