Because the normal distribution can take all real numbers (is continuous) but the binomial distribution can only take integer values (is discrete), a normal approximation to the binomial should identify the binonial event "8" with the normal interval "(7.5, 8.5)" (and similarly for other integer values). The figure below shows that for P(X > 7) we want the magenta region which starts at 7.5.

Example: If n=20 and p=.25, what is the probability that X is greater than or equal to 8?

- The normal approximation without the continuity correction factor yields

z=(8-20 × .25)/(20 × .25 × .75)^.5 = 1.55, hence P(X *greater than or equal to* 8) is approximately .0606 (from the table). - The continuity correction factor requires us to use 7.5 in order to include 8 since the inequality is weak and we want the region to the right. z = (7.5 - 5)/(20 × .25 × .75)^.5 = 1.29, hence the area under the normal curve (magenta in the figure above) is .0985.
- The exact solution is .1019 approximation

**Competencies: **Use the normal approximation with the continuity correction factor to approximate the probability of more than 40 successes if n=60 and p=.75.