Sample Solution

1. State whether the following statements are true or false and also give the reason in support of your answer:

Introduction

Justification

1. For $V_{\text{ran}}-V_{\text{prop}}$$V_{\text{ran}}-V_{\text{prop}}$V_(“ran”)-V_(“prop”)V_{\text{ran}} – V_{\text{prop}}$V_{\text{ran}}-V_{\text{prop}}$:
   $$V_{\text{ran}}-V_{\text{prop}}=\frac{\sum N_i{(\overline{x_i}-\overline{x})}^2}{nN}\ge 0$$$$V_{\text{ran}}-V_{\text{prop}}=\frac{\sum N_i{\left(\overline{x_i}-\overline{x}\right)}^2}{nN}\ge 0$$V_(“ran”)-V_(“prop”)=(sumN_(i)( bar(x_(i))-( bar(x)))^(2))/(nN) >= 0V_{\text{ran}} – V_{\text{prop}} = \frac{\sum N_i\left(\overline{x_i}-\bar{x}\right)^2}{n N} \geq 0$$V_{\text{ran}}-V_{\text{prop}}=\frac{\sum N_i{(\overline{x_i}-\overline{x})}^2}{nN}\ge 0$$
   This implies that $V_{\text{ran}}\ge V_{\text{prop}}$$V_{\text{ran}}\ge V_{\text{prop}}$V_(“ran”) >= V_(“prop”)V_{\text{ran}} \geq V_{\text{prop}}$V_{\text{ran}}\ge V_{\text{prop}}$.
2. For $V_{\text{prop}}-V_{\text{opt}}$$V_{\text{prop}}-V_{\text{opt}}$V_(“prop”)-V_(“opt”)V_{\text{prop}} – V_{\text{opt}}$V_{\text{prop}}-V_{\text{opt}}$:
   $$V_{\text{prop}}-V_{\text{opt}}=\frac{1}{nN}\left[\sum _{i=1}^k{N}_i{(S_i-\overline{S})}^2\right]\ge 0$$$$V_{\text{prop}}-V_{\text{opt}}=\frac{1}{nN}\left[\sum _{i=1}^k N_i{\left(S_i-\overline{S}\right)}^2\right]\ge 0$$V_(“prop”)-V_(“opt”)=(1)/(nN)[sum_(i=1)^(k)N_(i)(S_(i)-( bar(S)))^(2)] >= 0V_{\text{prop}} – V_{\text{opt}} = \frac{1}{n N}\left[\sum_{i=1}^k N_i\left(S_i-\bar{S}\right)^2\right] \geq 0$$V_{\text{prop}}-V_{\text{opt}}=\frac{1}{nN}\left[\sum _{i=1}^k{N}_i{(S_i-\overline{S})}^2\right]\ge 0$$
   This implies that $V_{\text{prop}}\ge V_{\text{opt}}$$V_{\text{prop}}\ge V_{\text{opt}}$V_(“prop”) >= V_(“opt”)V_{\text{prop}} \geq V_{\text{opt}}$V_{\text{prop}}\ge V_{\text{opt}}$.

Conclusion