»  Solutions to puzzles in my National Review Online Diary

  November 2010

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In my November diary I posed this brain-teaser.

A reader draws my attention to the following passage in Chapter 2 of Charles Murray's book Real Education. Charles is trying to illustrate what "below average" actually means when it comes to academic ability. For this purpose he borrows a number of test questions from the National Assessment of Educational Progress (NAEP), the program used by the federal Department of Education to track student accomplishment. Murray:
 … Now consider some items that more specifically identify what it means to be below average in math as an eighth-grader.

Example 2.  Amanda wants to paint each face of a cube a different color. How many colors will she need?

    (A) Three    (B) Four    (C) Six    (D) Eight

Twenty percent of eighth graders did not choose C. Approximately 27 percent did not know the right answer.
Charles is quite correct; but how did he get from the 20 percent of the penultimate sentence to the 27 percent of the last one?

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Solution

Let x be the number of test-takers who did not know the right answer. Assume they all guessed at random from the available choices. One in four of them would get lucky and pick the correct answer. The 20 percent observed is therefore only ¾ of x. Therefore x is, to the nearest whole number, 27.

This is, as several readers noted, only a first approximation. To test-takers who don't know the right answer, the four choices are probably not all equally guess-worthy. We have no information about that, though. Charles made a reasonable default assumption given the information he has.