91Ó°ÊÓ

Consider the given question,

According to the null and alternative hypotheses,

$H_0$denotes race distributions among the four U.S. regions are homogeneous.

$H_a$denotes race distributions among the four U.S. regions are non-homogeneous.

Specific level of significance is $\mathrm{Î\pm }=0.01$

The given table contains the expected frequencies corresponding to the observed frequencies.

All of the expected frequencies are greater than $1$, we can verify using the table.

At most $20\%$of the expected frequencies are less than $5$.

Hence, we can say all the assumptions are satisfied.

The test statistics,$x^2=26.897$.

The row variable has $4$values and the column variable has $3$values. Therefore, we can say $$\begin{gathered} r=4, \\ c=2 \end{gathered}$$.

According to the chi-square, the critical value is $16.812$.

We see that the value of the test statistics falls in the rejection region. Thus, we reject. The test results are significant.

At significance level of $1\%$, the data provide sufficient evidence, concluding a difference in race distributions among the four U.S. regions.