On a particular day let $X_1,X_2,X_3$ be the number of boys born before the first girl is born in hospitals $1,2,3$ respectively.If the observations are $X_1=0$ ,$X_2=3$ and $X_3=2$, find the most powerful test to test the null hypothesis that a girl and a boy are equally likely to be born against the alternative that a girl is less likely to be born than a boy.
My try: Let $p$ be the probability of a boy being born. We need to test the hypothesis $p=\frac{1}{2}$ against $p>\frac{1}{2}$ Now we know that $X_1,X_2,X_3 \sim $ Geometric$(p)$.But I cannot frame the most powerful test in this case on the basis of the observed values. Can anyone help?
We can use the likelihood ratio procedure. As $X_i$ are a sample from $\sf{Ge}(p)$ with $p$ unknown, we form the hypotheses $$H_0:p=\frac12,\quad H_1:p=p_1>\frac12.$$ Then the likelihood ratio is $$\frac{f(x\mid p_1)}{f(x\mid p)}=\frac{p_1^n}{(1/2)^n}\left(\frac{1-p_1}{1-1/2}\right)^{\sum X_i-n}>k$$ for some $k\in\Bbb R$. Letting $k'=k^{1/n}$, we get $$2p_1[2(1-p_1)]^{\bar X-1}>k'$$ and as $\bar X$ increases, the LHS decreases since $2(1-p_1)<1$. Thus we choose $c$ such that $P(\bar X<c\mid H_0)=\alpha$ and note that $\sum X_i$ follows a $\sf{NegBin(n,p)}$ distribution. It is possible to approximate this with a Normal distribution, but don't expect it to work too nicely as transformations from a discontinuous function to a continuous one are usually quite complicated.