Question

The Quinnipiac University Poll conducts nationwide surveys as a public service and of research. This problem is based on the results of once such poll. Independent simple random samples of $300$residents each in read (predominantly Republiclian), blue (predominantly Democratic) and purple (mixed) states were asked how satisfied they were with the way things are going today. The following table summarizes the responses.

At the $10\%$significance level, do the data provide sufficient evidence to conclude that the satisfaction-level distributions differ among residents of red, blue and purple states?

Short answer

We know, chi-square is $7.05,df=6,$ p-value is$0.3162$.
Also, $x^2=7.05$, where $7.05<10.6446$rejecting the null hypothesis.

On concluding, we can say that the satisfaction levels are not differing among the three categories of people.

Step-by-step solution

Step 1. Given information.

Consider the given question,

Step 2. Consider the null and alternative hypotheses.

Null Hypothesis is the satisfaction levels are not differing among the three categories of people, $$\begin{gathered} H_0:R=B \\ =P \end{gathered}$$

Alternate Hypothesis is the satisfaction levels are not differing among the three categories of people, $$\begin{gathered} H_1:R\ne B \\ \ne P \end{gathered}$$

According to the decision rule,

When P-value is less than the Level of significance then it results in the rejection of the null hypothesis.

When the test statistics value is greater than the tabulated value then reject the null hypothesis.

Step 3. Calculate the x2 test for the given data.

On calculating the $x^2$test,

Chi-square$=7.05$,

$df=6$,

p-value$=0.3162$

Step 4. Calculate the test statistics value.

Consider the above table,

$$\begin{gathered} x^2=\sum _{}\frac{{\left(O-E\right)}^2}{E} \\ =7.05 \end{gathered}$$

Hence, we know $$\begin{gathered} x^2=7.05, \\ P-value=0.3162 \end{gathered}$$.

Compare $x^2$-test value and critical value with $$\begin{gathered} \left(r-1\right)\times \left(c-1\right)=\left(4-1\right)\times \left(3-1\right) \\ =6 \end{gathered}$$at $10\%$level of significance is role="math" localid="1651939512728" $10.6446,7.05<10.6446$, fail to reject null hypothesis.

Therefore, on concluding, we can say that the satisfaction levels are not differing among the three categories of people.