I want to find the probability that my student is a random guesser. On a 360-item multiple choice test with four choices for each question, he got 28.5% or 103 of the questions correctly.
Here is what I have so far. As everyone knows, the expected score is 25% or 90 items. Assuming that he is indeed a random guesser, I used the binomial distribution to get the variance np(1-p) = 360(.25)(.75) = 67.5; hence, a standard deviation of 8.22. Further assuming that random guessers are normally distributed, his z-score is (103-90)/8.22 = 1.58, making him an outlier. This places him in the top 6% of random guessers. This suggest that either
(1) he is a very good guesser,
(2) he is a very lucky guesser, or
(3) he is NOT a random guesser at all.
Now I don't know what other concepts to use to find the probability that he is a random guesser. I don't even know if there is enough information; nor do I know whether all my computations and assumptions make any sense. I hope you can help. Cheers!
PS: I only had a 3-unit statistics course way back in college. "Dummifying" your explanations would surely be appreciated. Cheers! :-)