Suppose we want to test whether the mean of some random variable X is zero. To keep it simple, X1,...,Xn are i.i.d. N(mu, 1). Testing if mu=0.
1/🧵
Now suppose that IRL mu=0.001 (remember- IRL the null hypothesis never exactly holds!)
The z-test for testing mu=0 involves computing
Z=sqrt(n)*Xbar/sigma
and comparing to a N(0,1) distribution. And sigma=1 by my earlier assumption that X_1,...,X_n are i.i.d. N(mu,1).
2/
So Z=sqrt(n)*Xbar.
Suppose n=10^12 so sqrt(n)=1000000.
This means Z=1000000*Xbar.
And Xbar should be somewhere in the ballpark of 0.001 since it's the sample mean of a bunch of observations with mean mu=0.001.
3/
So Z=1000000*Xbar equals something humongous, much much bigger than the any reasonable quantile of a N(0,1) distribution. So we get a small p-value.
If we wanted to be fancy we could compute the actual power to reject the null at any given level alpha (p-value threshold).
4/
But I think intuition is enough for Twitter, so just notice that 1000000*Xbar is around 1000000*0.001=1000 and the probability of a N(0,1) random variable taking on a value of (at least) 1000 is basically 0. So that's a p-value of zero.
Hope this helps clarify!
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This new torch lab will make it easier for instructors to teach deep learning, without needing to troubleshoot a classroom full of keras installation errors. 👩🏫💻😭
Thanks to @dfalbel and @zkajdan for this valuable contribution!!! 🙏🙏🙏
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