he298c

2021-05-02

Suppose that each student in a sample had been categorized with respect to political views, marijuana usage, and religious preference, with the categories of this latter factor being Protestant, Catholic, and other. The data could be displayed in three different two-way tables, one corresponding to each category of the third factor. With pijk = P(political category i, marijuana category j, and religious category k), the null hypothesis of independence of all three factors states that pijk=pi..p.j.p..kp Let nijk denote the observed frequency in cell (i, j, k). Show how to estimate the expected cell counts assuming that H0 is true. Then use the general rule of thumb to determine the number of degrees of freedom for the chi-squared statistic.

Talisha

This might be different from other exercise. Notice that there are total of $3\cdot 3\cdot 3=27$ cells. The total sample size (students) is 1 and it is fixed, so there are 27— 1 = 26 total left cells. From the rule of thumb, one should see how many parameters are there to estimate for the chi-squared test.
As explained, pi.,pj.pk should be estimated, and since i,j,k = 1,2,3, and from the fact that
$\Sigma \left[3,i=1\right]\pi ,=\Sigma \left[3,j=1\right]pj=\Sigma \left[3,k=1\right]pk=1$,
one should estimate only total of six independent parameters (when two are determined, third can be found). Finally, the degrees of freedom are
df=27-1-6=20.

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