Answer :
### Step-by-Step Solution
#### Step 1: Formulate the Hypotheses
We want to test if the proportion of men (population 1) who support the candidate is different from the proportion of women (population 2) who support the candidate.
The appropriate hypotheses are:
- Null hypothesis ([tex]$H_0$[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative hypothesis ([tex]$H_a$[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
So the correct formulation is:
[tex]\[ H_0: p_1 = p_2 \quad ; \quad H_a: p_1 \neq p_2 \][/tex]
#### Step 2: Verify Sample Size Requirements
In a Z-test for proportions, the requirements are:
- [tex]\( n_1 \)[/tex] and [tex]\( n_2 \geq 30 \)[/tex]
- At least 5 successes and 5 failures in each sample.
From the given information:
- For men: [tex]\( n_1 = 80 \)[/tex], successes [tex]\( = 52 \)[/tex], failures [tex]\( = 80 - 52 = 28 \)[/tex]
- For women: [tex]\( n_2 = 84 \)[/tex], successes [tex]\( = 58 \)[/tex], failures [tex]\( = 84 - 58 = 26 \)[/tex]
Both sample sizes are greater than 30, and there are more than 5 successes and 5 failures in each group. Therefore, the sample size requirements are satisfied.
#### Step 3: Identify the Test Distribution
Since we are comparing proportions and have large sample sizes, we use the Z distribution.
- This is a two-tailed test.
- The distribution used is [tex]\( Z \)[/tex] since we are testing two proportions.
#### Step 4: Calculate Sample Proportions and Pooled Proportion
- Sample proportion for men ([tex]\( p_1 \)[/tex]):
[tex]\[ p_1 = \frac{52}{80} = 0.65 \][/tex]
- Sample proportion for women ([tex]\( p_2 \)[/tex]):
[tex]\[ p_2 = \frac{58}{84} \approx 0.6905 \][/tex]
- Pooled proportion ([tex]\( p \)[/tex]):
[tex]\[ p = \frac{52 + 58}{80 + 84} \approx 0.6707 \][/tex]
#### Step 5: Calculate the Standard Error
The standard error (SE) for the difference in proportions is given by:
[tex]\[ SE = \sqrt{p(1 - p) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
Substituting the values:
[tex]\[ SE \approx 0.0734 \][/tex]
#### Step 6: Compute the Z-Score
The Z-Score is calculated by:
[tex]\[ z = \frac{p_1 - p_2}{SE} \][/tex]
[tex]\[ z \approx \frac{0.65 - 0.6905}{0.0734} \approx -0.5513 \][/tex]
#### Step 7: Determine the P-Value
Using the Z-score, we can find the P-value:
[tex]\[ P(z \text{ is at least as extreme as } -0.5513) = 2 \times (1 - \text{CDF of } |z|) = 2 \times (1 - \text{CDF of } 0.5513) \approx 0.5814 \][/tex]
#### Step 8: Make a Decision
Compare the P-value with the significance level ([tex]\( \alpha \)[/tex]):
- [tex]\( \alpha = 0.10 \)[/tex]
- P-value [tex]\( \approx 0.5814 \)[/tex]
Since the P-value [tex]\( (0.5814) \)[/tex] is greater than [tex]\( \alpha (0.10) \)[/tex], we fail to reject the null hypothesis.
#### Conclusion
Given the data and our calculations, we do not have enough evidence to reject the null hypothesis at the [tex]\( \alpha = 0.10 \)[/tex] significance level. Therefore, we do not have sufficient evidence to conclude that the proportion of men who support the candidate is different from the proportion of women who support the candidate.
#### Step 1: Formulate the Hypotheses
We want to test if the proportion of men (population 1) who support the candidate is different from the proportion of women (population 2) who support the candidate.
The appropriate hypotheses are:
- Null hypothesis ([tex]$H_0$[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative hypothesis ([tex]$H_a$[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
So the correct formulation is:
[tex]\[ H_0: p_1 = p_2 \quad ; \quad H_a: p_1 \neq p_2 \][/tex]
#### Step 2: Verify Sample Size Requirements
In a Z-test for proportions, the requirements are:
- [tex]\( n_1 \)[/tex] and [tex]\( n_2 \geq 30 \)[/tex]
- At least 5 successes and 5 failures in each sample.
From the given information:
- For men: [tex]\( n_1 = 80 \)[/tex], successes [tex]\( = 52 \)[/tex], failures [tex]\( = 80 - 52 = 28 \)[/tex]
- For women: [tex]\( n_2 = 84 \)[/tex], successes [tex]\( = 58 \)[/tex], failures [tex]\( = 84 - 58 = 26 \)[/tex]
Both sample sizes are greater than 30, and there are more than 5 successes and 5 failures in each group. Therefore, the sample size requirements are satisfied.
#### Step 3: Identify the Test Distribution
Since we are comparing proportions and have large sample sizes, we use the Z distribution.
- This is a two-tailed test.
- The distribution used is [tex]\( Z \)[/tex] since we are testing two proportions.
#### Step 4: Calculate Sample Proportions and Pooled Proportion
- Sample proportion for men ([tex]\( p_1 \)[/tex]):
[tex]\[ p_1 = \frac{52}{80} = 0.65 \][/tex]
- Sample proportion for women ([tex]\( p_2 \)[/tex]):
[tex]\[ p_2 = \frac{58}{84} \approx 0.6905 \][/tex]
- Pooled proportion ([tex]\( p \)[/tex]):
[tex]\[ p = \frac{52 + 58}{80 + 84} \approx 0.6707 \][/tex]
#### Step 5: Calculate the Standard Error
The standard error (SE) for the difference in proportions is given by:
[tex]\[ SE = \sqrt{p(1 - p) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
Substituting the values:
[tex]\[ SE \approx 0.0734 \][/tex]
#### Step 6: Compute the Z-Score
The Z-Score is calculated by:
[tex]\[ z = \frac{p_1 - p_2}{SE} \][/tex]
[tex]\[ z \approx \frac{0.65 - 0.6905}{0.0734} \approx -0.5513 \][/tex]
#### Step 7: Determine the P-Value
Using the Z-score, we can find the P-value:
[tex]\[ P(z \text{ is at least as extreme as } -0.5513) = 2 \times (1 - \text{CDF of } |z|) = 2 \times (1 - \text{CDF of } 0.5513) \approx 0.5814 \][/tex]
#### Step 8: Make a Decision
Compare the P-value with the significance level ([tex]\( \alpha \)[/tex]):
- [tex]\( \alpha = 0.10 \)[/tex]
- P-value [tex]\( \approx 0.5814 \)[/tex]
Since the P-value [tex]\( (0.5814) \)[/tex] is greater than [tex]\( \alpha (0.10) \)[/tex], we fail to reject the null hypothesis.
#### Conclusion
Given the data and our calculations, we do not have enough evidence to reject the null hypothesis at the [tex]\( \alpha = 0.10 \)[/tex] significance level. Therefore, we do not have sufficient evidence to conclude that the proportion of men who support the candidate is different from the proportion of women who support the candidate.
