Questions: Find the critical values χL2 and χU2 for the given level of confidence and sample size. Round your answer to 2 decimal places. a) 90% confidence, n=20; χL2= χU2= b) 99% confidence, n=14; χL2= χU2= c) 95% confidence, n=25; χL2= χU2=

Find the critical values χL2 and χU2 for the given level of confidence and sample size. Round your answer to 2 decimal places.
a) 90% confidence, n=20;
χL2= χU2=
b) 99% confidence, n=14;
χL2= χU2=
c) 95% confidence, n=25;
χL2= χU2=

Transcript text: Find the critical values $\chi_{L}^{2}$ and $\chi_{U}^{2}$ for the given level of confidence and sample size. Round your answer to 2 decimal places. a) $90 \%$ confidence, $n=20$; $\chi_{L}^{2}=$ $\square$ $\chi_{U}^{2}=$ $\square$ b) $99 \%$ confidence, $n=14$; $\chi_{L}^{2}=$ $\square$ $\chi_{U}^{2}=$ $\square$ c) $95 \%$ confidence, $n=25$; \[ \chi_{L}^{2}=\square \chi_{U}^{2}=\square \]

Solution

Solution Steps

To find the critical values $\chi_{L}^{2}$ and $\chi_{U}^{2}$ for a given level of confidence and sample size, we use the chi-square distribution. The degrees of freedom for the chi-square distribution is $n - 1$, where $n$ is the sample size. For a given confidence level, the critical values are found by determining the chi-square values that correspond to the lower and upper tails of the distribution. Specifically, for a confidence level of $C$, the lower critical value $\chi_{L}^{2}$ corresponds to the $(1-C)/2$ percentile, and the upper critical value $\chi_{U}^{2}$ corresponds to the $(1+C)/2$ percentile.

Step 1: Determine Critical Values for 90% Confidence and $ n = 20 $

For a 90% confidence level with $ n = 20 $, the degrees of freedom is calculated as: \[ df = n - 1 = 20 - 1 = 19 \] The critical values are: \[ \chi_{L}^{2} = 10.1170 \quad \text{and} \quad \chi_{U}^{2} = 30.1435 \]

Step 2: Determine Critical Values for 99% Confidence and $ n = 14 $

For a 99% confidence level with $ n = 14 $, the degrees of freedom is: \[ df = n - 1 = 14 - 1 = 13 \] The critical values are: \[ \chi_{L}^{2} = 3.5650 \quad \text{and} \quad \chi_{U}^{2} = 29.8195 \]

Step 3: Determine Critical Values for 95% Confidence and $ n = 25 $

For a 95% confidence level with $ n = 25 $, the degrees of freedom is: \[ df = n - 1 = 25 - 1 = 24 \] The critical values are: \[ \chi_{L}^{2} = 12.4012 \quad \text{and} \quad \chi_{U}^{2} = 39.3641 \]

Final Answer

a) $ \chi_{L}^{2} = 10.1170 $, $ \chi_{U}^{2} = 30.1435 $
b) $ \chi_{L}^{2} = 3.5650 $, $ \chi_{U}^{2} = 29.8195 $
c) $ \chi_{L}^{2} = 12.4012 $, $ \chi_{U}^{2} = 39.3641 $

\[ \boxed{ \begin{align_} \text{a)} & \quad \chi_{L}^{2} = 10.1170, \quad \chi_{U}^{2} = 30.1435 \\ \text{b)} & \quad \chi_{L}^{2} = 3.5650, \quad \chi_{U}^{2} = 29.8195 \\ \text{c)} & \quad \chi_{L}^{2} = 12.4012, \quad \chi_{U}^{2} = 39.3641 \end{align_} } \]

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