ASTRONOMY FUNDAMENTALS - Chapter 8, Exercise 3 Solution
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Why a 5-Sigma Disagreement Is a Real Scientific Problem

PROBLEM
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The Hubble tension shows two independent measurement methods
disagreeing at a statistical significance greater than 5 sigma.
Explain why this level of disagreement is treated as a genuine
scientific problem rather than something likely explained by ordinary
measurement error.

SOLUTION
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In real scientific practice, "sigma" measures how far an observed
result sits from an expected value, in units of the real uncertainty
(the margin of error) in the measurement itself. A small disagreement -
within 1 or 2 sigma - is genuinely expected even between correct
measurements, simply because every real measurement carries some
random uncertainty; results that close together are considered
consistent with each other once that uncertainty is properly accounted
for.

A disagreement of more than 5 sigma is a very different, much stronger
claim. It means the gap between the two real measured values (the
early-universe ~67.7 km/s/Mpc figure and the late-universe ~73 km/s/Mpc
figure) is far too large to be reasonably explained by ordinary random
measurement uncertainty alone - the real probability of this specific
size of discrepancy occurring purely by chance, if both methods were
actually measuring the same true value correctly, is extremely small.

ANSWER: A disagreement above 5 sigma is treated as a genuine problem,
rather than routine measurement noise, precisely because random
measurement error becomes an implausible explanation at that level of
statistical significance. This strongly suggests that at least one of
the two measurement methods - or, more likely, something in the
underlying physical model connecting "early universe" measurements to
"late universe" ones - is missing something real that current
cosmology doesn't yet fully account for.

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WHY THIS WORKS AS AN ANSWER

This connects directly back to Chapter 1's own opening discipline
around evaluating evidence carefully. A small disagreement between two
measurements is routine and expected; a large, statistically
significant one - like the real Hubble tension - is a genuine signal
that something in the current scientific picture is incomplete, not
simply a case of one team measuring carelessly. This is exactly why
the Hubble tension remains an actively researched, genuinely unresolved
problem in cosmology today, rather than something already dismissed as
a rounding difference between two independent, real measurement
techniques.