Question

Consider the following situation: A normal distribution of a random variable, \(\mathrm{X}\), has a variance \(\sigma_{1}^{2}\), where \(\sigma_{1}^{2}\) is unknown. It is found however that experimental values of \(\mathrm{X}\) have a wide dispersion indicating that \(\sigma_{1}^{2}\) must be quite large. A certain modification in the experiment is made to reduce the variance. Let the post-modification random variable be denoted \(\mathrm{Y}\), and let \(\mathrm{Y}\) have a normal distribution with variance \(\sigma_{2}^{2}\). Find a completely general method of determining confidence intervals for ratios of variances, \(\sigma_{1}^{2} / \sigma_{2}^{2}\)

Short answer

To determine confidence intervals for ratios of variances, \(\sigma_{1}^{2} / \sigma_{2}^{2}\), use the F-distribution. Identify the degrees of freedom for each variance, \(df_1\) and \(df_2\). Find the F-critical values, \(F_{\alpha/2}\) and \(F_{1-\alpha/2}\), using the significance level, \(\alpha\), and the degrees of freedom. Calculate the confidence interval using the F-critical values and the sample variances, \(s_{1}^{2}\) and \(s_{2}^{2}\): \[ Confidence\;Interval = \left(\frac{s_{1}^{2}}{s_{2}^{2}} \times \frac{1}{F_{1-\alpha/2}}, \frac{s_{1}^{2}}{s_{2}^{2}} \times \frac{1}{F_{\alpha/2}}\right) \] The result is a \(100(1-\alpha)\)% confidence interval for the ratio of the two variances.

Step-by-step solution

Identify Key Variables

The key variables in this situation are the variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\). These represent the dispersion of the distributions from which \(X\) and \(Y\) are drawn, respectively.

Understand the F-distribution

The F-distribution can be used to find confidence intervals for ratios of variances. If \(X\) and \(Y\) are normally distributed and independent, the ratio of their variances (\(\sigma_{1}^{2} / \sigma_{2}^{2}\)) follows an F-distribution.

Identify the Degrees of Freedom

The degrees of freedom for each of the variances, \(df_1\) for \(\sigma_{1}^{2}\) and \(df_2\) for \(\sigma_{2}^{2}\), are typically equal to the sample sizes minus 1 from which the variances were computed (N-1, if N is the sample size).

Find the F-critical values

The F-critical values define the range (F(Lower) and F(Upper)) for the \(100(1-\alpha)\)% confidence interval and can be found from the F-distribution using the degrees of freedom defined above and the significance level \(\alpha\). These F critical values are denoted as \(F_{\alpha/2}\) for the lower bound and \(F_{1-\alpha/2}\) for the upper bound.

Calculate the confidence interval

Finally, the confidence interval for the ratio of variances \(\sigma_{1}^{2} / \(\sigma_{2}^{2}\) can be calculated using the F-critical values and the sample variances, \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\), which can be replaced by \(s_{1}^{2}\) and \(s_{2}^{2}\) respectively when they are estimated from a sample: \[ Confidence\;Interval = \left(\frac{s_{1}^{2}}{s_{2}^{2}} \times \frac{1}{F_{1-\alpha/2}}, \frac{s_{1}^{2}}{s_{2}^{2}} \times \frac{1}{F_{\alpha/2}}\right) \] Where \(s_{1}^{2}\) and \(s_{2}^{2}\) are the sample variances of \(X\) and \(Y\) respectively. The resulting interval represents a \(100(1-\alpha)\)% confidence interval for the ratio of the two variances.

Key concepts

F-distribution

The F-distribution plays a crucial role when comparing two sample variances. It's a distribution that arises when calculating the ratio of two independent chi-squared variables, each divided by their respective degrees of freedom. This makes it particularly useful for testing hypotheses about the equality of two variances.

• If we know that two sets of data are independent and normally distributed, we can use the F-distribution to test the hypothesis that their population variances are equal.
• The shape of the F-distribution is influenced by two parameters: the degrees of freedom for the numerator and the denominator.
• This distribution is positively skewed, and its values cannot be negative.

Understanding the F-distribution helps in calculating confidence intervals for the ratio of two variances, which is essential when we want to assess the effect of experimental changes.

Degree of Freedom

The degrees of freedom are a key concept in statistical procedures, influencing the shape of the F-distribution. Degrees of freedom refer to the number of values in a calculation that are free to vary.

Calculating Degrees of Freedom

• For a sample variance of a dataset, the degrees of freedom is calculated as the sample size minus one ( -1").
• In the context of the F-distribution, there are two separate degrees of freedom: one for the numerator and one for the denominator. These correspond to the sample sizes of each group minus one.

Having accurate degrees of freedom ensures that the F-distribution is correctly applied, which is crucial for determining the confidence intervals of the variance ratio.

Variance Ratio

The variance ratio is the quotient obtained by dividing one sample variance by another. It's a useful measure when comparing the spread of two different datasets.

Understanding Variance Ratio

• Variance indicates how much individual data points in a dataset deviate from the mean. A larger variance means more spread out data.
• By calculating the variance ratio, we can determine if the experiments or changes applied have successfully reduced the variability as intended.

The F-distribution uses this variance ratio to calculate confidence intervals, helping determine if observed differences in data variability are statistically significant.

Sample Variance

Sample variance is an estimate of the variance within a population, derived from a sample of data. It captures the average squared deviation from the sample mean.

Calculating Sample Variance

• To calculate sample variance, sum up the squared differences between each data point and the sample mean, then divide by the degrees of freedom.
• The formula is: \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \), where \( x_i \) are the data points, \( \bar{x} \) is the sample mean, and \( n \) is the sample size.

Utilizing sample variance in calculations allows for estimation of population variance, important for constructing confidence intervals using the F-distribution.