Chapter 8.3, Problem 6P

Critical Thinking: Brain Teaser A requirement for using the normal distribution to approximate the $\hat{p}$ distribution is that both up> 5 and nq > 5. Since we usually do not know p, we estimate p by $\hat{p}$ and q by $\hat{q}=1-\hat{p}$. Then we require that $n\hat{p}>5\text{ and }n\hat{q}>5$. Show that the conditions $n\hat{p}>5\text{ and }n\hat{q}>5$ 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 $n\hat{p}>5$. replace $\hat{p}$ by r/n and solve for r. In the inequality $\begin{array}{l}n\hat{q}>5\\ \end{array}$. replace $\hat{q}\text{ by }(n-r/n)$ and solve for $n-r$.