I am testing to see if the means of two groups G and R are different. I cannot use a t-test because the data is not normal so I am using the Wilcoxon rank sum test which seems like the non-parametric version of the t-test for difference in means.
I am using R and the detail of the Wilcoxon test states: https://stat.ethz.ch/R-manual/R-patched/library/stats/html/wilcox.test.html
"Otherwise, if both x and y are given and paired is FALSE, a Wilcoxon rank sum test (equivalent to the Mann-Whitney test: see the Note) is carried out. In this case, the null hypothesis is that the distributions of x and y differ by a location shift of mu and the alternative is that they differ by some other location shift (and the one-sided alternative "greater" is that x is shifted to the right of y)."
here are the means
mean(G) = 6.1 man(R) = 8.3
So when I run
wilcox.test(G,R ,mu=0)$p.value  0.06497015
with a p-value of $.064. >.05$ I conclude that there is not sufficient evidence to show the means are different BUT
when I run a one sided test I get different results
> wilcox.test(G,R ,alternative = "less")$p.value  0.03248507
with a p-value of .032... I reject the NULL that mean(G) > mean(R) and I conclude that mean(G) < mean(R)
How should I interpret these results. The two sided test says there is no difference and the one sided test says there is. Is this just a close call. i.e. .06 is near .05 ?
I have one more question if I do a two sided test and let's say the p value is .0006
wilcox.test(G,R ,mu=0)$p.value = .00006
this means the means of G and R are significantly different from zero correct?
It doesn't actaully tell me that difference is actaully mean(G) = 6.1 minus mean(R) = 8.3 correct?
I can only say the means are different but not by how much??