A binomial sample of size n has y = 0 successes.

a. Show that the confidence interval for π based on the likelihood function is [0.0, 1 – exp( –z²_a/2/2n)]. For a = 0.05, use the expansion of an exponential function to show that this is approximately [0,2/n].

b. For the score method, show that the confidence interval is [0. Z²_a/2/(n + z²_a/2 )], or approximately [0, 4/(n + 4)] when α = 0.05.

c. For the Clopper—Pearson approach, show that the upper bound is 1 – (α/2)^1/n, or approximately – log(0.025)/n = 3.69/n when α = 0.05.

d. For the adaptation of the Clopper–Pearson approach using the mid-P-value, show that the upper bound is 1 – α^1/n, or approximately – log(0.05)/n = 3/n when α = 0.05.