The students at Hypothetical High School decide to protest standardized testing by purposely failing the state’s multiple choice test. In spite of their best efforts they still answer 1% of the questions correctly. Can you tell or estimate how well they would have scored based on how bad they did?
The details.
Test has 1000 questions. 4 choices for each question. 1 right and 3 wrong.
There are 1000 students in the class.
Each student is trying to answer every question incorrectly.
The average student answered 1% of the questions correctly.
I’m guessing that any score much lower than 25% indicates some knowledge.
Is there enough data to tell how well they could have done? Maybe with probability or statistics?
You have to make some assumptions, but overall this is a fairly trivial calculation:
Let x = the number of questions the average student knew (and purposely gave the wrong answer for)
Let y = the number of questions the average student did not know and guessed on.
Assuming the average students got wrong the questions they guesssed on 75% of the time (which is the statistical liklihood assuming random guessing…big assumption here), you can make the following equations:
1000 (total question on test) - x - 0.75 y = 10 (the number of questions answered correctly)
We also know that x + y =1000
so you can combine these two equations to make:
1000 - x - 0.75 (1000-x) = 10
using some algebra
x = 960
Plugging 960 for x into x + y =1000 means y = 40
So the total number of questions the average student would of gotten right is :
960 + 0.25 (40) = 970 or the average score would of been 97%
Hopefully I did not just answer your homework for you, and if I did you at least followed how I did it.
So either they know at least one answer is wrong, in which case they have a 100% chance of getting it wrong, or they’re unsure, in which case they have a 75% chance of getting it wrong. They got 99% of the answers wrong, so that would mean that we expect that they knew a wrong answer about 96% of the time.
Note that this doesn’t tell us whether or not they knew the right answer, just that they could identify at least one wrong answer.
First, let’s make the simplifying assumption that, if they’re trying to do as well as possible, they’ll get the correct answer on all the questions they do know and 25% of the questions they don’t know. And that if they’re trying to do as poorly as possible, they’ll get the correct asnwer on none of the questions they do know and 25% of the questions they don’t know.
In that case, knowing the correct answer to 96% of the questions would mean they get 0(96) + .25(4) = 1 percent of the questions correct.
And then, if they had been trying to do as well as possible, they would have gotten 97% correct.
My simplifying assumptions may not be realistic. If you don’t know which answer is correct, but you can rule out one or more of the choices as being obviously wrong, you have a 100% chance of getting the wrong answer if you’re trying to, but your chance of guessing correctly is 33.3% or 50%.
I don’t have time to work on this at the moment but I’m going to add an assumption for you:
Knowledge is all-or-nothing. If they know the answer, their probability of marking a wrong one is 100%. If they do not know the answer, the probability is 75%. That is, there are no throw-away or guessable choices.
Average random answer score would be 250.
Average student score was 10.
Students must have known correct answer to (250-10)/250 = 96% of questions.
That leaves 40 they didn’t know the answer to, for which random choice would give them an average score of 10, which is what they had.
Agreed. If you can make this assumption, then the math works out as several others have already posted.
The assumption is not consistent with typical standardized tests, though. Many multiple choice questions have one or more possible answers that can be readily eliminated, even if the test-taker doesn’t know the correct answer. Thus, students intentionally trying to score as poorly as possible could choose an obviously incorrect answer on some questions (perhaps most questions) even if they don’t know the correct one.
If this is a homework question, this is likely to be irrelevant. If it’s simply curiosity about a hypothetical situation, it makes a difference.
Quoting for emphasis and repetition. A major strategy in most multiple choice tests is eliminating answers you know are incorrect, and if this test is designed like most standardized tests there will be at least 1, if not 2 answers that an average student will know are incorrect.
If students get 99% “wrong”, then while there clearly are an average of 4% of questions in which they had no clue which answer was correct, on the other 96% you only know they determined one answer to be wrong. Thus it is feasible that the students would score no better than 32% on average. Determining an actual expected value would depend on the distribution of the answers among the students’ knowledge bases; just knowing that they’re unsure on 4% does not give you any information on how many questions they can eliminate 2 or 3 answers from.
Yeah, that’s exactly what I was going to say. (no really, you beat me to it. Bravo)
Plus everyone knows if “all of the above” is an option, it’s probably right. So they’d all try to avoid choosing that one.
In all seriousness there is no way to accurately answer the OP’s question as it stands. Now if you want to take the thought exercise a step further try the following addendum:
After the first round, the students get to take the test a second time. They cannot choose any of the answers they chose previously for any of the questions, and they are still trying to get as many wrong answers as possible. This time, they average 10% correct answers. The test is administered a third time, with the same rules (both sets of previous answers eliminated*), and they average 15% correct answers. How well would they have done on the test?
*I’m trying to figure out if this is solvable if only the answers from the 2nd round are eliminated but the 1st round options are back. It’s late and I’m tired so maybe someone else can determine that.
Just to allay everyone’s fears, this is not a homework question. I am well past HS age. My brain just comes up with what ifs. Thanks for all the replies so far.
A properly designed multiple choice test should have at least one plausible wrong answer that someone with superficial knowlege would jump on. When I took the American Chemical Society final exam in organic chemistry there were several students who, although trying, did worse than random.
