Question

Out of 10 trials, a receiver got four cards correct. (This is identical to question 5). What is the lower bound of the Agresti-Coull confidence interval for the proportion of correctly identified cards? Round off to TWO digits after the decimal point.

Answer 1

Solution:

The Agresti – Coull lower bound for confidence interval for the proportion is given as below:

Lower limit = [(p̂ + (z^2/2n) - z*sqrt[(p̂q̂/n) + (z^2/4n^2)]]/[ 1 + (z^2/n)]

We are given

n = 10,

x = 4,

p̂ = x/n = 4/10 = 0.4,

q̂ = 1 - p̂ = 1 – 0.4 = 0.6

We assume confidence level = 95%

So, z = 1.96 (by using z-table)

Lower limit = [(0.4 + (1.96^2/2*10) – 1.96*sqrt[(0.4*0.6/10) + (1.96^2/4*10^2)]]/[ 1 + (1.96^2/10)]

Lower limit = [(0.4 + (1.96^2/20) – 1.96*sqrt[(0.4*0.6/10) + (1.96^2/4*10^2)]]/[ 1 + (1.96^2/10)]

Lower limit = [(0.4 + 0.19208 – 1.96*sqrt[(0.4*0.6/10) + (1.96^2/4*10^2)]]/[ 1 + (1.96^2/10)]

Lower limit = [(0.59208– 1.96*sqrt((0.4*0.6/10) + (1.96^2/400))]/[ 1 + (1.96^2/10)]

Lower limit = (0.59208- 0.359295)/1.38416

Lower limit = 0.168178

Lower limit = 0.17