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I am reading Chapter 4 (Testing and Confidence Regions: Basic Theory) of Mathematical Statistics by Bickel and Doksum. In Example 4.1.1, the authors use the study of sex bias in graduate admissions at Berkeley as an example to illustrate the difficulty of specifying a stochastic model and formulating a task as a statistical hypothesis. Specifically, the authors state:

[The Graduate Division of Berkeley] initially tabulated $N_{m1},N_{f1}$, the numbers of admitted male and female applicants, and the corresponding numbers $N_{m0},N_{f0}$ of denied applicants. If $n$ is the total number of applicants, it might be tempting to model $(N_{m1},N_{m0},N_{f1},N_{f0})$ by a multinomial, $\mathcal{M}(n,p_{m1},p_{m0},p_{f1},p_{f0})$, distribution. But this model is suspect because in fact we are looking at the population of all applicants here, not a sample.

My question: I am having trouble understanding why using a multinomial model is an issue. What do the authors mean by saying that we are looking at the population of all applicants instead of a sample?

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    $\begingroup$ It sound like their complaint is that you don’t need to do any modeling; you have everything. I see some merit to this but also find this, without running to my copy of the book to see the context, a bit extreme. It’s reasonable to run a test to see if a significantly higher proportion of male students were admitted than female as some evidence supporting a claim of discrimination against the women. $\endgroup$
    – Dave
    Jul 27 at 3:29

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