All my observations are identical. Use a non-parametric test?

Question

I was doing test for normality on variables and came across one that had identical values for each observation. I can't do a normality test. I plotted the kernel density and it looks normal, however a qq plot says different. I have not come across this situation before and am not sure how to proceed. Can I assume normality and use t.test or should I go non-parametric? I need to test for differences with another sample. Forgive me if the answer is obvious. Cheers.

Here is an example. Say we have 10 stores selected at random in a city that all sell the same item. The price of that item at some time point is say $50.00 for each store (sample 1). Some time later, a material used to make that item becomes slightly harder to come by. After the availability of the material changes, the price of the item is checked at each of the original 10 stores, with the prices this time being say 50,50,50,50,50,50,51,50,50,49 (sample 2). I want to test if there is a difference in mean price before and after the availability of the material changed. First I check for normality, but this is difficult for sample 1 as they are all the same price. The usual tests for normality will not work for sample 1. So, I am unsure if I can compare the mean difference using a t.test or if I need to compare location shift using a non-parametric test.

Answer 1

A couple of points:

1. The KDE is not designed to assess normality, so don't use it that way.
2. It is clearly not reasonable to assume that your data (e.g. the vector of 10 price differences $d_i:=p_{2i}-p_{1i}$ where many of the $d_i$'s are 0) is normally distributed. You should, as a rule, prefer the sign test (specially as part of your working assumptions that the price differences are likely skewed). Furthermore, in a sign test, the large number of $d_i$ that are exactly 0 will simply 'drop' from the calculations, making this a non issue.