Suppose that independent samples of sizes n1 and n2 are

Chapter 8, Problem 125SE

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Suppose that independent samples of sizes \(n_{1} \text { and } n_{2}\) are taken from two normally distributed populations with variances \(\sigma_{1}^{2} \text { and } \sigma_{2}^{2}\), respectively. If \(S_{1}^{2} \text { and } S_{2}^{2}\) denote the respective sample variances, Theorem  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  distribution with \(n_{1}-1\) numerator degrees of freedom and \(n_{2}-1\) denominator degrees of freedom.
b Use the  -distributed quantity from part (a) as a pivotal quantity, and derive a formula for a \(100(1-\alpha) \%\) confidence interval for \(\sigma_{2}^{2} / \sigma_{1}^{2}\)

Equation Transcription:

 

 

   

 

 

     

Text Transcription:

n1 and n2

\sigma_1^2 and \sigma_2^2  

s1^2 and s2^2  

(n1-1)s_1^2/\sigma_1^2  

(n2-1)s_2^2/\sigma_2^2  

\chi^2

n1-1  

n2-1

\chi^2

n1-1  

n2-1

100(1- \alpha)%      

\sigma_2^2/\sigma_1^2