Can anyone help with a statistics problem?

Question

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!!!**

Its not a take home test, its a test review sheet. The final is next week (Summer A). So any help would be greatly appreciated.

Answer 1

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

Answer 2

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.

Answer 3

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

Answer 4

~So, the test was a take-home, huh? Nice try. Do it yourself.

Answer 5

1,2,3,4,5