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Is there a way to test whether our data from sample is similiar to population data?

Let's say that we conducted a poll about political preferences with 2% marigin of error and 95% confidence level. Can we check reliably whether we had a proper sample?

I know about chi square tests. Let's say we have a party, which got 36% of 10000000 votes (3600000) and poll had said that they ought to get 35,5% (3550000). Chi-square result is about 704, which seems far too big.

I've heard that chi square shouldn't be used for large samples, so are there any other tests I can use?

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Traditional statistical tests like chi-square-tests and binomial test are meant to investigate point hypotheses. If you test for x = 35.5%, you test for x = 35.50000000000...% and with a large sample (such as your $10^7$) things get quite precise and the slightest difference will yield a $p$ very close to zero.

First, you should not take the point estimation of 35.5% from your poll, but the 95% or 99% confidence interval from your poll and compare that to the 36%. This is still comparing to the "same" population, not "similar" populations. You will have to define what "similar" should actually mean.

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