1.25 We noted in Section 1.4.2 that the midpoint of the score confidence interval (1.14)

for π is the sample proportion after adding 22/2 observations to the sample, half of each type. This motivates a simple confidence interval,

$$ѱ Ζα/2√元(1 – ñ)/n*$$, where n* = n + 2/2.

Show that the variance (1 – 元)/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. [Hint: Use Jensen's inequality.] Thus, this interval, which is sometimes referred to as the Agresti-Coull confidence interval, contains the score interval. [Agresti and Coull (1998) and Brown et al. (2001) showed that it performs much better than the Wald interval. It does not have the score interval's disadvantage (Exercise 16.32) of poor coverage near 0 and 1. With 95% confidence, this motivates a simple method that uses the Wald method after adding 2 observations of each type (Agresti and Coull 1998, Agresti and Caffo 2000); this is sometimes called the plus four confidence interval.]