Critical Thinking: Brain Teaser A requirement for using the normal distribution to approximate the p distribution is that both np 5 and nq 5. Since

Critical Thinking: Brain Teaser A requirement for using the normal distribution to approximate the pˆ distribution is that both np 5 and nq 5.

Since we usually do not know p, we estimate p by pˆ and q by qˆ 5 1  pˆ. Then we require that npˆ 5 and nqˆ 5.

Show that the conditions npˆ 5 and nqˆ 5 are equivalent to the condition that out of n binomial trials, both the number of successes r and the number of failures n  r must exceed 5.

Hint:

In the inequality npˆ 5, replace pˆ by r/n and solve for r. In the inequality nqˆ 5, replace qˆ by n  r /n and solve for n  r.