I would argue if "only" is appropriate in this context. Usually these tests are not symmetric in this way, so we don't expect really that if our assumptions are false, the negated null hypothesis will just as easily give significantly important data in favour of it being not true. The most basic example is showing linear correlation. The contrary is pretty much impossible to significantly proof, as variables correlated with ratio are linearly correlated, and indistinguishable in practice. Previously we were trying to show that ρ just is not equal to zero, so exclude one case, designing the experiment to expose |ρ| being big enough to make significant difference. But to show that there is no correlation it is not enough to expose |ρ| as small, there is just no positive ε small enough small enough to |ρ|<ε showing lack of linear correlation significantly, and the exact answer usually has probability 0. This asymmetry is not related to this particular statistic ρ (the same applies for any testing if point lies outside compact subset, which is much easier than outside open subspace if the data lie around point from its closure) but is something to be expected in every study. It might be less extreme, but there is a reason for using experimental methods very often to show that some random variable is bigger than second (for example that ordinary running is on average faster than running while juggling at the same time) and we don't just do the same with null hyppothesis reversed where we expect that there is no real difference between them.
The 0.25 success rate stated as that do not mean it is low and it may as well extremely large ~SpectralFlux 04:51, 7 August 2025 (UTC)
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