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The table provided shows voting data for two candidates, A and B, across urban and rural areas. It seems that a chi-square (\( \chi^2 \)) test is to be used to examine whether the area (urban vs. rural) is related to voting preference.

### Step-by-Step Solution:

1. **Set up the null hypothesis (\(H_0\))**: The null hypothesis is that voting preference is independent of area, meaning that there's no significant difference in candidate preference between urban and rural voters.

2. **Observed Frequencies**:
   - Urban voters for A: 550
   - Rural voters for A: 620
   - Urban voters for B: 450
   - Rural voters for B: 380

3. **Expected Frequencies**:
   We calculate the expected frequencies for each cell using the formula:
   \[
   E = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}
   \]
   where the grand total here is 2000.

4. **Calculating the Chi-Square Statistic**:
   Use the formula:
   \[
   \chi^2 = \sum \frac{(O - E)^2}{E}
   \]
   where \(O\) is the observed frequency and \(E\) is the expected frequency for each cell.

5. **Degrees of Freedom**:
   For a 2x2 table, degrees of freedom (df) is \((\text{rows} - 1) \times (\text{columns} - 1) = 1\).

6. **Compare with Critical Value**:
   According to the provided information, the critical value of \( \chi^2 \) at 1 df and 5% significance level is 3.84.

Would you like a detailed calculation of each step? 

### Related Questions
1. What does the chi-square test reveal about the relationship between area and voting preference?
2. How are expected frequencies calculated in contingency tables?
3. Why do we use degrees of freedom in a chi-square test?
4. What would happen if the observed data changed significantly?
5. How does the level of significance impact the outcome of the hypothesis test?

**Tip**: Always double-check calculations, as even small errors in expected frequencies can alter the chi-square value significantly.

Use a chi-square (χ²) test to examine if the nature of the area (urban vs. rural) is related to voting preference in an election between candidates A and B.

Explore the chi-square test of independence to determine if area type (urban vs. rural) affects voting preference between candidates A and B. This example includes observed and expected frequencies, calculation of the chi-square statistic, and comparison with critical values.

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