Null Hypothesis testing: any effect of sample size to population size ratio?

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#1

I am going through last week videos on Null Hypothesis. As shown in the screenshot below, sample size, null hypothesised mean and population variance controls test statistic z and hence the rejection and acceptance of Null Hypothesis.

I am looking for some intuition why the ratio of sample size to population size doesn’t have any role here. Whether the population is 1 billion or 1 thousand, sample size seems to have the same effect I would think a larger population would require a larger sample size for confidently rejecting or accepting a NULL Hypothesis , but there is no such relation reflected in the formulae presented.
Are there any related videos (past week or this week itself) which discuss this?

Screenshot 2020-10-03 at 1.59.43 PM
Screenshot 2020-10-03 at 1.59.43 PM1281×724 141 KB

#2

As per my understanding, there’s a role to play for ratio of sample size to population size.
The numerator in RHS consists of terms x̅ and μ. and both of them have a direct relation with sample size and population size.
I don’t have a concrete argument though, will try to find one from a credible source.

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#3

That will be will be very helpful. I agree that \bar{x} and \mu are calculated from population (size) in some way, its just not very explicit and doesn’t give any indication of population size (small or big) by itself.
Arguments that I read about good sample size (10% not exceeeding 1000) based on population size are mostly empirical.

#4

Yes, but my point was let’s say if we have a lower ratio for (sample size : population), then our estimated mean (x_bar) would deviate alot from the population mean (mu).
In such a case, it will directly affect the z-score as well.
On the other side, if we have a large ratio (sample size : population), then (x_bar) and (mu) will be more close.
That’s just my viewpoint regarding this.

#5

True :+1: That’s one way I can look at it.