A test consists of 910 true or false questions. If the student guesses on each question, what is the expected standard deviation of the number of correct answers?

1 Answer
Bill K.
Dec 10, 2015

#sqrt{910 * 0.5 * 0.5}=sqrt{227.5} approx 15.08#.

Explanation:

The number #X# of correct guesses in #n=910# trials is a binomial random variable with probability #p=0.5# of success. The standard deviation of such a variable is #sqrt{np(1-p)}#. In this case, that's #sqrt{910 * 0.5 * 0.5}=sqrt{227.5} approx 15.08#.

Such a variable would be well-modeled by a Normal distribution with mean #np=910 * 0.5 = 455# and standard devation #sqrt{227.5 approx 15#. Using the 68-95-99.7 rule-of-thumb , about 68% of the people who guessed on such an exam would score between 440 and 470, about 95% of such people would score between 425 and 485, and about 99.7% of such people would score between 410 and 500.

These are approximations and we are not making a distinction between these ranges including the endpoints or not. The "half-unit (continuity) correction " could be used along with a table of Normal probabilities to try to make these estimates more precise.

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