Let's re-evaluate the situation given that the test statistic is \( F = 1.81 \).

1. **State the null and alternative hypotheses:**
   \[
   H_0: \sigma_1^2 = \sigma_2^2 \quad (\text{The variances are equal})
   \]
   \[
   H_1: \sigma_1^2 \neq \sigma_2^2 \quad (\text{The variances are not equal})
   \]

2. **Calculate the test statistic (F-ratio):**
   Given \( F = 1.81 \).

3. **Determine the critical value:**
   For \( n_1 = 25 \) and \( n_2 = 25 \), the degrees of freedom are \( df_1 = 24 \) and \( df_2 = 24 \). For a two-tailed test at the \(\alpha = 0.05\) significance level, the critical values for \( F_{(24,24)} \) are:
   \[
   F_{0.025, 24, 24} = 2.053
   \]
   \[
   F_{0.975, 24, 24} = \frac{1}{2.053} \approx 0.4873
   \]

4. **Compare the test statistic to the critical value:**
   The test statistic \( F = 1.81 \) falls within the range \( 0.4873 < F < 2.053 \).

5. **Decision:**
   Since the test statistic \( F = 1.81 \) is within the acceptance region, we fail to reject the null hypothesis. There is not enough evidence to conclude that there is a significant difference in the variances of the salaries between the two counties.

### Summary:
- **Test statistic (F):** 1.81
- **Decision:** Fail to reject \( H_0 \)
- **Conclusion:** There is no significant difference in the variability of salaries between the two counties at the \(\alpha = 0.05\) level.

Would you like details or have any questions? Here are some related questions you might find useful:

1. How do you interpret the results of an F-test for variances?
2. What assumptions must be met to perform an F-test?
3. How can you check if the variances are normally distributed before performing the F-test?
4. What is the impact of sample size on the F-test?
5. Can you perform an F-test with unequal sample sizes?
6. What are some alternatives to the F-test if the assumptions are not met?
7. How does the significance level (\(\alpha\)) affect the decision in an F-test?
8. What is the relationship between the F-test and ANOVA?

**Tip:** When conducting hypothesis tests, always double-check your calculations and ensure the assumptions of the test are satisfied to ensure valid results.

Explore the F-test for variance with an example test statistic of 1.81. Learn how to interpret the results, calculate critical values, and make decisions based on hypothesis testing. Detailed analysis and conclusion included.

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