Chapter 12.3, Problem 11E

When Y_1i, for i = 1, 2,…, n, and Y_2i, for i = 1, 2,…, n, represent independent samples from two populations with means μ₁ and μ₂ and variances ${\sigma }_1^2$ and ${\sigma }_2^2$, respectively, we determined that ${\sigma }_{(\overline{Y_1}-{\bar{Y}}_2)}^2=(1\text{/}n)({\sigma }_1^2+{\sigma }_2^2)$. If the samples were paired and we computed the differences, D_i, for i = 1, 2,…, n, we determined that $\sigma (2\text{/}D)=(1\text{/}n)({\sigma }_1^2+{\sigma }_2^2-2\rho {\sigma }_1{\sigma }_2)$.

1. a When is ${\sigma }_{({\bar{Y}}_1-{\bar{Y}}_2)}^2$ greater than σ(2/D)?
2. b When is ${\sigma }_{({\bar{Y}}_1-{\bar{Y}}_2)}^2$ equal to σ(2/D)?
3. c When is ${\sigma }_{({\bar{Y}}_1-{\bar{Y}}_2)}^2$ less than σ(2/D)?
4. 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?