Part 4 of 8
Points: 0.13 of 1
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 (a) through (d) below.
Sketch the graph of the normal distribution with the indicated probability shaded
A.
B.
C.
(x).
The normal distribution cannot be used to approximate the binomial distribution.
(b) Find the probability that the number who have encountered fraudulent charges on their credit cards is at least 66 .
(Round to four decimal places as needed)

Step 1 ：Given that the number of trials (n) is 100 and the probability of success (p) is 0.63, we first need to check if the normal distribution can be used to approximate the binomial distribution. The rule of thumb is that the normal distribution can be used if both np and n(1-p) are greater than or equal to 5.

Step 2 ：Calculate np and n(1-p): \(np = 100 * 0.63 = 63\) and \(n(1-p) = 100 * (1-0.63) = 37\). Both values are greater than 5, so we can use the normal distribution to approximate the binomial distribution.

Step 3 ：Next, we need to find the probability that the number who have encountered fraudulent charges on their credit cards is at least 66. This can be done by calculating the z-score for 66 and finding the area under the normal distribution curve to the right of this z-score.

Step 4 ：Calculate the mean and standard deviation of the distribution: mean = np = 63 and standard deviation = \(\sqrt{np(1-p)} = \sqrt{63 * 0.37} = 4.83\).

Step 5 ：Calculate the z-score for 66: \(z = \frac{66 - mean}{std_dev} = \frac{66 - 63}{4.83} = 0.62\).

Step 6 ：Finally, find the area under the normal distribution curve to the right of this z-score. This is the probability that the number who have encountered fraudulent charges on their credit cards is at least 66. The probability is 0.2672.

Step 7 ：Final Answer: The probability that the number who have encountered fraudulent charges on their credit cards is at least 66 is \(\boxed{0.2672}\).