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This question involves hypothesis testing for the difference between two proportions.

### Problem Summary:
- **Group 1**: 80% are ready to sell their votes for a small amount. 
- **Group 2**: 40% are ready to sell their votes for a large amount.
- Both groups have 1,000 members each.
- We are testing at a 5% significance level to see if there's a significant difference between the two proportions (two-sided test).

### Solution Outline:
1. **Define the Hypotheses:**
   - Null Hypothesis (\( H_0 \)): \( p_1 = p_2 \) (No difference in the proportion of people ready to sell their votes in both groups).
   - Alternative Hypothesis (\( H_a \)): \( p_1 \neq p_2 \) (There is a difference in the proportions).

2. **Calculate Sample Proportions:**
   - Proportion for Group 1: \( \hat{p}_1 = 0.80 \)
   - Proportion for Group 2: \( \hat{p}_2 = 0.40 \)

3. **Determine the Pooled Proportion (\( \hat{p} \))**:
   \[
   \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{0.80 \times 1000 + 0.40 \times 1000}{1000 + 1000} = \frac{800 + 400}{2000} = 0.60
   \]

4. **Calculate the Standard Error (SE)**:
   \[
   SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.60 \times 0.40 \times \left( \frac{1}{1000} + \frac{1}{1000} \right)} = \sqrt{0.24 \times 0.002} = \sqrt{0.00048} \approx 0.0219
   \]

5. **Calculate the Test Statistic (Z)**:
   \[
   Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.80 - 0.40}{0.0219} \approx \frac{0.40}{0.0219} \approx 18.26
   \]

6. **Determine the Critical Value and Make a Decision**:
   - For a two-tailed test at a 5% significance level, the critical Z-value is approximately \( \pm 1.96 \).
   - Since the calculated \( Z \approx 18.26 \) is far greater than 1.96, we reject the null hypothesis.

### Conclusion:
There is a statistically significant difference between the two proportions at the 5% level of significance.

Would you like further details on any part of this solution or have additional questions? 

### Further Related Questions:
1. How would the test change if it was one-sided instead of two-sided?
2. What would happen if the sample sizes were different for each group?
3. How do we interpret the p-value in this context?
4. What are Type I and Type II errors in the context of this test?
5. How would the conclusion change if the significance level was reduced to 1%?

#### Tip:
In hypothesis testing, always clearly define your null and alternative hypotheses before starting any calculations.

A social experiment shows that in a group 80% of people are ready to sell their votes for money when they are offered a small amount. In another group, 40% of people are ready to sell their votes when they are offered a huge sum of money. In both cases, 1000 members each were participated. Test at 5% level of significance (Two-sided) that there is a difference between two proportions.

Explore a hypothesis test to determine if there's a statistically significant difference between two proportions. This example evaluates if the proportion of people willing to sell votes for a small amount differs from those willing to sell for a large sum. The test uses a two-sided Z-test at a 5% significance level, with sample sizes of 1000 each.

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