How do I choose between a Wilcoxon, Wilcoxon Signed Rank, or t-test with matched observations?

[#1]

I'm new to JMP, and trying to determine which test is appropriate for my study. I have matched biological measurements at two time points (before and after a meal) for 25 individual donors. Each donor has a single data point per time point. The population has a large standard deviation for most of these measurements, and usually the results do not have a normal distribution. Some observations have a few outliers. I have attached a sample graph depicting my data.

Is it appropriate to analyze the data with a simple t-test or Wilcoxon test via the Fit Y by X menu? Or should I perform the analysis using the Specialized Modeling > Matched Pairs > Wilcoxon Signed Rank approach? The main question in this study is whether or not the biological measurement (ie, yield) changes after eating a meal. I'm not particularly interested in donor-specific responses, more the trend of the population as a whole. Any advice on which test to use, and how to execute that test in JMP, would be greatly appreciated!

Thanks,
Jen

Yield, before and after a meal. Each dot represents one donor, line connecting donor yield before/after. Red lines indicate SD

[#2]

Maybe I'm misunderstanding, but if your question is "The main question in this study is whether or not the biological measurement (ie, yield) changes after eating a meal."  Wouldn't you really want to compare the delta between before and after, not before vs after?  While your sample data may be non-normal (it looks like a gamma distribution to me), your delta distribution is most likely normal with a more reasonable std deviation.  Assuming this is the case you could just test the delta distribution mean against the null hypothesis that it's mean is 0 (or there are no changes after eating a meal.) 

[#3]

Great explanation, thank you for making that so clear! I also have additional measurements with three time points (morning, noon, evening) - in this case, would it be valid to compare the deltas between each pair individually (T1-T2... T2-T3... T1-T3)? Or is there a different test that I should be using? Thank you so much for your guidance, I'm new to this and it is very helpful! 

[#4]

That all depends on the question you are trying to answer.  If you are trying ask "Which meal affects yield the most: breakfast, lunch or dinner?", then you would do a t-test of the deltas, of course this would require 6 measurements.  Since you have three, I'm guessing the question is, "Is there a difference in yeild at different times of day?"  In this case, I would do both a parametric test (t-test) checking the power to determine if it has any merits based on the central limit theorem and a non-parametric test (Wilcoxen, ect.) to cover my basis.  You can also apply a transformation to the data (e.g. log(yeild)) to see if it approximates normallity better.