python compute sample size for one-sample Z-test

Question

I'm trying to see if there's an easy way to calculate minimum sample size required for a one-sample Z-test to reject the null hypothesis.

I know that we can reject the null hypothesis (ie the A/B test is successful) if

1 - scipy.stats.norm.cdf((x-mu)/(s/np.sqrt(n)) < alpha

where x is the sample mean, mu is the population mean, s is the population standard deviation and n the sample size.

Is there a way in python to solve the equation above for n ?

Answer 1

Distributions in scipy.stats have an inverse of the cdf function, which is called ppf . ppf stands for "percentage point function" but this is a misnomer because it actually deals with quantiles , not percentiles. We can use this function and the fact that ppf(cdf(x)) = x to rearrange and solve your equation.

Assuming that alpha < 0.5 , there are two cases to cases to consider. If x <= mu then there are no solutions, otherwise we can rearrange the equation to:

np.sqrt(n) > s * scipy.stats.norm.ppf(1 - alpha) / (x - mu)

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An aside:

If alpha >= 0.5 then the solutions are strange: either any n will do (if x >= mu ) or you get an upper bound on n (if x < mu ). But that's what you get for treating your null hypothesis with such contempt!