Chi Square Distribution: Videos & Practice Problems

Confidence intervals for variance require understanding the chi-squared 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-squared distribution is essential because it provides the critical values needed for these intervals.

The chi-squared distribution is characterized by its asymmetry and right skewness, unlike the symmetric t-distribution. This asymmetry means that critical values cannot be found by simply negating one value to find the other, as is possible with symmetric distributions. Instead, two distinct critical values must be identified: the left critical value (χ²_L) and the right critical value (χ²_R), both of which are positive and range from zero to infinity.

When using chi-squared tables, the critical values correspond to specific tail areas. For a confidence level of 1 - α, the two tails each have an area of α/2. However, because the chi-squared distribution is not symmetric, the area used to find χ²_R is α/2 to the right of the critical value, while the area used to find χ²_L is 1 - α/2 to the right of the critical value. This distinction is crucial for correctly identifying the critical values.

The degrees of freedom (df) for the chi-squared distribution when estimating variance is calculated as n - 1, where n is the sample size. To find the critical values for a 95% confidence interval with a sample size of 31, for example, first calculate α = 1 - 0.95 = 0.05. Then, α/2 = 0.025 and 1 - α/2 = 0.975. Using a chi-squared table, locate the critical value χ²_R corresponding to an area of 0.025 to the right with 30 degrees of freedom, which is approximately 46.98. Similarly, find χ²_L corresponding to an area of 0.975 to the right with 30 degrees of freedom, which is approximately 16.79.

These critical values are then used to construct the confidence interval for the population variance. The interval is calculated using the formula:

where s² is the sample variance, and χ²_L and χ²_R are the left and right critical values, respectively. This formula accounts for the asymmetry of the chi-squared distribution and ensures the interval accurately reflects the variability of the population variance.

Understanding the chi-squared distribution's properties and how to correctly identify critical values is fundamental for constructing reliable confidence intervals for variance. This knowledge extends statistical inference beyond means and proportions, enabling more comprehensive data analysis and interpretation.

When working with chi-squared distributions, it is common to estimate chi-squared values using a chi-squared table, especially when given a probability statement such as the probability that the chi-squared statistic exceeds a certain value. For example, if the probability that \(X^2\) is greater than a chi-squared value is 0.025, this corresponds to the area to the right of the chi-squared value on the distribution curve.

To find the chi-squared value for a given probability and degrees of freedom (df), you first locate the probability in the top row of the chi-squared table, then move down to the row corresponding to your degrees of freedom. However, tables often list degrees of freedom only up to 30 individually, then increment by tens (e.g., 40, 50, 60). If your exact degrees of freedom, such as 45, is not listed, there are two common methods to estimate the chi-squared value.

The first method is to use the chi-squared value for the next lowest degrees of freedom available in the table. For instance, if you need the value for 45 degrees of freedom but only have 40 and 50, you use the value for 40 degrees of freedom. This approach is more conservative and less precise but is straightforward. For example, the chi-squared value at 0.025 significance level for 40 degrees of freedom is approximately 59.342.

The second method involves interpolation, which provides a more accurate estimate by calculating a weighted average between the two closest degrees of freedom values. Since 45 is exactly halfway between 40 and 50, you can take the average of the chi-squared values for 40 and 50 degrees of freedom. If the chi-squared value for 40 degrees of freedom is 59.342 and for 50 degrees of freedom is 71.420, the interpolated value for 45 degrees of freedom is:

This interpolation method assumes a linear relationship between the chi-squared values and degrees of freedom in this range, which is a reasonable approximation for many practical purposes.

In summary, when the exact degrees of freedom are not available in a chi-squared table, you can either use the next lowest degrees of freedom value for a conservative estimate or interpolate between the two nearest values for a more precise estimate. Understanding how to estimate chi-squared values is essential for hypothesis testing and confidence interval calculations involving chi-squared distributions.

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 \(α = 1 - C\). For example, a 90% confidence level corresponds to \(α = 0.10\). This value is then split into two tails for a two-sided test, so each tail has an area of \(α/2\). The left chi-squared critical value corresponds to the upper tail probability \$1 - α/2\(, while the right chi-squared critical value corresponds to the lower tail probability \)α/2\(.

The degrees of freedom (df) for the chi-squared distribution are determined by the sample size n as \)df = n - 1\(. This parameter significantly influences the shape and critical values of the chi-squared distribution. As the degrees of freedom increase, the chi-squared distribution shifts to the right and begins to resemble a normal distribution more closely.

For instance, with a sample size of 17 and a 90% confidence level, the degrees of freedom are 16. The critical values can be found using chi-squared tables or statistical software by locating the values at \)α/2 = 0.05\( and \)1 - α/2 = 0.95\( for 16 degrees of freedom. These values are approximately \)χ^2_{0.05,16} = 26.30\( and \)χ^2_{0.95,16} = 7.96\(. Increasing the sample size to 61 (thus \)df = 60\() while keeping the confidence level at 90% results in higher critical values: \)χ^2_{0.05,60} ≈ 79.08\( and \)χ^2_{0.95,60} ≈ 43.19\(. This demonstrates that larger sample sizes increase the chi-squared critical values, reflecting a distribution that is more spread out and shifted rightward.

Changing the confidence level also affects the chi-squared critical values. For a 99% confidence level with a sample size of 17 (\)df = 16\(), the significance level is \)α = 0.01\(, and the tail probabilities are \)α/2 = 0.005\( and \)1 - α/2 = 0.995\(. The corresponding critical values are approximately \)χ^2_{0.005,16} = 34.27\( and \)χ^2_{0.995,16} = 5.14$. Compared to the 90% confidence level, these values are further apart, indicating a wider confidence interval due to the higher confidence requirement.

In summary, increasing the sample size raises the degrees of freedom, which shifts the chi-squared distribution to the right and increases the critical values. Increasing the confidence level decreases the significance level, resulting in critical values that are further apart, thus widening the confidence interval. Understanding these relationships is essential for accurately constructing chi-squared confidence intervals and interpreting statistical results.