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This test: 100 points
This question: 3 points

A recent survey conducted on a random sample of adults 18 years of age or older living in a certain country asked their reaction to the word "socialism" and which political party they most associate with. Results of the survey are given in the table below. Complete parts (a) through (c).

[tex]\[
\begin{array}{ccc}
& \text{Positive} & \text{Negative} \\
\text{Democrats} & 61 & 368 \\
\text{Independents} & 230 & 286 \\
\text{Republicans} & 145 & 409 \\
\end{array}
\][/tex]

Does the evidence suggest individuals within each political affiliation react differently to the word "socialism"? Use the [tex]\(\alpha = 0.05\)[/tex] level of significance. State the hypotheses.

A. [tex]\(H_0: \text{Political party and reaction are dependent.}\)[/tex]
[tex]\(H_1: \text{Political party and reaction are independent.}\)[/tex]

B. [tex]\(H_0: \text{Political party and reaction are independent.}\)[/tex]
[tex]\(H_1: \text{Political party and reaction are dependent.}\)[/tex]

C. [tex]\(H_0: p_D = p_R = p_I\)[/tex]
[tex]\(H_1: \text{At least one of the proportions is different from the others.}\)[/tex]

D. [tex]\(H_0: O_D = E_D \text{ and } O_R = E_R \text{ and } O_I = E_I\)[/tex]
[tex]\(H_1: \text{At least one mean is different from what is expected.}\)[/tex]

Sagot :

GinnyAnsweridn
Let's address this problem step by step.

### Step 1: State the Hypotheses

We need to determine which hypothesis is correct based on the provided choices:
- A. [tex]\(H_0\)[/tex]: Political party and reaction are dependent.
[tex]\(H_1\)[/tex]: Political party and reaction are independent.
- B. [tex]\(H_0\)[/tex]: Political party and reaction are independent.
[tex]\(H_1\)[/tex]: Political party and reaction are dependent.
- C. [tex]\(H_0\)[/tex]: [tex]\(p_D = p_R = p_I\)[/tex]
[tex]\(H_1\)[/tex]: At least one of the proportions is different from the others.
- D. [tex]\(H_0\)[/tex]: [tex]\(O_D = E_D\)[/tex] and [tex]\(O_R = E_R\)[/tex] and [tex]\(O_I = E_I\)[/tex]
[tex]\(H_1\)[/tex]: At least one mean is different from what is expected.

For a Chi-Square Test of Independence:
- The null hypothesis [tex]\(H_0\)[/tex] should state that the political party and reaction are independent.
- The alternative hypothesis [tex]\(H_1\)[/tex] should state that the political party and reaction are dependent.

Thus, the correct set of hypotheses is:
- B. [tex]\(H_0\)[/tex]: Political party and reaction are independent.
[tex]\(H_1\)[/tex]: Political party and reaction are dependent.

### Step 2: Perform the Chi-Square Test of Independence

Given the data from the table:
#### Observed Frequencies:
| | Positive | Negative |
|-------------------|----------|----------|
| Democrats | 61 | 368 |
| Independents | 230 | 286 |
| Republicans | 145 | 409 |

We have calculated the Chi-Square statistic and p-value from the table given.
- Chi-square statistic: 108.25665226578609
- p-value: 3.1071816671054687e-24

### Step 3: Determine the Conclusion at [tex]\(\alpha = 0.05\)[/tex]

- Significance level ([tex]\(\alpha\)[/tex]): 0.05

Compare the p-value to the significance level:
- If [tex]\( \text{p-value} < \alpha \)[/tex], we reject the null hypothesis.
- If [tex]\( \text{p-value} \geq \alpha \)[/tex], we fail to reject the null hypothesis.

In this case, the p-value [tex]\(3.1071816671054687e-24\)[/tex] is significantly smaller than the significance level [tex]\(0.05\)[/tex].

### Conclusion

Since the p-value is much smaller than [tex]\(\alpha = 0.05\)[/tex], we reject the null hypothesis. Thus, there is sufficient evidence to suggest that individuals within each political affiliation react differently to the word "socialism".

This means:
- The correct answer is [tex]\(\mathbf{B}\)[/tex].
- [tex]\(H_0\)[/tex]: Political party and reaction are independent.
- [tex]\(H_1\)[/tex]: Political party and reaction are dependent.
- Our conclusion: Reject [tex]\(H_0\)[/tex]. There is significant evidence that political party and reaction are dependent.
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