Probability Question for David Sklansky

[Borodog]

Others are welcome as well, but David is the master of these sorts of problems, so I'd like to hear his response.

Two of my students missed Test 4 and had to take a makeup. They took the test at the same time, in the same room, sitting at the same table. The test on the left is the male's, the test on the right the female's. He got 14/19, she 15/19. His previous test scores are significantly better than hers. In case you can't tell from the picture, on her test, on number 11 she had originally selected D, but then erased it and chose C. The girl is of above average attractiveness.

What is the probability that cheating occurred?

[vhawk01]

Wouldn't he need to know some estimate of the likelihood of each person getting any given question correctly? You could easily craft a test where this was 100:1 and a near lock to be cheating.

[Borodog]

[ QUOTE ]
Wouldn't he need to know some estimate of the likelihood of each person getting any given question correctly? You could easily craft a test where this was 100:1 and a near lock to be cheating.

[/ QUOTE ]

Good point. The average on the test was 10.3/19, with a standard deviation of 3.6. None of the questions were "gimmes." There were several other students that took the makeup, at the same time, at the same table, and their tests looked completely different. 2 students scored 8s, but their 8s were very different 8s, with little overlap between the questions they got wrong.

[Borodog]

Also remember, it's not just a matter of getting the same problems correct; it's also a matter of getting the incorrect problems incorrect in the same way.

[surftheiop]

Not that im going to guess on a percentage but 2 ?s (seeing as accusing someone of cheating in college is really serious).
Did they know each other well? If they did would studying together/in the same group be a feasible explanation for them making similar mistakes? Also on the ones they got wrong, did most of the class choose the same wrong answers if they got the problems in question wrong or were their wrong answers more unique? (This is based on my experience taking highschool tests when it seems most people whol get a question wrong do so for the same reason)

[Borodog]

[ QUOTE ]
Not that im going to guess on a percentage but 2 ?s (seeing as accusing someone of cheating in college is really serious).
Did they know each other well? If they did would studying together/in the same group be a feasible explanation for them making similar mistakes?

[/ QUOTE ]

Could be, but they are in 2 different sections.

[ QUOTE ]
Also on the ones they got wrong, did most of the class choose the same wrong answers

[/ QUOTE ]

No.

[ QUOTE ]
if they got the problems in question wrong or were their wrong answers more unique? (This is based on my experience taking highschool tests when it seems most people whol get a question wrong do so for the same reason)

[/ QUOTE ]

[DougShrapnel]

Zero, but I'd estimate that teamwork occured 1. [img]/images/graemlins/smile.gif[/img]

[PairTheBoard]

I would question the students and go for a confession. Give them a plea bargain. If they confess let them retake the test with both getting the lowest score between them on the retake - or maybe with points taken off their own score. Tell them if they don't confess you consider the evidence strong for cheating and you will have to take the matter to higher authorities. Opening it up for class discussion might be one such higher authority. You can also take into consideration your read on whether they are lying if they refuse to confess.

As for the strength of the evidence for cheating, you could argue that even assuming they both knew the answers to the same 14 questions - which is a longshot already - the chance that they would randomly guess the same way among the 4 incorrect choices on the 4 questions they both missed is 1 in 256. Without thinking about it too much that looks like about 99.6% confidence that the wrong guesses were copied by one of them.

PairTheBoard

[PairTheBoard]

btw, there are Statistics Experts here who could probably give you a more rigorous statistical inference than my 1 chance in 256. However, I don't think Sklansky is one of them. He's much better at computing odds on God.

PairTheBoard

[Wyman]

[ QUOTE ]
btw, there are Statistics Experts here who could probably give you a more rigorous statistical inference than my 1 chance in 256. However, I don't think Sklansky is one of them. He's much better at computing odds on God.

PairTheBoard

[/ QUOTE ]

Zing!

I'd expect that they could dismiss some wrong answers, and thus there would be a higher probability (than 1/256) that they randomly guess the same wrong answers.

I have a facebook account, and I gave a quiz once and suspected cheating. I looked on facebook, and one of their "status updates" was "[Name] is a cheater."

I love the guilty conscience. The student had dropped before the next class.

FWIW, I agree with confronting them. Tell them your job is not to accuse them of or judge them for cheating. Your job is to take any situation which has overwhelming evidence of cheating and present it to the appropriate higher-ups. This way you come off as the coach/helper/advocate, rather than the complete adversary (as much as this is possible). Alternately, I've also simply said to the class: "There was clear evidence of cheating on this quiz/exam. I will be presenting these cases to the [honor council, dean, dept chair, or whatever-the-heck you have]. If you'd like to fess up before-hand, we can possibly work something out."

Seems to work.