Safe margin of error. The margin of error z∗

^p(1

−

^p)/n is 0 when

^p is 0 or 1 and is largest when ^p is 1/2. To see this, calculate ^p(1 − ^p) for

^p

= 0, 0.1, 0.2, . . . , 0.9, and 1. Plot your results vertically against the values of ^p horizontally. Draw a curve through the points. You have made a graph of

^p(1

−

^p). Does the graph reach its highest point when ^p

= 1/2? You see that taking ^p = 1/2 gives a margin of error that is always at least as large as needed.