You measure the Km of a kidney enzyme (in nM) before and after a treatment. Each experiment was done with renal tissue from a different animal.

Using a conventional paired t test, the 95% confidence interval for the mean difference between control and treated Km value is -0.72 to 9.72, which includes zero. The P value 0.07. The difference between control and treated is not consistent enough to be statistically significant. This makes sense because the paired t test looks at differences, and the differences are not very consistent.

The ratios are much more consistent, so it makes sense to perform the ratio t test. The  geometric mean of the ratio treated/control is 2.02, with a 95% confidence interval ranging from 1.88 to 2.16. The data clearly show that the treatment approximately doubles the Km of the enzyme.

Analyzed with a paired t test, the results were ambiguous. But when the data are analyzed with a ratio t test, the results are very persuasive – the treatment doubled the Km of the enzyme.

The P value is 0.0005, so the effect of the treatment is highly statistically significant.

The P value answers this question:

If there really were no differences between control and treated values, what is the chance of obtaining a ratio as far from 1.0 as was observed? If the P value is small, you have evidence that the ratio between the paired values is not 1.0.