Find the critical values $$ \chi_{1-\alpha / 2}^{2} $$ and $$ \chi_{\alpha / 2}^{2} $$ for a $$ 99 \% $$ confidence level and a sample size of $$ n=10 $$. $$ \chi_{1-\alpha / 2}^{2}=1.735 $$ (Round to three decimal places as needed.) $$ \chi_{\alpha / 2}^{2}=\square $$ (Round to three decimal places as needed.)

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To find the critical values $$ \chi_{1-\alpha / 2}^{2} $$ and $$ \chi_{\alpha / 2}^{2} $$ for a $$ 99\% $$ confidence level with a sample size of $$ n = 10 $$, we first need to determine the degrees of freedom. The degrees of freedom for a chi-squared distribution in this context is given by:

$$
df = n - 1 = 10 - 1 = 9
$$

For a $$ 99\% $$ confidence level, the significance level $$ \alpha $$ is:

$$
\alpha = 1 - 0.99 = 0.01
$$

This means that $$ \alpha/2 = 0.005 $$.

Now, we need to find the critical values:

1. **Finding $$ \chi_{1-\alpha/2}^{2} $$**:
   - We are given that $$ \chi_{1-\alpha/2}^{2} = \chi_{0.995}^{2} = 1.735 $$.

2. **Finding $$ \chi_{\alpha/2}^{2} $$**:
   - We need to find $$ \chi_{0.005}^{2} $$ for $$ df = 9 $$.

Using a chi-squared distribution table or calculator, we can find the critical value for $$ \chi_{0.005}^{2} $$ with $$ 9 $$ degrees of freedom.

From the chi-squared distribution table, the critical value $$ \chi_{0.005}^{2} $$ for $$ df = 9 $$ is approximately:

$$
\chi_{0.005}^{2} \approx 21.666
$$

Thus, rounding to three decimal places, we have:

$$
\chi_{\alpha / 2}^{2} = 21.666
$$

In summary, the critical values are:

$$
\chi_{1-\alpha / 2}^{2} = 1.735
$$
$$
\chi_{\alpha / 2}^{2} = 21.666
$$
The critical values are $$ \chi_{1-\alpha / 2}^{2} = 1.735 $$ and $$ \chi_{\alpha / 2}^{2} = 21.666 $$.

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