The rationale behind using the paired -test is that it takes meaningfully linked data into account when calculating the -value. The paired -test works by first calculating the difference between each individual pair. Then a mean and variance are calculated for all the differences among the pairs. Finally, a one-sample -test is carried out where the null hypothesis is that the mean of the differences is equal to zero. Furthermore, the paired -test can be one- or two-tailed, and arguments for either are similar to those for two independent means. Of course, standard programs will do all of this for you, so the inner workings are effectively invisible. Given the enhanced power of the paired -test to detect differences, it is often worth considering how the statistical analysis will be carried out at the stage when you are developing your experimental design. Then, if it's feasible, you can design the experiment to take advantage of the paired -test method.
The F statistic is computed by taking the ratio of what is called the "between treatment" variability to the "residual or error" variability. This is where the name of the procedure originates. In analysis of variance we are testing for a difference in means (H0: means are all equal versus H1: means are not all equal) by evaluating variability in the data. The numerator captures between treatment variability (i.e., differences among the sample means) and the denominator contains an estimate of the variability in the outcome. The test statistic is a measure that allows us to assess whether the differences among the sample means (numerator) are more than would be expected by chance if the null hypothesis is true. Recall in the two independent sample test, the test statistic was computed by taking the ratio of the difference in sample means (numerator) to the variability in the outcome (estimated by Sp).
How to Write a Testable Hypothesis.
It is also worth pointing out that there is another way in which the -test could be used for this analysis. Namely, we could take the ratios from the first three blots (3.33, 3.41, and 2.48), which average to 3.07, and carry out a one-sample two-tailed -test. Because the null hypothesis is that there is no difference in the expression of protein X between wild-type and backgrounds, we would use an expected ratio of 1 for the test. Thus, the -value will tell us the probability of obtaining a ratio of 3.07 if the expected ratio is really one. Using the above data points, we do in fact obtain = 0.02, which would pass our significance cutoff. In fact, this is a perfectly reasonable use of the -test, even though the test is now being carried out on ratios rather than the unprocessed data. Note, however, that changing the numbers only slightly to 3.33, 4.51, and 2.48, we would get a mean of 3.44 but with a corresponding -value of 0.054. This again points out the problem with -tests when one has very small sample sizes and moderate variation within samples.