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One sunny Friday, one of the stricter teachers of Lazy Larry's class decided to give her students a surprisingly mundane poll homework. Every student was required to talk to 100 people, dividing them into 7 groups based on the time they spend using the Internet, and writing down the percentage of people in each group.

Then, after heartily enjoying himself until Sunday, Larry came to his senses. His grades (especially at math) were barely high enough for him to pass his classes, so he had to get out and somehow manage to complete this freaking homework in this rainy day. Of course, he could just write made-up results, but he figured that it would've backfired on him, because teacher was more than smart enough to just ask him about the people and places he went to. In short, this "cunning plan" would take a lazy kid like him too much preparation to succeed.

He realized that he had to be so subtle that not even his not-quite-undeserved bad reputation would get in the way, so he couldn't get it done without getting out and talking to actual people. After he was finished with it, his homework read like this:

0 - 1 hours......1 - 2 hours......2 - 3 hours......3 - 5 hours......5 - 6 hours.......6 - 7 hours.....7+hours
....8%...................12%..................20%.................24%...............22%...............10%.................4%


The next day, the teacher looked at it for a minute and gave him an F, to his surprise. What did she notice that was wrong?



Notes:

  • For each group, the lower limit applies but the upper one doesn't. For example, someone spending 2 hours a day belongs to the 3rd group, not the 2nd.

  • There are only 15 students in Larry's class (himself included).

  • The teacher hates being made a fool of and just gives an F whenever she catches someone trying to cheat in his/her homework.

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  • 1
    $\begingroup$ The 3-5 hours looks strange, but definitely not impossible - there could easily be a drop like that in the middle. $\endgroup$ – YowE3K Jan 9 '17 at 22:58
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    $\begingroup$ @ChrisCudmore - I was thinking that a lot of people use the internet a lot, and a lot of people use it a little, so there could easily be a dip in the middle between the two extremes. And with only a sample size of 100 people, that dip could be quite pronounced. There's got to be more to this question than that. (I'm thinking that rand al'thor might have the right idea.) $\endgroup$ – YowE3K Jan 9 '17 at 23:20
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    $\begingroup$ I'm a bit confused about the wording of the problem. Did Larry actually talk to people? Is that where he got these numbers? $\endgroup$ – MikeQ Jan 9 '17 at 23:34
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    $\begingroup$ If the accepted answer is the correct answer, you really need to edit the question to make it clear that the teacher isn't qualified to teach that particular subject. Or to say that the teacher is lazier than Larry. Just because there is only a 1 in 5 chance that one of their students would have such an outcome is no reason to assume that student cheated. (Especially when someone who just made up the numbers is more likely, not less likely, to have numbers that fitted the teacher's naive expectations.) $\endgroup$ – YowE3K Jan 10 '17 at 15:16
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    $\begingroup$ @Nautilus - I dved your question, but it wasn't out of hate - it was simply a case of the answer not being determinable based on the question. The question was tagged with [logical-deduction] but it can not be deduced from the question (as it is currently written) that the teacher did not understand probability. $\endgroup$ – YowE3K Jan 10 '17 at 19:29
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He only asked 25 people. All of his percentages are even. The odds of getting all even percentages when asking 100 people are low. (edit - Oops, he asked only 50 people)

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    $\begingroup$ 2.5 people in 6-7 hours? It would have to be 50 people. And the odds of getting all even numbers with 50 people are (I think) 1 in 64. $\endgroup$ – YowE3K Jan 9 '17 at 23:17
  • $\begingroup$ Where do you see him only asking 25 people? $\endgroup$ – TrojanByAccident Jan 9 '17 at 23:28
  • $\begingroup$ @YowE3K Oops, yep, it has to be 50. But she knows Larry is lazy :-) I think the odds legitimize the F. $\endgroup$ – Hubba_Dubba Jan 9 '17 at 23:46
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    $\begingroup$ Still, asking only 50 people when asked for 100 doesn't really justify for straight F. Also, with a sizable number of students in a class, it's not entirely unlikely to have one student having all numbers even. If this is the intended solution, then the puzzle is bad =/ $\endgroup$ – justhalf Jan 10 '17 at 1:57
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    $\begingroup$ With a 1 in 64 chance of having all even numbers and a class of, say, 30 children, there is a 38% chance that at least one child would be given an F simply because they were unlucky enough to have had a random sample that the teacher didn't like. $\endgroup$ – YowE3K Jan 10 '17 at 3:15
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Larry is not timing people using the internet, he is just asking. Therefore most people would give answers in whole hours. So if many people say 1 hr for day, it would occupy the 1st and 2nd groups equally.

As the percents add up to 100 that is not the case. So most people are randomly placed between two groups

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  • $\begingroup$ Let's just say the lower limit for the group is included, and the upper one isn't. $\endgroup$ – Nautilus Jan 10 '17 at 8:44
  • $\begingroup$ This is one of the most common mistakes I see made when querying date data. I thought that's how you came up with the puzzle $\endgroup$ – Andrey Jan 10 '17 at 15:03
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The middle bucket spans two hours, while the rest span one. Seeing as the fat bucket is near the center of what appears to be a normal ish distribution, one would expect it to be closer to the sum of the adjacent buckets.

In other words, if we were to split it evenly, it would be 20-12-12-22 in the center 4 of 8 buckets, which is not the type of distribution expected.

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  • 1
    $\begingroup$ -1, because unless I'm misunderstanding what you're saying, that wouldn't be a good enough reason to just flat out give him an F $\endgroup$ – TrojanByAccident Jan 10 '17 at 0:07
  • $\begingroup$ You'd better dv the others as well. They are proposing unlikely but still entirely possible data distributions as evidence of data tampering as well. I don't see any complaints there. $\endgroup$ – Chris Cudmore Jan 10 '17 at 0:25
  • $\begingroup$ I think this may be the intended answer, but the distribution does not look that bad: imgur.com/a/jNvi3 In fact, if a class of 30 students were all asked to do this homework, it would be likely for at least one person to get something which does not resemble a normal distribution $\endgroup$ – Yang Li Jan 10 '17 at 0:26
  • $\begingroup$ @ChrisCudmore I did, actually, just wasn't sure what to write $\endgroup$ – TrojanByAccident Jan 10 '17 at 0:39

protected by Deusovi Jan 9 '17 at 23:18

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