Question

The Quinnipiac University Pol conducts nationwide surveys as a public service and for research. This problem is baed on the results of one such poll that asked independent random samples of American adults in urban, suburban, and rural regions, "Do you support or oppose requiring background checks for all gun buyers?" Here are the results.

At the $1\%$significance level, do the data provide sufficient evidence to conclude that a difference exists in the proportions of supporters among the three regions?

Short answer

We know, $$\begin{gathered} H_0:p_1=p_2 \\ =p_3 \end{gathered}$$(proportions of supporters are equal).

$H_a:$Not all the proportions of supporters are equal, $\alpha =0.01$.

The test statistics is $11.520$and the critical value is $9.210$where $P<0.005\left(P=0.003\right)$. Also $H_0$rejects at $1\%$significance level, the data provide sufficient evidence to conclude that a difference exists in a proportions of supporters among the three regions.

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.

$H_{\alpha }$denotes race distributions among the four U.S. regions are non-homogeneous.

The level of significance is $\alpha =0.01$.

Step 3. Find the expected frequency and test statistics.

On finding the expected frequency and test statistics,

Pearson Chi-square$=11.520,$

$$\begin{gathered} df=2, \\ p-value=0.003 \end{gathered}$$

Therefore, the value of Pearson chi-square is $11.520$.

Therefore, the p-value is$0.003$.

Step 4. Consider the rejection rule.

Consider the rejection rule,

If $P-value\le \alpha$, then reject the null hypothesis.

Here, the P-value is lesser than the level of significance, $P-value\left(=0.003\right)<\alpha \left(=0.01\right)$

Therefore, the null hypothesis is rejected at $1\%$level.

The results are statistically significant at the above given level of significance.

Hence, the data provide sufficient evidence to conclude that a difference exists in a proportions of supporters among the three regions at the $1\%$significance level.