91Ó°ÊÓ

Suppose that independent samples of sizes \(n_{1}\) and \(n_{2}\) are taken from two normally distributed populations with variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2},\) respectively. If \(S_{1}^{2}\) and \(S_{2}^{2}\) denote the respective sample variances, Theorem 7.3 implies that \(\left(n_{1}-1\right) S_{1}^{2} / \sigma_{1}^{2}\) and \(\left(n_{2}-1\right) S_{2}^{2} / \sigma_{2}^{2}\) have \(\chi^{2}\) distributions with \(n_{1}-1\) and \(n_{2}-1\) df, respectively. Further, these \(\chi^{2}\) -distributed random variables are independent because the samples were independently taken. a. Use these quantities to construct a random variable that has an \(F\) distribution with \(n_{1}-1\) numerator degrees of freedom and \(n_{2}-1\) denominator degrees of freedom. b. Use the \(F\) -distributed quantity from part (a) as a pivotal quantity, and derive a formula for a \(100(1-\alpha) \%\) confidence interval for \(\sigma_{2}^{2} / \sigma_{1}^{2}\)

Step by step solution

01

Define the Chi-square distributed variables

According to the problem, \( \left(n_1-1\right) S_1^2 / \sigma_1^2 \) and \( \left(n_2-1\right) S_2^2 / \sigma_2^2 \) are independently Chi-square distributed with \( n_1-1 \) and \( n_2-1 \) degrees of freedom, respectively. These can be defined as \( X_1 = \left(n_1-1\right) S_1^2 / \sigma_1^2 \) and \( X_2 = \left(n_2-1\right) S_2^2 / \sigma_2^2 \).

02

Construct the F-distributed random variable

The \( F \)-distribution is formed by the ratio of two independent Chi-square variables divided by their respective degrees of freedom. Thus, the ratio \( F = \frac{X_1 / (n_1-1)}{X_2 / (n_2-1)} = \frac{\left(S_1^2 / \sigma_1^2\right)}{\left(S_2^2 / \sigma_2^2\right)} \) follows an \( F \)-distribution with \( n_1-1 \) numerator and \( n_2-1 \) denominator degrees of freedom.

03

Formulate the pivotal quantity

The \( F \)-statistic as a pivotal quantity is \( \frac{(S_1^2/\sigma_1^2)}{(S_2^2/\sigma_2^2)} \), which allows us to estimate \( \sigma_2^2/\sigma_1^2 \) by rearranging the formula to \( \frac{(S_1^2/S_2^2)}{F} \).

04

Derive the confidence interval

For a \( 100(1-\alpha)\% \) confidence interval, we use the \( F \)-distribution critical values \( F_{\alpha/2} \) and \( F_{1-\alpha/2} \). Solving \( \frac{(S_1^2/S_2^2)}{F_{1-\alpha/2}} < \frac{\sigma_2^2}{\sigma_1^2} < \frac{(S_1^2/S_2^2)}{F_{\alpha/2}} \) gives us the interval \[ \frac{S_1^2}{S_2^2}F_{\alpha/2, n_1-1, n_2-1}^{-1} < \frac{\sigma_2^2}{\sigma_1^2} < \frac{S_1^2}{S_2^2}F_{1-\alpha/2, n_1-1, n_2-1} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chi-square Distribution

The chi-square distribution is a fundamental concept in statistics, which you'll encounter frequently in various statistical analyses. It's primarily used in hypothesis testing and constructing confidence intervals in the context of variance and standard deviation. Think of the chi-square distribution like a special case of the gamma distribution, but dedicated to variances. Here's what makes the chi-square distribution interesting:

• It's only defined for positive values because variances can't be negative.
• As the degrees of freedom increase, the chi-square distribution becomes more symmetrical and spread out.
• The distribution can be thought of as modeling the sum of the squares of standard normal variables (hence its name).

In practice, the chi-square distribution allows us to understand how much the observed variance in a sample could differ from the actual population variance. In the context of comparing two independent samples, as discussed in the exercise, the chi-square plays a key role in forming the foundation for the F-distribution. When you divide one chi-square distributed random variable by another, after normalizing them by their respective degrees of freedom, you get a random variable that follows the F-distribution.

Confidence Interval

Let's talk about confidence intervals. They're an essential tool in statistics that provide a range of values which are likely to contain a population parameter. You can think of confidence intervals as a way to express uncertainty in your estimation. Key points to remember about confidence intervals:

• The confidence level, often expressed as a percentage, signifies how sure you are that the interval includes the true parameter. For example, a 95% confidence interval means you can be 95% certain the range contains the true ratio of variances.
• The wider the interval, the more uncertainty you have about the parameter's exact value. Conversely, a smaller interval indicates more precise estimation but might increase the risk of missing the true value.
• Construction of confidence intervals often involves critical values from a statistical distribution, like the chi-square, t, or F-distribution.

In the exercise, a confidence interval is constructed for the ratio of population variances. This interval leverages properties of the F-distribution, where the interval bounds are determined by critical values that correspond to the desired confidence level. By putting your sample data into this interval, you can state how confident you are about the ratio of the population variances.

Degrees of Freedom

Degrees of freedom is a concept that appears in a broad range of statistical procedures. It refers to the number of independent values that can vary in an analysis without breaking any constraints involved. It's like having a level of flexibility in your data. Here are some quick facts about degrees of freedom:

• In any calculation of a statistic, the degrees of freedom are determined by the sample size minus the number of parameters estimated. For example, if you calculate a sample variance, the degrees of freedom is the sample size minus one.
• Degrees of freedom affect the shape of various statistical distributions, such as the chi-square and t-distributions, influencing the breadth and central tendency of these distributions.
• They're crucial in calculating statistical significance, leading to valid inferences from sample data about the population.

Within the context of this exercise, each chi-square distributed variable comes with degrees of freedom, which aids in structuring the F-distribution when these variables are combined. The numerator and denominator degrees of freedom ultimately dictate the behavior of the F-distribution, thereby influencing how you evaluate the relationship between variances from independent samples.