Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be two independent random samples from the respective normal distributions \(N\left(\mu_{1}, \sigma_{1}^{2}\right)\) and \(N\left(\mu_{2}, \sigma_{2}^{2}\right)\), where the four parameters are unknown. To construct a confidence interval for the ratio, \(\sigma_{1}^{2} / \sigma_{2}^{2}\), of the variances, form the quotient of the two independent chi-square variables, each divided by its degrees of freedom, namely $$ F=\frac{\frac{(m-1) S_{2}^{2}}{\sigma_{2}^{2}} /(m-1)}{\frac{(n-1) S_{1}^{2}}{\sigma_{1}^{2}} /(n-1)}=\frac{S_{2}^{2} / \sigma_{2}^{2}}{S_{1}^{2} / \sigma_{1}^{2}} $$

Short answer

The solution to the exercise involves obtaining the F-statistic for the two sample variances. Once obtained, the F-statistic can be used to derive the confidence interval for the ratio of the variances by kullaning percentile values for the F-distribution.

Step-by-step solution

Identify the Samples and the Variables

The given random samples are \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\), and these are from normal distributions \(N\left(\mu_{1}, \sigma_{1}^{2}\right)\) and \(N\left(\mu_{2}, \sigma_{2}^{2}\right)\) respectively. The goal is to construct a confidence interval for the ratio of the variances, \(\sigma_{1}^{2} / \sigma_{2}^{2}\).

Compute the Given Expression

The given expression to compute is: \(F=\frac{\frac{(m-1) S_{2}^{2}}{\sigma_{2}^{2}} /(m-1)}{\frac{(n-1) S_{1}^{2}}{\sigma_{1}^{2}} /(n-1)}\). Simplifying this we get: \(F=\frac{S_{2}^{2}}{S_{1}^{2}}\). This simplification is crucial for getting a hold on how F is related to the variances of the sample variances of the two populations.

Relation to F-distribution

The value computed in the prior step is an F-statistic, and given the samples are independent, it follows an F-distribution with \(m-1\) and \(n-1\) degrees of freedom. Constructing a confidence interval would involve using the percentile values for the F-distribution.

Construct the Confidence Interval

Using percentiles and confidence level \(C\), the confidence interval for \(\sigma_{1}^{2} / \sigma_{2}^{2}\) can be constructed. It will be \([FL, FU]\) where \(FL = (n-1)F_{(1-C)/2,m-1,n-1}\) and \(FU = (n-1)F_{1-(1-C)/2,m-1,n-1}\).

Key concepts

F-distribution

The F-distribution is pivotal when comparing two samples and their variances to understand their relationship. Specifically, the F-distribution emerges from the ratio of two independent chi-squared distributions, each divided by their respective degrees of freedom. This ratio creates what is known as the F-statistic.

In the context of our exercise, the F-statistic is used to construct a confidence interval for the variance ratio. Understanding the shape and properties of the F-distribution is essential because it helps us determine the critical values when estimating a confidence interval. The F-distribution is skewed to the right and its shape changes depending on the degrees of freedom of the two distributions being compared.

In constructing a confidence interval for the variance ratio, the upper and lower bounds are determined by the critical values of the F-distribution, which correlate to the desired level of confidence. Critical values are selected from F-distribution tables or can be computed using statistical software.

Independent Random Samples

Independent random samples are a cornerstone of inferential statistics, particularly when comparing groups or calculating confidence intervals. When we draw independent samples from two populations, as indicated in our exercise with samples \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\), we are gathering data such that the selection of an individual in one sample does not influence the selection in the other sample.

Independence is crucial for the use of an F-distribution because it underpins the assumption that the variances of the two groups can actually be compared in a meaningful way. Violation of this assumption could lead to unreliable results and incorrect confidence intervals.

In practice, ensuring independence typically involves random selection procedures and proper experimental design. This principle not only allows for the appropriate analysis of variance but also for it to be generalizable to the broader population.

Normal Distribution

The normal distribution is a foundational probability distribution in statistics, often referred to as the bell curve because of its characteristic shape. Its significance lies in the Central Limit Theorem, which implies that regardless of the original population distribution, the means of sufficiently large independent samples will be approximately normally distributed.

In the exercise we're discussing, the samples come from normal distributions, which is an important detail. This is because many statistical procedures, including those for estimating confidence intervals for variance ratios, rely on the assumption that the data follows a normal distribution. The normality assumption allows us to conclude that the sample variances follow a chi-squared distribution, and their ratio, an F-distribution, under the given conditions.

Normal distributions are defined by two parameters: the mean and the variance. While the means \(\mu_1\) and \(\mu_2\) are unknown in our samples, we can estimate them, along with the variance, from the data. The estimation of these parameters is integral to conducting any further analysis and making inferences about the population variance ratio.