1. ## Comparing Populations

I have a population (X1) that's made of a sequence of single digits (0-9). My second population is a psuedo-random Uniform[0,9] distribution (X2). I would like to use the following test statistic:

With the following null/alternative hypothesis:

H0: There is no difference between population X1 and population X2
H1: There is a difference between the two populations

Essentially, I'm trying to show that the first population is also psuedo-random by comparing it distirbution that I know is psuedo-random.

Is this possible?

2. It sounds to me that you should consider the $\displaystyle \chi^2$ test statistic with the following null/alternative hypothesis:

H0: There is no difference between the distribution in population X1 and population X2
H1: There is a difference between the distribution of the two populations

Call X1 the observed and X2 the predicted

I'm having trouble picturing your data. Can you share the data set?

3. The observed is the first 10000 digits of pi, and the predicted is 10000 U[0,9] I generated using R. I want to know if the digits of pi are psuedo-random. I already used the chi-square test statistic, along with the kolmogov-smirnov test. I'm trying to "come up" with my own test and I wanted to know if the above test statistic was suitable for it.

4. Well your test statistic suggests you are looking for a significant difference in means but the context of your post (hypothesis formualtion) says you are trying to find if the populations are distributed the same way. So I don't think the test statistic fits the formulation.

The two data sets are fixed, i.e. won't be changed as you can't take different samples, so no need for a test, they are the same or they are not!

5. Is it possible to test if the digits of pi were generated by U[0,9]?

6. Make a histogram of the points generated by $\pi$, is it the same as U[0,9]?

7. They are somewhat similiar, but not enough to make a claim for it. I kinda wanted a test of some sort test statistic (like that one in my first post) because I don't know if the digits of pi are actually distributed as U[0,9].

8. Originally Posted by statmajor
They are somewhat similiar, but not enough to make a claim for it. I kinda wanted a test of some sort test statistic (like that one in my first post) because I don't know if the digits of pi are actually distributed as U[0,9].
They are very similar I agree, personally I would just test this with a $\displaystyle \chi^2$ and infer from this.

9. I already did (along with the kolmongov-Smirnov test) and it passed both tests. I was just wondering if I could use the population comparison test in this situation.

10. I don't think your proposed test statistic suits the test and I cannot think of any tests that work right now, you may need to wait for another forum member to add some extra intelligence.

matheagle, Mr Fantastic and Moo are some of the better stats gurus around here, maybe you could politely ask any of them?

11. I think I might.