# A question about hypothesis testing and maximum likelihood ratio test

Assume there are 2000 students, $m$ boys and the rest are girl. Now we take a sample of 5 students which contains only 1 boy. We claim that there are more girls than boys in the population. So what we want to do is to check the maximum likelihood under $H_1$ $H_0$, but the problem is I don't know how to get the maximum likelihood. Some said that it attains max when $m=400$ under $H_0$ but I don't quite get how they get.

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What are your hypotheses? $H_0:$ as many boys as girls and $H_1:$ more girls? –  MånsT Apr 5 '12 at 6:06
I don't understand why you want to use ML at all. Can't you just use $\frac{P(1 boy, 4 girls|m>1000)}{P(1 boy, 4 girls|m\le 1000)}$? –  fabee Apr 5 '12 at 6:19

For both of these, you need the likelihood function. In this case it is the probability function of a random variable with a hypergeometric distribution, since $X$, the number of boys sampled, is hypergeometric. Then
$P(X=k)=\frac{\binom{m}{k}\binom{2000-m}{5-k}}{\binom{2000}{5}}$
To obtain the maximum likelihood, maximize this with respect to $m$ for $k=1$. Your claim seems to be that the maximum is given by $m=200$, i.e. when the proportion of boys is 400/2000=1/5=the observered proportion.
To obtain the likelihood ratio test, compute $-2\ln\frac{P_{H_0}(X=1)}{P_{H_1}(X=1)}$. Your hypotheses should be statements about $m$.