QUANTUM PHYSICS FUNDAMENTALS - Chapter 9, Exercise 1 Solution
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Checking a Marginal Bell Test Result

PROBLEM
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A hypothetical Bell test measures S = 2.05 with a margin of error of
+/- 0.10. Does this result genuinely, statistically violate the
classical bound of S <= 2? Explain your reasoning.

SOLUTION
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As in the chapter's own worked example, checking a real violation
means subtracting the margin of error from the measured value and
seeing whether the result stays above the classical bound of 2.

    2.05 - 0.10 = 1.95

Since 1.95 is LESS than 2, the lower bound of this hypothetical
result's own real uncertainty range overlaps with, and dips below,
the classical bound.

ANSWER: No - this hypothetical result does NOT genuinely, statistically
violate the classical bound. Even though the measured value (2.05) is
technically above 2, its own margin of error is large enough that a
value consistent with the classical local-hidden-variable prediction
(S = 2, or even slightly below) cannot be ruled out. A real,
convincing Bell test needs its measured value to stay above the
classical bound even after accounting for its own full margin of
error, exactly as the chapter's own real Hensen et al. 2015 example
did (2.42 - 0.20 = 2.22, still comfortably above 2).

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WHY THIS WORKS AS AN ANSWER

This distinction - between a raw measured number and a result that is
genuinely, statistically significant once real experimental
uncertainty is accounted for - is exactly why real Bell test
experiments are reported with careful error bars, and why physicists
did not consider the question fully settled until results like
Hensen et al.'s own real, comfortably-significant 2015 measurement
were achieved. A marginal result like this hypothetical one would
correctly be treated as inconclusive, not as a genuine violation,
under the same real statistical standard this chapter's own worked
example applied.