I hate to spoil people's fun but maybe there are better uses for all those analytical resources. Including brains

Apparently it takes 100s of 1000s of coin tosses to "prove' bias 

Fascinating question

Sorry for commenting it seems like something that is almost certainly true but unprovable

Sorry for stepping outside of statistics

I am at the stage of my life when grandchildren are appearing and people ask you to guess the sex. I could answer that “my family produces boys so you will also.” Is it true though that pr(male) varies over the population (or over time)? With simple assumptions, this is empirically detectable. If some of the population only produce girls and the rest only boys then we would see this clearly in the data. The overall population distribution would be over-dispersed.

You may or may not want to read my notes below, but my main question is whether any members know the answer to the key question in the title. If you have a few key references that would be great. Or perhaps you could direct me to a colleague who finds this issue of interest.

Here is my unresearched thinking:

Null hypothesis would be that pr(male)=p is the same for all mothers/couples. This is almost certainly slightly greater than 0.5 and may differ across age (evidence suggests not) but even if it did, we could then define p to be the integral with respect to the standard age distribution of mothers.

The data will be the sequence of males and females born to all mothers in a population. The data sets will be large. Gender order will potentially be important, see below.

The number of sons and daughters a woman has depends on

• Desire/propensity/ability to have total number τ of children. This will be determined by factors such as economics, physical limitations, promiscuity and other factors that need not be modelled (but could be).
• Parental desire for a particular sex or combination of sexes
• • Primogeniture: Mother/parents will keep having children until they get a male. This will not affect the proportion of males in the population because of the martingale stopping rule property. But you could see it in in the sequence of data (for instance last child is male) and estimate how many use this rule. Proportion of population who desire males can be μ. Proportion who desire a daughter could be δ.
  • Gender mix: Will stop once you have both sexes. This would lead to a under dispersion of we condition on family size=2 (see first link above). Proportion of population could be B (for both). There is evidence that this is the case. Conditional on a family of 2, there is an over-abundance of mixed sex.

Parents’ stopping rules based on gender make this interesting and complex. So, at the very least there are parameters p, τ,  μ, δ, B. This looks like a really interesting classical inference problem to me. What distribution would be convenient for heterogeneity of pr(male)=p to define the alternative? What would be a LR test? Is it largely a test of over-dispersion relative to the expected distribution under the estimated parental stopping rule? Is it tractable?

I fear it might be difficult to test if p is constant with much power since: if half the population want daughters and the other half sons, you will end up with over-dispersion, which is the same general pattern you expect if p is not constant. But maybe everything can be untangled with enough data and the data set will be large.

Kind regards and seasonal greetings!

CL