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 $n-\; 1$. This method is essential for constructing accurate confidence intervals for variance, complementing techniques for means and proportions.

Critical Values: Chi Square Distribution

Confidence intervals for variance require understanding the chi-square distribution, which differs significantly from the normal and t distributions used for means and proportions. Variance, denoted as σ², represents the square of the standard deviation and is a key parameter in statistics. To construct confidence intervals for variance, the chi-square distribution is essential because it provides the critical values needed for these intervals.

The chi-square distribution is distinctively asymmetric and right-skewed, unlike the symmetric t-distribution. This asymmetry means that critical values cannot be found by simply mirroring one tail’s value to the other. Instead, two separate critical values must be identified: one for the left tail (χ²_L) and one for the right tail (χ²_R). Both critical values depend on the degrees of freedom, which for variance estimation is calculated as n - 1, where n is the sample size.

When using chi-square tables, the areas referenced correspond to the right side of the critical value, not the left. For the right critical value χ²_R, the area to the right is α/2, where α is the significance level (1 minus the confidence level). For the left critical value χ²_L, the area to the right is 1 - α/2. This distinction is crucial because the chi-square distribution’s skewness means the left and right critical values are not symmetric.

For example, to find the critical values for a 95% confidence interval with a sample size of 31, first calculate α = 1 - 0.95 = 0.05. Then, α/2 = 0.025. The right critical value χ²_R corresponds to the area 0.025 to its right, and the left critical value χ²_L corresponds to the area 0.975 (1 - 0.025) to its right. With degrees of freedom 30 (31 - 1), the chi-square table lookup yields χ²_R ≈ 46.98 and χ²_L ≈ 16.79.

Understanding these critical values allows for the construction of confidence intervals for variance using the formula:

\[\left( \frac{(n-1)s^2}{\chi^2_R}, \frac{(n-1)s^2}{\chi^2_L} \right)\]

where s² is the sample variance. This interval estimates the range within which the true population variance lies with the specified confidence level.

Mastering the chi-square distribution and its critical values is fundamental for accurate variance estimation and enhances statistical inference beyond means and proportions.

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)

Critical Values: Chi Square Distribution Example 1

When working with chi-squared distributions, it is common to use a chi-squared table to find or estimate critical values based on a given probability and degrees of freedom. For example, if you need to find the chi-squared value where the probability that the chi-squared statistic exceeds this value is 0.025, you look up the area to the right of the chi-squared value in the table. The degrees of freedom (df) are essential in locating the correct row in the table.

In many cases, chi-squared tables list values for degrees of freedom up to around 30, then increment by tens (e.g., 40, 50, 60). If your specific degrees of freedom, such as 45, is not listed, there are two common approaches to estimate the chi-squared value:

First, you can use the chi-squared value corresponding to the next lowest degrees of freedom available in the table. For instance, if df = 45 is not listed, use the value for df = 40. This method provides a conservative estimate, which may be less precise but is straightforward. For example, the chi-squared value for df = 40 at a right-tail probability of 0.025 might be approximately 59.342.

Second, a more accurate method is interpolation between the two closest degrees of freedom values. Since 45 is halfway between 40 and 50, you can estimate the chi-squared value by averaging the values for df = 40 and df = 50. If the chi-squared values are 59.342 for df = 40 and 71.420 for df = 50, the interpolated value for df = 45 is calculated as:

\[\frac{59.342 + 71.420}{2} = 65.381\]

This interpolation assumes a roughly linear increase between the two points, providing a more precise estimate than simply using the next lowest degrees of freedom.

Understanding how to estimate chi-squared critical values when exact degrees of freedom are not listed is crucial for hypothesis testing and confidence interval calculations involving chi-squared distributions. Using either the next lowest degrees of freedom or interpolation ensures that you can proceed with statistical analysis even when tables are limited.

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)

Critical Values: Chi Square Distribution Example 2

When calculating chi squared values for confidence intervals, the confidence level and sample size play crucial roles in determining the critical values from the chi squared distribution. The confidence level, denoted as c, helps define the significance level α, which is calculated as \$1 - c\(. For example, a 90% confidence level corresponds to an alpha of 0.10. This alpha is then split into two tails for a two-sided test, so each tail has an area of \)\frac{\alpha}{2}\(. The left chi squared critical value corresponds to the complement of the upper tail, calculated as \)1 - \frac{\alpha}{2}\(, while the right chi squared critical value corresponds to \)\frac{\alpha}{2}\(.

The degrees of freedom (df) for the chi squared distribution are determined by the sample size minus one, i.e., \)\text{df} = n - 1\(. These degrees of freedom are essential for locating the correct critical values in chi squared tables. For instance, with a sample size of 17, the degrees of freedom are 16.

Using a 90% confidence level and a sample size of 17, the critical chi squared values are found by looking up the values at \)\frac{\alpha}{2} = 0.05\( and \)1 - \frac{\alpha}{2} = 0.95\( for 16 degrees of freedom. These values are approximately \)26.30\( (right tail) and \)7.96\( (left tail). Increasing the sample size to 61 (df = 60) while keeping the confidence level at 90% shifts these critical values to approximately \)79.08\( and \)43.19\(. This increase in degrees of freedom causes the chi squared distribution to shift rightward and become more symmetric, resembling a normal distribution. Consequently, the critical values increase and spread further apart.

When the confidence level increases, for example from 90% to 99%, while keeping the sample size constant at 17, the alpha decreases to 0.01, and the tail areas become \)0.005\( and \)0.995\(. The corresponding chi squared critical values for 16 degrees of freedom are approximately \)34.27\( (right tail) and \)5.14$ (left tail). Compared to the 90% confidence level, these values are further apart, reflecting a wider confidence interval due to the higher confidence level.

In summary, increasing the sample size (and thus degrees of freedom) generally increases the chi squared critical values and shifts the distribution to the right, making it more symmetric. Increasing the confidence level decreases alpha, which results in critical values that are further apart, indicating a wider confidence interval. Understanding how these parameters affect chi squared values is essential for constructing accurate confidence intervals in statistical inference.

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 asymmetric and only takes positive values, starting at zero and extending to positive infinity. It is important for constructing confidence intervals for variance because variance is always positive, and the chi-square distribution models the variability of sample variances. Unlike the normal or t distributions, the chi-square distribution is right-skewed, which means critical values must be found separately for the lower and upper tails. This distribution allows us to calculate the range within which the true population variance is likely to fall with a certain confidence level, making it essential for statistical analysis involving variance.

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

To find the critical values for a chi-square distribution, you first determine the significance level $\alpha$ from the confidence level by calculating $\alpha =\; 1\; -$ confidence level. Then, divide $\alpha$ by 2 to get $\alpha /2$. The chi-square distribution is asymmetric, so you find two critical values: the right critical value $r_{}$ using the area $\alpha /2$ to the right, and the left critical value $l_{}$ using the area $1\; -\alpha /2$ to the right. You look these values up in a chi-square table using the degrees of freedom, which is $n\; -\; 1$. These two critical values define the confidence interval for the variance.

Why is the chi-square distribution asymmetric and how does this affect confidence interval calculations?

The chi-square distribution is asymmetric because it is based on the sum of squared standard normal variables, which cannot be negative, resulting in a distribution that is skewed to the right. This asymmetry means that the distribution is not symmetric around its mean, unlike the normal or t distributions. Consequently, when calculating confidence intervals for variance, you cannot simply find one critical value and mirror it for the other tail. Instead, you must find two separate critical values corresponding to the lower and upper tails using different areas in the chi-square table. This ensures accurate interval estimation for variance, which is always positive.

How do you calculate degrees of freedom for the chi-square distribution in variance estimation?

Degrees of freedom (df) for the chi-square distribution in variance estimation are calculated as the sample size minus one, expressed as $\mathrm{df\; =\; n\; -\; 1}$. This adjustment accounts for the fact that one parameter (the sample mean) is estimated from the data before calculating the variance. Using $n\; -\; 1$ ensures that the chi-square distribution correctly models the variability of the sample variance, which is essential for constructing accurate confidence intervals for the population 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, divide by 2: $\alpha /2\; =\; 0.025$. The degrees of freedom are $31\; -\; 1\; =\; 30$. To find the right critical value $r_{}$, 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 $l_{}$, use the area $1\; -\; 0.025\; =\; 0.975$ to the right, which corresponds to approximately 16.79. These two values form the bounds of the confidence interval for the variance.