For a given sample proportion p, showthat a value 0 for which the test statistic z =

For a given sample proportion p, showthat a value π0 for which the test statistic z = (p − π0)/

√

[π0(1 − π0)/n] takes some fixed value z0 (such as 1.96) is a solution to the equation (1 + z2 0/n)π2 0

+ (−2p − z2 0/n)π0 + p2 = 0. Hence, using the formula x = [−b ±

√

(b2 − 4ac)]/2a for solving the quadratic equation ax2 + bx + c = 0, obtain the limits for the 95% confidence interval in Section 1.3.4 for the probability of success when a clinical trial has nine successes in 10 trials.