Math Help - Comparison of Multiple Means

1. Comparison of Multiple Means

My data set is the first 10,000 digits of Pi, and I want to know if they are psuedo-random numbers.

My idea is that I seperate the data into 10 inteveralls of 1000 digits, and find their corresponding mean.

I believe, that if the data is truly psuedo-random, then those 10 sample means should be relatively the same.

Is my line of thinking correct?

Thank you

2. This sounds like a different way of attacking it since http://www.mathhelpforum.com/math-he...tml#post628139

I would use an ANOVA here, but are these really samples? If you have 10,000 digits into 10 groups and your are finding the mean of the whole 1000 then this is the population. Of the 1000 if you only took a random 50 for example then that would be a sample.

3. Originally Posted by pickslides
This sounds like a different way of attacking it since http://www.mathhelpforum.com/math-he...tml#post628139
Probably because it is. I'm trying to come up with some new approach on how to test to test for randomness., and for some reason, I'm really hung up on this "mean" idea.

Originally Posted by pickslides
I would use an ANOVA here, but are these really samples? If you have 10,000 digits into 10 groups and your are finding the mean of the whole 1000 then this is the population. Of the 1000 if you only took a random 50 for example then that would be a sample.
Good point, I didn't think about that.

How would I use an ANOVA table here? I've used it in the past, but only when doing regression analysis, and not sure how to apply it here.

4. A one way ANOVA tests the following hypothesis formulation

$\displaystyle H_0:\mu_1 =\mu_2= \mu_3= \dots =\mu_n$

$\displaystyle H_A:\text{At least one } ~\mu_n ~~ \text{is different}$

The Anova test statistic is $\displaystyle F_{calc}$ follow this example

One-way ANOVA Example

Reject $\displaystyle H_0$ if $\displaystyle F_{calc}>F_{crit }$

5. Thanks for the example, it really helped out.

So failing to reject the null hypothesis would imply that the digits of pi are independently generated or uniformly generated?

I was also thinking of instead of splitting into 10 groups, and sample from them, I would just sample 1000 digits (with replacement) and do this maybe 5 times. Would that still work?

6. Originally Posted by statmajor

So failing to reject the null hypothesis would imply that the digits of pi are independently generated or uniformly generated?

It would infer the population means are not significantly different. If different population means are equal can the populations still have a different distribution?

Spoiler:
Yes!

Originally Posted by statmajor

I was also thinking of instead of splitting into 10 groups, and sample from them, I would just sample 1000 digits (with replacement) and do this maybe 5 times. Would that still work?
Seems like a pretty big sample size, i.e 5 lots of n=1,000 given N=10,000. The whole idea of taking a sample is to save time and effort.

I think you have to split them into defined groups as each particular group has its own population mean.

By just using the one group (or population) and many sample sets on that one group, the test reverts back to a simple t-test for determing the signifcance of that one population's mean.

Hope this all makes sense, my head is starting to spin...

7. Originally Posted by pickslides
It would infer the population means are not significantly different. If different population means are equal can the populations still have a different distribution?
But I'm not sure what the underlying distribution of the digits of pi. It looks like a U[0,9], but I don't have a way of showing that.

I think that if there is little difference between the different population means, thet that would imply that the digits have been uniformly distributed.

Hopefilly I'm not the reason why you're head is spinning, and if I am, my apologies.