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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Maximum likelihood and likelihood ratios are not the same thing. Maximum likelihood is an estimation method, whereas likelihood ratios are used to construct tests. 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 $m$. |
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