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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