Statistical method to prove that variations are not important? I'm checking the effect a specific substance has in the elongation of the root of variousplants of the Solanum genus. I had my plants grow in soil with different concentration of hormones. My results are quite obscure from what I expected, so I'm guessing that at very low concentrations the substance doesn't have any effect. However, I did noticed a small difference in length from  those samples that grew up without the substance. Is there any statistical method to prove that the variations I observe are not important?
 A: The first problem is, how do you quantify "not important?"
For example, if we run a clinical trial comparing the treatment effect of a new experimental drug against an existing drug that is the current standard of care, we might find that for a very large trial with thousands of patients, the experimental drug is better by only 1% compared to the existing drug, whereas the existing drug already achieves an 85% response rate.  In that context, a clinician would almost certainly decide that the new drug's efficacy isn't any better (or worse) than the existing treatment.  But with thousands of patients, you could statistically detect that 1% treatment difference.  It just isn't clinically significant.
Similarly, in your case, you have to tell us what extent of variability is regarded as "not important."  Is a variability of as much as 10% from the mean in either direction the threshold of "importance?"  20%?  Or is it measured in absolute units rather than a ratio; e.g., a variability of +/- 5mm is not important?  This threshold is something that is decided a priori and with the use of specific knowledge about the subject.  It can be informed by previous studies or analyses; but it is almost always a decision based on a combination of subject-specific knowledge and prior statistical analysis.
Once you decide this threshold, you would perform a equivalence test.  But depending on whether the threshold is a ratio or a difference, the form of the statistical hypothesis is different.  For example, for a difference, the hypothesis would be structured like $$H_0 : |\mu - \mu_0| > \Delta \quad \text{vs.} \quad H_a : |\mu - \mu_0| \le \Delta,$$ where $\Delta$ is the equivalence margin that establishes the extent to which the true mean is allowed to differ from the hypothesized mean.
