To determine the critical values for a 90% confidence interval using the chi-square distribution with 25 degrees of freedom, follow these steps:
1. Understand the confidence level and the significance level:
- The confidence level is given as 90%, which means we are looking for the values that capture the central 90% of the distribution.
- The significance level ([tex]\(\alpha\)[/tex]) is complementary to the confidence level and is calculated as [tex]\(1 - \text{confidence level}\)[/tex]. Thus, [tex]\(\alpha = 1 - 0.90 = 0.10\)[/tex].
2. Determine the critical regions:
- Since we need the critical values to capture the central 90% of the distribution, the remaining 10% is split equally between the two tails of the chi-square distribution.
- Therefore, each tail will have an area of [tex]\(\alpha/2 = 0.10/2 = 0.05\)[/tex].
3. Use the chi-square distribution table or a computational tool:
- Locate the critical value that corresponds to the lower 5% (left tail) and the upper 5% (right tail) within a chi-square distribution with 25 degrees of freedom.
4. Find the lower critical value:
- The lower critical value is found at the point where the cumulative distribution function (CDF) equals 0.05 for 25 degrees of freedom.
- This value is approximately 14.611.
5. Find the upper critical value:
- The upper critical value corresponds to the point where the CDF equals 0.95 (since 1 - 0.05 = 0.95) for 25 degrees of freedom.
- This value is approximately 37.652.
6. Round the values to three decimal places:
- According to the problem statement, we should round both critical values to three decimal places.
Therefore, the critical values for a 90% confidence interval using the chi-square distribution with 25 degrees of freedom are [tex]\(14.611\)[/tex] and [tex]\(37.652\)[/tex].
