Answer :
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.
### 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.