Chi Square Distribution: Videos & Practice Problems

Confidence intervals for variance require the chi-square distribution, which is asymmetric and only takes positive values. Critical values, $\chi r_{}$ and $\chi l_{}$, are found separately using areas $\alpha /2$ and $1\; -\; \alpha /2$ to the right of these values, respectively. Degrees of freedom equal $n\; -\; 1$. For a 95% confidence interval with sample size 31, critical values are approximately 46.98 and 16.79. This method is essential for constructing confidence intervals for variance in statistical analysis.

Use a table to find or estimate ${\chi }^2$ such that:
$P(X^2>{\chi }^2)=0.010$ (Area to the right)
$df=50$

• A. 27.99
• B. 29.71
• C. 76.15
• D. 79.49

Answer: (C)

Use a table to find or estimate ${\chi }^2$ such that:
(Area to the left)
$df=50$

• A. 27.99
• B. 29.71
• C. 76.15
• D. 79.49

Answer: (B)

Find the left and right ${\chi }^2$-values for a 99% confidence interval with a sample size of 25.
$\alpha =$ ________ $df=$ _______
$\frac{\alpha }{2}=$ _________ ${\chi }_R^2=$ _______
$1-\frac{\alpha }{2}=$ ________ ${\chi }_L^2=$ _______

• A. $\chi_{L}^2=$ 9.89 ; $\chi_{R}^2=$ 45.56
• B. $\chi_{L}^2=$ 45.56; $\chi_{R}^2=$ 9.89
• C. $\chi_{L}^2=$ 10.86 ; $\chi_{R}^2=$ 42.98
• D. $\chi_{L}^2=$ 10.52 ; $\chi_{R}^2=$ 46.93

Answer: (A)

Here’s what students ask on this topic:

What is the chi-square distribution and why is it important for confidence intervals of variance?

The chi-square distribution is a probability distribution that is used primarily for inference about population variances. It is important because it is the basis for constructing confidence intervals for variance. Unlike the normal or t-distributions, the chi-square distribution is asymmetric and only takes positive values, starting at zero and extending to positive infinity. This asymmetry means that when finding critical values for confidence intervals, we must consider two separate critical values, ${\chi }_l$ and ${\chi }_r$, corresponding to the left and right tails of the distribution. These critical values are found using areas ${\alpha }_{\frac{1}{2}}$ and $1-{\alpha }_{\frac{1}{2}}$ to the right of the values, respectively. Understanding this distribution is essential for accurate statistical analysis of variance.

How do you find the critical values for a chi-square distribution when constructing a confidence interval for variance?

To find the critical values for a chi-square distribution, first determine the significance level $\alpha$ from the confidence level using $\alpha =\; 1\; -\; ext\{confidence\; level\}$. Then calculate ${\alpha }_{\frac{1}{2}}=\; rac\{\alpha \}\{2\}$. The degrees of freedom are $n\; -\; 1$, where $n$ is the sample size. The right critical value ${\chi }_r$ is found by looking up the chi-square value corresponding to the area ${\alpha }_{\frac{1}{2}}$ to the right. The left critical value ${\chi }_l$ is found by looking up the chi-square value corresponding to the area $1\; -{\alpha }_{\frac{1}{2}}$ to the right. These two values define the interval for the variance with the desired confidence level.

Why is the chi-square distribution asymmetric and how does this affect finding critical values?

The chi-square distribution is asymmetric because it is based on the sum of squared standard normal variables, which are always positive or zero. This results in a distribution that is right-skewed and only takes positive values, starting at zero and extending to infinity. Due to this asymmetry, the critical values for confidence intervals cannot be found by simply taking the positive and negative of a single value, unlike symmetric distributions such as the t-distribution. Instead, two separate critical values must be found: one for the left tail and one for the right tail, each corresponding to different areas under the curve. This requires looking up two different areas in the chi-square table, making the process slightly more complex.

How do degrees of freedom affect the chi-square distribution and confidence intervals for variance?

Degrees of freedom (df) in the chi-square distribution are calculated as $n\; -\; 1$, where $n$ is the sample size. The degrees of freedom affect the shape of the chi-square distribution: as df increases, the distribution becomes more symmetric and approaches a normal distribution. For confidence intervals of variance, the degrees of freedom determine which row to use in the chi-square table to find the critical values. Using the correct degrees of freedom is essential because it directly influences the critical values ${\chi }_l$ and ${\chi }_r$, and thus the accuracy of the confidence interval for the variance.

Can you provide an example of finding chi-square critical values for a 95% confidence interval with a sample size of 31?

For a 95% confidence interval with $n\; =\; 31$, first calculate the significance level: $\alpha \; =\; 1\; -\; 0.95\; =\; 0.05$. Then, ${\alpha }_{\frac{1}{2}}=\; 0.025$. The degrees of freedom are $31\; -\; 1\; =\; 30$. To find the right critical value ${\chi }_r$, look up the chi-square value with 30 degrees of freedom and area 0.025 to the right, which is approximately 46.98. For the left critical value ${\chi }_l$, look up the chi-square value with 30 degrees of freedom and area 0.975 to the right (which is 1 - 0.025), approximately 16.79. These values define the confidence interval for the variance.