91Ó°ÊÓ

Worker A

$\begin{array}{rcl}\mathbf{Meanforsubtask}\mathbf{1}& \mathbf{=}& \frac{\mathbf{30}\mathbf{+}\mathbf{28}\mathbf{+}\mathbf{31}\mathbf{+}\mathbf{38}\mathbf{+}\mathbf{25}\mathbf{+}\mathbf{29}\mathbf{+}\mathbf{30}}{\mathbf{7}}\\ & \mathbf{=}& \frac{\mathbf{211}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{30}\mathbf{.}\mathbf{14}\\ & & \\ \mathbf{Meanforsubtask}\mathbf{2}& \mathbf{=}& \frac{\mathbf{2}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{3}\mathbf{+}\mathbf{3}\mathbf{+}\mathbf{2}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{3}}{\mathbf{7}}\\ & \mathbf{=}& \frac{\mathbf{21}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{3}\\ & & \\ \mathbf{Overalltimetaken}& \mathbf{=}& \mathbf{Meanforsubtask}\mathbf{1}\mathbf{+}\mathbf{Meanforsubtask}\mathbf{2}\\ & \mathbf{=}& \mathbf{30}\mathbf{.}\mathbf{14}\mathbf{+}\mathbf{3}\\ & \mathbf{=}& \mathbf{33}\mathbf{.}\mathbf{14}\\ & & \end{array}$

Therefore, the overall time taken by Worker A to complete the work under study is 33.14 mins.

Worker B

$\begin{array}{rcl}\mathbf{Meanforsubtask}\mathbf{1}& \mathbf{=}& \frac{\mathbf{31}\mathbf{+}\mathbf{30}\mathbf{+}\mathbf{32}\mathbf{+}\mathbf{30}\mathbf{+}\mathbf{29}\mathbf{+}\mathbf{30}\mathbf{+}\mathbf{31}}{\mathbf{7}}\\ & \mathbf{=}& \frac{\mathbf{213}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{30}\mathbf{.}\mathbf{43}\\ & & \\ \mathbf{Meanforsubtask}\mathbf{2}& \mathbf{=}& \frac{\mathbf{7}\mathbf{+}\mathbf{2}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{5}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{4}}{\mathbf{7}}\\ & \mathbf{=}& \frac{\mathbf{29}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{4}\mathbf{.}\mathbf{14}\\ & & \\ \mathbf{Overalltimetaken}& \mathbf{=}& \mathbf{Meanforsubtask}\mathbf{1}\mathbf{+}\mathbf{Meanforsubtask}\mathbf{2}\\ & \mathbf{=}& \mathbf{30}\mathbf{.}\mathbf{43}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{14}\\ & \mathbf{=}& \mathbf{34}\mathbf{.}\mathbf{57}\\ & & \end{array}$

Therefore, the overall time taken by Worker A to complete the work under study is 34.57 mins.

localid="1668432137286" $\begin{array}{rcl}\mathbf{VarianceforworkerA}& \mathbf{=}& \frac{\mathbf{∑}\mathbf{(}\mathbf{x}\mathbf{-}\stackrel{\mathbf{¯}}{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}}\\ & \mathbf{=}& \frac{\mathbf{94}\mathbf{.}\mathbf{8572}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{13}\mathbf{.}\mathbf{55}\\ & & \\ \mathbf{Standarddeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{13}\mathbf{.}\mathbf{55}}\\ & \mathbf{=}& \mathbf{3}\mathbf{.}\mathbf{68}\\ & & \\ \mathbf{VarianceforworkerB}& \mathbf{=}& \frac{\mathbf{∑}\mathbf{(}\mathbf{x}\mathbf{-}\stackrel{\mathbf{¯}}{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}}\\ & \mathbf{=}& \frac{\mathbf{5}\mathbf{.}\mathbf{7143}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{0}\mathbf{.}\mathbf{816}\\ & & \\ \mathbf{Standarddeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{0}\mathbf{.}\mathbf{816}}\\ & \mathbf{=}& \mathbf{0}\mathbf{.}\mathbf{9}\\ & & \end{array}$

Worker A's standard deviation for completing subtask 1 is 3.68 minutes.

The standard deviation for completing subtask 1 for Worker B is 0.9 minutes.

localid="1668431443790" $\begin{array}{rcl}\mathbf{VarianceforworkerA}& \mathbf{=}& \frac{\mathbf{∑}\mathbf{(}\mathbf{x}\mathbf{-}\stackrel{\mathbf{¯}}{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}}\\ & \mathbf{=}& \frac{\mathbf{4}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{0}\mathbf{.}\mathbf{57}\\ & & \\ \mathbf{Standarddeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{0}\mathbf{.}\mathbf{57}}\\ & \mathbf{=}& \mathbf{0}\mathbf{.}\mathbf{75}\\ & & \\ \mathbf{VarianceforworkerB}& \mathbf{=}& \frac{\mathbf{∑}\mathbf{(}\mathbf{x}\mathbf{-}\stackrel{\mathbf{¯}}{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}}\\ & \mathbf{=}& \frac{\mathbf{26}\mathbf{.}\mathbf{8527}}{\mathbf{7}}\\ & \mathbf{=}& \mathbf{3}\mathbf{.}\mathbf{8361}\\ & & \\ \mathbf{Standarddeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{3}\mathbf{.}\mathbf{8361}}\\ & \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{96}\\ & & \end{array}$

Worker A's standard deviation for completing subtask 2 is 0.75 minutes.

The standard deviation for completing subtask 2 for Worker B is 1.96 minutes.

The overall time taken by B is more than A.

The standard deviation for subtask 1 is more significant for Worker A than B. Greater standard deviation implies that A's time to complete subtask 1 varies more, and hence there is no certainty as to when the work will get done. Therefore, type B workers should be given subtask 1.

Type A workers should be given subtask 2 because their standard deviation is smaller than worker Bs, implying that there will be a certainty that the work will get done quickly and the time taken will not vary much.