d Based on the discussion in the text and your answers to

Chapter 12, Problem 11E

(choose chapter or problem)

When \(Y_{1 i}\), for \(i=1,2, \ldots, n\), and \(Y_{2 i}\), for \(i=1,2, \ldots, n\), represent independent samples from two populations with means \(\mu_{1}\) and \(\mu_{2}\) and variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\), respectively, we determined that \(\sigma_{\left(\bar{Y}_{1}-\bar{Y}_{2}\right)}^{2}=(1 / n)\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)\). If the samples were paired and we computed the differences, \(D_{i}\), for \(i=1,2, \ldots, n\), we determined that \(\sigma(2 / D)=(1 / n)\left(\sigma_{1}^{2}+\sigma_{2}^{2}-2 \rho \sigma_{1} \sigma_{2}\right)\)


a. When is \(\sigma_{\left(\bar{Y}_{1}-\bar{Y}_{2}\right)}^{2}\) greater than \(\sigma(2 / D)\)?


b.  When is \(\sigma_{\left(\bar{Y}_{1}-\bar{Y}_{2}\right)}^{2}\) equal to \(\sigma(2 / D)\)?


c. When is \(\sigma_{\left(\bar{Y}_{1}-\bar{Y}_{2}\right)}^{2}\) less than \(\sigma(2 / D)\)?


d. Based on the discussion in the text and your answers to parts (a)-(c), when would it be better to implement the matched-pairs experiment and when would it be better to implement the independent samples experiment?

Equation Transcription:

Text Transcription:

Y_1i

i = 1, 2, ... , n

Y_2i

mu_1

mu_2

sigma_1^2

sigma_2^2

sigma_(Y bar_1 - Y bar_2)^2 = (1/n)(sigma_1^2 + sigma_2^2)

D_i

sigma(2/D) = (1/n) (sigma_1^2 + sigma_2^2 - 2p sigma_1 sigma_2)

sigma_(Y bar_1 - Y bar_2)

sigma(2/D)

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