Find the critical values x1-a/2 and x12 for a 80% confidence level and a sample size of n = 10. X1-a/2 =U (Round to three decimal places as needed.) (Round to three decimal places as needed.)

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### Finding Critical Values for Chi-Square Distribution

In this exercise, you will determine the critical values \(\chi^2_{1-\alpha/2}\) and \(\chi^2_{\alpha/2}\) for an 80% confidence level and a sample size of \(n = 10\).

#### Critical Value Calculation

To find the critical values, you need to refer to the chi-square distribution table and use the following formulas:

\[ \chi^2_{1-\alpha/2} = \_ \]
(Round to three decimal places as needed.)

\[ \chi^2_{\alpha/2} = \_ \]
(Round to three decimal places as needed.)

Given:
- Confidence Level: 80%
- Sample Size: \(n = 10\)

### Steps:

1. **Determine \(\alpha\)**:
   
   Since the confidence level is 80%, \(\alpha = 1 - 0.80 = 0.20\).

2. **Identify \(\alpha/2\) and \(1-\alpha/2\)**:
   
   - \(\alpha/2 = 0.20 / 2 = 0.10\)
   - \(1 - \alpha/2 = 1 - 0.10 = 0.90\)

3. **Degrees of Freedom (df)**:
   
   The degrees of freedom (df) is given by the formula \(df = n - 1\).
   Here, \(df = 10 - 1 = 9\).

4. **Use Chi-Square Distribution Table**:

   You will look up the critical values in the chi-square distribution table for df = 9 with \(\alpha/2 = 0.10\) and \(1 - \alpha/2 = 0.90\).

- \(\chi^2_{1-\alpha/2} = \text{Critical value at 0.90 for df = 9}\)
- \(\chi^2_{\alpha/2} = \text{Critical value at 0.10 for df = 9}\)

### Conclusion:
After determining the appropriate values from the chi-square table, you will have:
- \(\chi^2_{1-\alpha/2} = \_\_\_\)
- \(\chi^2_{\alpha/2} = \_\_\_\)

Remember to round your final answers to three decimal
Transcribed Image Text:### Finding Critical Values for Chi-Square Distribution In this exercise, you will determine the critical values \(\chi^2_{1-\alpha/2}\) and \(\chi^2_{\alpha/2}\) for an 80% confidence level and a sample size of \(n = 10\). #### Critical Value Calculation To find the critical values, you need to refer to the chi-square distribution table and use the following formulas: \[ \chi^2_{1-\alpha/2} = \_ \] (Round to three decimal places as needed.) \[ \chi^2_{\alpha/2} = \_ \] (Round to three decimal places as needed.) Given: - Confidence Level: 80% - Sample Size: \(n = 10\) ### Steps: 1. **Determine \(\alpha\)**: Since the confidence level is 80%, \(\alpha = 1 - 0.80 = 0.20\). 2. **Identify \(\alpha/2\) and \(1-\alpha/2\)**: - \(\alpha/2 = 0.20 / 2 = 0.10\) - \(1 - \alpha/2 = 1 - 0.10 = 0.90\) 3. **Degrees of Freedom (df)**: The degrees of freedom (df) is given by the formula \(df = n - 1\). Here, \(df = 10 - 1 = 9\). 4. **Use Chi-Square Distribution Table**: You will look up the critical values in the chi-square distribution table for df = 9 with \(\alpha/2 = 0.10\) and \(1 - \alpha/2 = 0.90\). - \(\chi^2_{1-\alpha/2} = \text{Critical value at 0.90 for df = 9}\) - \(\chi^2_{\alpha/2} = \text{Critical value at 0.10 for df = 9}\) ### Conclusion: After determining the appropriate values from the chi-square table, you will have: - \(\chi^2_{1-\alpha/2} = \_\_\_\) - \(\chi^2_{\alpha/2} = \_\_\_\) Remember to round your final answers to three decimal
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