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Part BCSIR NET December 2023

CRD, RBD, LSD essentials

In a Latin square design, the degrees of freedom for the error sum of squares is 42. Then the degrees of freedom for the sum of squares due to treatments is

  1. A.6
  2. B.7✓ Correct
  3. C.8
  4. D.9

The idea, in plain words

In this design every count is a formula in the single side length of the square. The error count given is one such formula, so solve it for the side length first, then read off the treatment count from the same side length.

Solution

For a p × p Latin square the error d.f. is (p − 1)(p − 2) = 42 ⇒ p = 8

so treatments have p − 1 = 7 d.f.

Why each option is right or wrong

Checked against 1 symbolic computation
  • A.WrongExecution slip

    False.

    6 would be p − 2, or the treatment degrees of freedom for a 7 × 7 square.

    Solving (p − 1)(p − 2) = 42 gives p = 8, not 7 — at p = 7 the error degrees of freedom would be 6 × 5 = 30.

  • B.Correct

    True.

    A p × p Latin square has error degrees of freedom (p − 1)(p − 2).

    SymPy solves (p − 1)(p − 2) = 42 and returns p = 8 (discarding the negative root −5).

    Treatments then have p − 1 = 7 degrees of freedom, the same as rows and columns.

    The trigger: in a Latin square everything is expressed in p

    so recover p from whatever is given before answering.

  • C.WrongBoundary and endpoint

    False.

    8 is p itself, not p − 1.

    Every source of variation in a Latin square — rows, columns, treatments — carries p − 1 degrees of freedom, one fewer than the number of levels.

  • D.Wrong

    False.

    9 would correspond to p = 10, whose error degrees of freedom are 9 × 8 = 72, not 42.

    Checking your candidate p against the given error degrees of freedom takes one multiplication and rules out three of these four options.

2 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.

More on this topic

The chapter behind this: Design of experiments essentials — free to read

From Sampling and Design of Experiments › CRD, RBD, LSD essentials

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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