QUESTION $6$

Consider the simulation results for simple processing system with one machine. The results are from $520$ minute independent, statistically identical runs.

$\backslash$table$[[,$Replication,,,$],[$Performance Measure,$1,2,3,4,5,$Mean,$\backslash$table$[[$Standard$],[$Deviation$]],\backslash$table$[[95\%$ Half$],[$Width$]]],[$Total production,$5,3,6,2,3,3\mathrm{.}80,1\mathrm{.}64,2\mathrm{.}04],[$Average waiting time in queue,$2\mathrm{.}53,1\mathrm{.}19,1\mathrm{.}03,1\mathrm{.}62,0\mathrm{.}00,1\mathrm{.}27,0\mathrm{.}92,1\mathrm{.}14],[$Average total time in system,$6\mathrm{.}44,5\mathrm{.}10,4\mathrm{.}16,6\mathrm{.}71,4\mathrm{.}26,5\mathrm{.}33,1\mathrm{.}19,1\mathrm{.}48],[$# of parts in queue,$0\mathrm{.}79,0\mathrm{.}18,0\mathrm{.}36,0\mathrm{.}16,0\mathrm{.}05,0\mathrm{.}31,0\mathrm{.}29,0\mathrm{.}36],[$Drill press utilization,$0\mathrm{.}92,0\mathrm{.}59,0\mathrm{.}90,0\mathrm{.}51,0\mathrm{.}70,0\mathrm{.}72,0\mathrm{.}18,0\mathrm{.}23]]$

$($a$)$ Explain what is meant by these replications being statistically identical and independent.

$($b$)$ Why multiple replications are necessary?

$($c$)$ Explain why variation in results exists.

Suppose the machine has been upgraded and can now operate twice as fast. The model was run again and the results from $520-$minute statistically identical and independent runs are as follows.

$\backslash$table$[[,$Replication,,,$],[$Performance Measure,$1,2,3,4,5,$Mean,$\backslash$table$[[$Standard$],[$Deviation$]],\backslash$table$[[95\%$ Half$],[$Width$]]],[$Total production,$6,4,6,4,5,5\mathrm{.}00,1\mathrm{.}00,1\mathrm{.}24],[$Average waiting time in queue,$7\mathrm{.}38,2\mathrm{.}10,3\mathrm{.}52,2\mathrm{.}81,2\mathrm{.}93,3\mathrm{.}75,2\mathrm{.}09,2\mathrm{.}60],[$Average total time in system,$10\mathrm{.}19,5\mathrm{.}61,5\mathrm{.}90,6\mathrm{.}93,5\mathrm{.}57,6\mathrm{.}84,1\mathrm{.}95,2\mathrm{.}42],[$# of parts in queue,$2\mathrm{.}88,0\mathrm{.}52,1\mathrm{.}71,0\mathrm{.}77,2\mathrm{.}25,1\mathrm{.}63,0\mathrm{.}99,1\mathrm{.}23],[$Drill press utilization,$1\mathrm{.}00,0\mathrm{.}92,1\mathrm{.}00,0\mathrm{.}95,1\mathrm{.}00,0\mathrm{.}97,0\mathrm{.}04,0\mathrm{.}05]]$

$($d$)$ Comparing the confidence intervals in the second experiment with those in the first, can you determine whether the upgrade to the machine has resulted in a significant improvement to the firm's operations?