> wald = function(x, n, alpha) {
    z = -qnorm(alpha/2)
    phat = x/n
    se = sqrt(phat * (1 - phat)/n)
    c(phat - z * se, phat + z * se)
}

> wald(x = 23, n = 89, alpha = 0.05)
[1] 0.1674777 0.3493762

> prop.test(x = 23, n = 89, correct = F)

	1-sample proportions test without continuity correction

data:  23 out of 89, null probability 0.5 
X-squared = 20.7753, df = 1, p-value = 5.165e-06
alternative hypothesis: true p is not equal to 0.5 
95 percent confidence interval:
 0.1788154 0.3580294 
sample estimates:
        p 
0.2584270 


> agresti.coull = function(x, n, alpha) {
    z = -qnorm(alpha/2)
    ntilde = n + z^2
    ptilde = (x + z^2/2)/ntilde
    se = sqrt(ptilde * (1 - ptilde)/ntilde)
    c(ptilde - z * se, ptilde + z * se)
}

> agresti.coull(x = 23, n = 89, alpha = 0.05)
[1] 0.1782826 0.3585622