::Also, the first sentence of the article reads: "Blount, Borland, and Lenski[1] claimed that a key evolutionary innovation was observed during a laboratory experiment. That claim is false." A small correction: There were several claims in the paper. The 'key evolutionary innovation' was acquiring the ability to utilize citrate as a food source. That claim was demonstrated multiple times. The claim, which pertains to this statistics discussion was that the Cit+ phenotype arose in a multi-step process, first requiring a rare, pre-adaptive mutation before additional mutation(s) lead to the subsequent development of citrate utilization.--[[User:Argon|Argon]] 20:46, 5 March 2009 (EST)
:::My biology-degreed wife assures me that mutation does not necessarily mean that evolution occurred. What the paper claimed is that evolution (a “key innovation”) occurred in the lab. The key innovation supposedly increased the mutation rate. In the experiments, the observed mutation rate increased after generation 31,000, but not enough to make a statistically significant claim that the rate is not constant. The analysis in the paper was similar to flipping a coin ten times, counting six heads and claiming that the coin must be biased against tails. In reality, there’s nothing surprising about a fair coin producing slightly more of one outcome than the other. Just like there's nothing surprising about there being slightly more mutations in later generations than early generations given the null hypothesis (constant mutation rate). [[User:SJohnson|SJohnson]] 10:46, 9 March 2009 (EDT)
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But that point (the foregoing paragraph) has no bearing at all on the null hypothesis, as you describe it. The mutation appeared, so that means the hypothesis that the mutation can't happen is disproved. Very simple. [[User:FredFerguson|FredFerguson]] 21:10, 8 March 2009 (EDT)
:I never said that “the null hypothesis is that this mutation cannot happen”. The chi-square test statistic I'm using wouldn’t be defined if the null hypothesis mutation rate was zero because the <math>E\left[n_{i,j}\right]</math> term in the denominator of the statistic (see above equation) would be zero.
:The test statistic from the paper is the average of the generation numbers of observed mutations. For experiment one this number is
::<math>
\frac{1}{4}\left(30500+31500+2\times32500\right)= 31750.
</math>
:The same number is shown in Table 2 of the paper. [[User:SJohnson|SJohnson]] 10:46, 9 March 2009 (EDT)