Find the critical values chi squared Subscript Upper L and chi squared Subscript Upper R for the given confidence level c and sample size n.
cequals0.8​,  nequals27
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Part 1
chi squared Subscript Upper Lequals

enter your response here ​(Round to three decimal places as​ needed.)

To find the critical values of the chi-squared distribution, \(\chi^2_L\) (left-tail critical value) and \(\chi^2_R\) (right-tail critical value), for a given confidence level \(c\) and sample size \(n\), we follow these steps:

### Step 1: Degrees of Freedom
The degrees of freedom (df) is calculated as:

\[
\text{df} = n - 1
\]
Given \(n = 27\), we have:

\[
\text{df} = 27 - 1 = 26
\]

### Step 2: Confidence Level
The confidence level \(c = 0.80\) means that the central 80% of the chi-squared distribution is retained. Thus, the area in each tail is:

\[
\frac{1 - c}{2} = \frac{1 - 0.80}{2} = 0.10
\]

So, the left tail area is 0.10, and the right tail area is also 0.10.

### Step 3: Finding Critical Values
We need to find the critical values \(\chi^2_L\) and \(\chi^2_R\) such that:

- \(\chi^2_L\) corresponds to a cumulative area of 0.10 (left tail),
- \(\chi^2_R\) corresponds to a cumulative area of 0.90 (right tail).

Using a chi-squared distribution table or statistical software:

- For \(\chi^2_L\), with \(df = 26\) and a left-tail area of 0.10, we find \(\chi^2_L = 16.151\).
- For \(\chi^2_R\), with \(df = 26\) and a right-tail area of 0.90, we find \(\chi^2_R = 37.652\).

### Final Answer:

\[
\chi^2_L = 16.151 \quad \text{and} \quad \chi^2_R = 37.652
\]

Would you like further explanation, or do you have any questions?

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Here are 5 related questions to consider:
1. How do you interpret confidence intervals using the chi-squared distribution?
2. What are some applications of the chi-squared test in statistics?
3. How does changing the confidence level affect \(\chi^2_L\) and \(\chi^2_R\)?
4. What is the relationship between sample size and the degrees of freedom in a chi-squared distribution?
5. How can we use a chi-squared table to approximate critical values?

**Tip:** The chi-squared distribution becomes more symmetric as the degrees of freedom increase. For small degrees of freedom, the distribution is highly skewed.

Find the critical values chi squared Subscript Upper L and chi squared Subscript Upper R for the given confidence level c and sample size n. c=0.8, n=27. Part 1: chi squared Subscript Upper L = (Round to three decimal places as needed.)

Find the chi-squared critical values for a confidence level of 0.80 and a sample size of 27. Learn how to calculate left and right critical values using the chi-squared distribution with 26 degrees of freedom. Includes key formulas and steps to find chi-squared values.

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