Can anyone help with a statistics problem?

An instructor is planning on giving a multiple choice test. Recognizing that students may use a guessing strategy, she wants to deduct points for providing the wrong answer. The test will consist of 25 problems, each with 5 choices. The instructor will award 4 points for a correct answer, but wants the expected value of the grade to be 0 for a student who gueses. How would the penalty work?





What if the test consisted of 20 problems with 4 choices each? 5 points for a correct answer. What is the penalty?





What if the test had 30 problems with 3 choices each? 3 and 1/3 points for a correct answer. What's the penalty?





What if the test had 50 true/false type problems. 2 choices and 2 points for each correct answer. What's the penalty?





**If anyone can help to solve this problem, please tell me if there's some formula you have to use to solve it. I have a statistics book, but have no idea which formula, if any, I should use to solve it. Thanks!!!**

Can anyone help with a statistics problem?
First one:





25 * [1/5 (4) + 4/5 (x)] = 0


20 + 20x=0


20x = -20


x = -1





Second one:


20 [1/4 (5) + 3/4 (x) ] = 0


25 + 15x = 0


15x = -25


x = -25/15


x = -1 2/3





Third one:


30 [ 1/3 (3 1/3) + 2/3 (x )] = 0


100/3 +20x = 0


20x = -33 1/3


x = -1 2/3





50 [ 1/2 (2) + 1/2 (x) ] = 0


50 + 50x = 0


50x = -50


x = -1
Reply:wrong answer in part 4 Report Abuse

Reply:Let me try to teach you how to fish instead of handing you one. Think about these kind of problems intuitively, and they're pretty easy.





For each situation:


1) Figure out how many right and wrong answers a random guesser will get for a given number of questions (the number of questions you pick doesn't matter, but if you use the number of choices, it makes the math easy)





For the first example with 5 choices, you would expect that for every 5 questions, the guesser will get 1 right and 4 wrong. A simple equation for this would be:





num_wrong = num_choices - 1


num_correct = 1





2) Now weight the wrong answers so that they exactly balance out the correct answers.





(pts per correct answer) x (#correct) = (pts per wrong answer) x (#wrong)





For the first example this is: (4)(1) = (x)(4)


Solve for x:


x = (4)(1)/4


x = 1


Therefore you take away 1 point for each wrong answer.








For the true/false example, it would be:


num_wrong = 2-1 = 1


num_correct = 1





Therefore 1 right answer for each wrong answer.





Balance them out: (2)(1) = (x)(1)


x = 2


lose 2 points for each wrong answer





The math for the other two is more tricky, but the concept is the same.
Reply:let m = number of choices for each question


Q = score of correct answer


and let P = Penalty for error





P = -Q/(m-1)





independent of the number of questions.





Q=4, m=5, P=-1


Q=5, m=4, P=-5/3


Q=10/3, m=3, P=-5/3


Q=2 m=2, P=-2
Reply:1,2,3,4,5
Reply:~So, the test was a take-home, huh? Nice try. Do it yourself.