Problem
Determine whether you can use the normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use the binomial distribution to find the indicated probabilities. A survey of adults in a region found that $63 \%$ have encountered fraudulent charges on their credit cards. You randomly select 100 adults in the region. Complete parts through (d) below. Determine whether a normal distribution can be used to approximate the binomial distribution. Choose the correct answer below. A. No, because $n q<5$ B. Yes, because both $n p \geq 5$ and $n q \geq 5$ C. No, because $\mathrm{np}<5$. (a) Find the probability that the number who have encountered fraudulent charges on their credit cards is (a) exactly 66, (b) at least 66, and (c) fewer than 66 . (Round to four decimal places as needed.)
Solution
Step 1 ：Given that n = 100, p = 0.63, and q = 1 - p = 0.37
Step 2 ：Calculate np and nq: np = 100 * 0.63 = 63.0, nq = 100 * 0.37 = 37.0
Step 3 ：Since both np and nq are greater than 5, we can use the normal distribution to approximate the binomial distribution
Step 4 ：\(\boxed{\text{B. Yes, because both } np \geq 5 \text{ and } nq \geq 5}\)
