Question

The U.S. Census Bureau complies data on the U.S. population by region and race and publishes its finding in Current Population Reports. Independent simple random samples of residents in the four U.S. regions gave the following data on race.

At the $1\%$significance level, do the data provide sufficient evidence to conclude that a difference exists in race distributions among the four U.S. regions?

Short answer

In $H_0$, no difference exists in race distributions among the four U.S. regions.

In $H_a$, a difference exists in race distributions among the four U.S. regions.

The value of $\alpha$is $0.01$,localid="1651935588360" $x^2=26.897$and critical value is$16.812$where$P<0.005$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

Step 2. Consider the null and alternative hypotheses.

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 $\alpha =0.01$

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.

Step 3. Calculate the chi-square.

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.