Question

Question 2 Each day thousands of international travelers pass through Dulles International Airport. Before leaving the airport each international traveler must pass through Customs. Customs agents want to be sure that travelers do not bring illegal items into the country. However, they do not have time to search each person's luggage. Instead, they require each person to press a button after they show their passport. Each a red or green bulb lights up. If the red light shows the passenger will be searched by a Customs agent. A green light means, "go ahead." Customs agents claim that the proportion of all travelers who will be stopped is 0.30, because the light has a probability 0.30 of showing red on any push of the button. To test this claim, a concerned citizen watches a random sample of 100 travelers push the button Only 20 get a red light. Assume that the Customs agents' claim is true. Find the probability that the proportion of travelers who get a red light is as small or smaller than the result in this sample. Show your work. (10 points) A. B. Based on your results in part a, do you believe the Customs agents' claim. Explain. (4 points)

Answer 1

Here define a random variable X

X;- Number of passengers who is shown a red light

taken a sample of size 100 it is observed that 20% of passengers is shown the red light

now here we may consider red light as success and green light as faliures

probability of showing red light is 0.30 since it is assumed that the custom agent's assumption is correct

hence it is a bernoulli trials since 1) probability of success remains 0.3 for all 100 persons 2) each time the light is green or red is independent of other trials

so x follows Binomial distribution with size 100 and proportion 0.3

now 0.2 proportion of persons in the sample is shown the red light

so probability that 0.2 or less proportion of people is shown the red light is

$P(X\leq 20)=0.0164$

B) Now on the basis of the assumption of the custom agent it has 1.64% chance that red light is shown to 20 or less than 20 people where in reality on the basis of 100 sample of passengers 20 people is shown the red light. So we can say that the custom agent's assumption is not true.