Why are degrees of freedom important in a Chi Square Test for Independence?

Find the critical values ${\chi }_R^2$ and ${\chi }_L^2$ for a confidence level of $c=0.95$ and a sample size of $n=12$.

The weights (in $\text{kg}$) of $8$ randomly selected sealed sugar packs were recorded as follows:

$1.0,1.2,0.9,1.1,1.0,0.8,1.3,1.1$

Assuming the weights follow a normal distribution, construct a $99\%$ confidence interval for the population standard deviation ($\sigma$). Also, interpret the results.

A random sample of $10$ adult males had their total cholesterol levels (in $\text{mg/dL}$) measured:

$180,190,195,200,205,210,195,200,205,190$

Use this sample to construct a $95\%$ confidence interval estimate of the population standard deviation of cholesterol levels.

A study records the monthly water consumption (in liters) of households in a neighborhood. A small sample contains the values $150$, $180$, $170$, $160$. Ten bootstrap samples are generated from this data:

Using the same bootstrap samples, construct an $80\%$ confidence interval estimate of the standard deviation of the monthly water consumption.

Find the critical values ${\chi }_{1-\frac{\alpha }{2}}^2$ and ${\chi }_{\frac{\alpha }{2}}^2$ for a confidence level of $95\%$ with sample size $n=15$.