Question

The 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.

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? Why or why not?

A) No, there is not convincing evidence that the TSA officers did not carry out a truly random selection. If none of the 10
passengers chosen was seated in first class there is a 0/10 = 0% chance that the TSA officers did not carry out a random
selection

B) Yes, there is convincing evidence that the TSA officers did not carry out a truly random selection. Everyone knows that First class passengers get treated better than passengers who fly coach.

C) No, there is not convincing evidence that the TSA officers did not carry out a truly random selection. If the selection is truly random, there is about a 15% chance that no one in first class will be selected. So it is not surprising that a single random selection would contain no first class passengers.

D) Yes, there is convincing evidence that the TSA officers did not carry out a truly random selection. 15% of the passengers chosen should have been from first class.

E) Yes, there is convincing evidence that the TSA officers did not carry out a truly random selection. Every random sample should contain at least one first class passenger.

Answer 1

C) No, there is not convincing evidence that the TSA officers did not carry out a truly random selection. If the selection is truly random, there is about a 15% chance that no one in first class will be selected. So it is not surprising that a single random selection would contain no first-class passengers.

Step-by-step explanation:

In this scenario, we have 76 passengers on the flight, with 12 in first class and 64 in coach class. The TSA officers randomly select 10 passengers for extra security screening. Out of 100 simulation trials, in 15 of them, none of the 10 selected passengers were from first class.

To assess whether the TSA officers carried out a truly random selection, we need to consider the probability of not selecting any first-class passengers. If the selection is truly random, the probability of not selecting a first-class passenger in a single random selection is (64/76) * (63/75) * ... * (55/67), which is approximately 15%.

Given that in 15 out of 100 simulation trials, none of the selected passengers were from first class, this outcome aligns with the expected probability if the selection is truly random.

Answer 2

The simulation results showing no first-class passengers selected in 15 out of 100 trials do not necessarily indicate a lack of randomness in TSA's procedure due to the low proportion of first-class passengers.

Step-by-step explanation:

The student's question is about determining whether there is convincing evidence that the TSA officers did not carry out a truly random selection of passengers for additional screening. Given that in 15 out of 100 trials of a simulation, no first-class passengers were chosen, this does not necessarily provide convincing evidence against true randomness. The selection of first-class passengers is expected to be low due to their small proportion (12 out of 76 passengers), but not impossible. Hence, finding that none were selected in 15% of the trials aligns with the likelihood of such an event occurring by chance, and does not imply that the procedure was not random.