Question

Airport securityThe Transportation Security Administration (TSA) is responsible for airport safety. On some flights, TSA officers randomly select passengers for an extra security check prior to boarding. One such flight had $76$passengers—$12$in first class and $64$in coach class. Some passengers were surprised when none of the $10$passengers chosen for screening were seated in first class. We want to perform a simulation to estimate the probability that no first-class passengers would be chosen in a truly random selection.

a. Describe how you would use a table of random digits to carry out this simulation.

b. Perform one trial of the simulation using the random digits that follow. Copy the digits onto your paper and mark directly on or above them so that someone can follow what you did.

c. In $15$of the $100$trials of the simulation, none of the $10$passengers chosen was seated in first class. Does this result provide convincing evidence that the TSA officers did not carry out a truly random selection? Explain your answer.

Short answer

a. We assign a unique number from $13$to $76$to each passengers in the coach class.

Now, we choose a row from the random digits table. Then we select the first two-digit number. If the number is between $01$and $76$then select the corresponding passenger

b. The first two-digit number is then chosen. Select the corresponding passenger if the number is between $01$and $76$, otherwise ignore the number and choose the next two-digit number. We'll do it again until you've chosen ten different passengers

c. Numerous trials resulted in no first-class passengers, it is likely that no first-class passengers will be selected, and thus there is no conclusive proof that the TSA officers did not select passengers at random.

Step-by-step solution

Part (a) Step 1 : Given Information

We have to describe how you would use a table of random digits to carry out this simulation.

Part (a) Step 2 : Simplification

In the question the TSA wants to check the security of the passengers by taking out the random sample. We need to describe how we would use a table of random digits to carry out this simulation. Thus, we have $76$passengers: $12$in first class and $64$in coach class. And $10$of the $76$passengers are chosen for screening. Then we assign a unique number from $1$to $12$to each passengers in first class and we assign a unique number from $13$to $76$to each passengers in the coach class.

Now, we choose a row from the random digits table. Then we select the first two-digit number. If the number is between $01$and $76$then select the corresponding passenger, else ignore the number and select the next two-digit number. We will repeat it until you have selected ten unique passengers.

Part (b) Step 1 : Given Information

We have to perform one trial of the simulation using the random digits given.

Part (b) Step 2 : Simplification

The TSA intends to assess the security of the passengers by taking a random sample in the query.

As a result, there are $76$passengers, including $12$in first class and $64$in coach. And $10$of the $76$passengers are chosen for screening out of a total of $76$. Then, for each passenger in first class, we assign a unique number ranging from $1$to $12$, and for each passenger in coach class, we assign a unique number ranging from$13$ to $76$.

$71487,09984,29077,14863,61683,47052,62224,51025$

The first two-digit number is then chosen. Select the corresponding passenger if the number is between $1$and $76$, otherwise ignore the number and choose the next two-digit number. We'll do it again until you've chosen ten different passengers. as an example,

$71$⇒ Select passenger$71$

$48$⇒ Select passenger $48$

$70$⇒ Select passenger$70$

$99$⇒ Ignore

$84$⇒ Ignore

$29$⇒ Select passenger $29$

$07$⇒ Select passenger$07$

$71$⇒ Ignore

$48$⇒ Ignore

$63$⇒ Select passenger $63$

$61$⇒ Select passenger$61$

$68$⇒ Select passenger $68$

$34$⇒ Select passenger$34$

$70$⇒ Ignore

$52$⇒ Select passenger$52$

We then note that the sample contains passengers $71,48,70,29,07,63,61,68,34,52$while only passenger $07$ is a first class passenger.

Part (c) Step 1 : Given Information

We have to explain if in $15$ of the$100$ trials of the simulation, none of the ten passengers chosen was seated in the first class does this result provide convincing evidence that the TSA officers did not carry out a truly random selection.

Part (c) Step 2 : Simplification

The TSA intends to assess the security of the passengers by taking a random sample in the query. Now, none of the ten passengers picked for the simulation were seated in first class in $15$of the$100$ trials. Because numerous trials resulted in no first-class passengers, it is likely that no first-class passengers will be selected, and thus there is no conclusive proof that the TSA officers did not select passengers at random.