Determine if one-point samples are significantly different from a set-of-values sample Well, I have a sample x made up of 28 cases, sample y -> one case, sample z -> one case. I would like to determine if y and/or z are significantly different from x.
It's not possible to take more measurements for neither y or z.
Data look as follows:
x             y            z

0.72981354    37.13        67.4
1.518328503
1.655629139
2.563412415
3.027129131
4.480875623
4.545650835
4.762517086
5.065794539
5.643197549
6.674587177
7.019766786
7.687576876
8.321129011
9.798331225
19.22282133
25.73062098

I would like to know what would be the most appropriate statistical test to use?
I know that in order to use one-way ANOVA, I need at least two measurements per sample. So, is there another way to do it?
 A: If your $x$ cases are i.i.d. and you are willing to assume a parametric distribution $F_X(\cdot|\theta)$, you could 


*

*estimate the parameters $\theta$ using the observations $x$ and then

*calculate the "tail probability" that an observation is as large as or greater than $y$: $1-F_X(y|\theta)$.


If the tail probability is very small, $y$ seems unlikely to have been generated by $X$. However, the estimation uncertainty associated with $\theta$ will make the "tail probability" uncertain. You could account for that by, for example, 


*

*obtaining bootstrap samples from the original sample and 

*estimating $\theta$ and calculating $F_X(y|\theta)$ for each bootstrap sample.


You would obtain a "cloud" of "tail probabilities", so you would have some idea about their distribution. But that could create a false sense of reliability; recall that you only have one $y$! Therefore, the inference will be quite uncertain anyway.
However, it seems that $x$ are not i.i.d. as they are increasing (unless you have ordered them increasingly before posting), so the above method will not work as is. Perhaps you could modify and adapt the underlying idea, though.
