$1.$ Suppose you have width measurements from a part produced using an injection molding process that is difficult to adjust settings to correct mean deviations. Assume specification for critical width dimension is $140+3$ mm$.$ For this process, individual parts are produced in four different cavities $($thus$,$ each cycle, you obtain four parts$).$ For each part, a critical width dimension is measured. They have $50$ different part sample measurements taken from $18$ batches. Currently, the production department is using an Individual$/$Moving Range chart to show that their process is in a state of statistical control. The design engineering group, however, feels strongly that the production operation team is not sufficiently controlling the process. They have asked you to examine the historical data to assess the current state. Given the available data $($see data below for Injection Mold$-$Width$),$

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a$)$ First construct an I$/$MR Chart and assess the current state

b$)$ Next, create additional SPC charts$/$settings or other analysis as deemed appropriate. Note: May try different charts under different assumptions along with any related analysis necessary to make an assessment if the process is stable and$/$or capable $($Cpk $>1\mathrm{.}33).$

c$)$ Given your analysis, recommend a chart for future process monitoring?

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Control Chart $($provide I$-$MR control chart using Minitab and comment on stability$)$

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SampleNum

BatchID

Width

$1$

$1$

$141\mathrm{.}136$

$2$

$1$

$139\mathrm{.}232$

$3$

$2$

$141\mathrm{.}474$

$4$

$2$

$141\mathrm{.}445$

$5$

$2$

$140\mathrm{.}493$

$6$

$3$

$140\mathrm{.}161$

$7$

$3$

$141\mathrm{.}051$

$8$

$4$

$137\mathrm{.}155$

$9$

$4$

$137\mathrm{.}289$

$10$

$4$

$140\mathrm{.}873$

$11$

$4$

$140\mathrm{.}616$

$12$

$5$

$139\mathrm{.}666$

$13$

$5$

$138\mathrm{.}904$

$14$

$6$

$138\mathrm{.}945$

$15$

$6$

$137\mathrm{.}313$

$16$

$6$

$138\mathrm{.}438$

$17$

$7$

$139\mathrm{.}635$

$18$

$7$

$140\mathrm{.}596$

$19$

$8$

$143\mathrm{.}073$

$20$

$8$

$142\mathrm{.}845$

$21$

$9$

$139\mathrm{.}081$

$22$

$9$

$138\mathrm{.}256$

$23$

$9$

$137\mathrm{.}608$

$24$

$9$

$140\mathrm{.}241$

$25$

$10$

$141\mathrm{.}29$

$26$

$10$

$142\mathrm{.}88$

$27$

$10$

$142\mathrm{.}801$

$28$

$11$

$143\mathrm{.}6$

$29$

$11$

$141\mathrm{.}544$

$30$

$12$

$137\mathrm{.}01$

$31$

$12$

$138\mathrm{.}535$

$32$

$12$

$136\mathrm{.}234$

$33$

$12$

$137\mathrm{.}98$

$34$

$13$

$141\mathrm{.}9$

$35$

$13$

$143\mathrm{.}866$

$36$

$13$

$141\mathrm{.}888$

$37$

$14$

$140\mathrm{.}134$

$38$

$14$

$138\mathrm{.}872$

$39$

$14$

$138\mathrm{.}954$

$40$

$15$

$143\mathrm{.}895$

$41$

$15$

$140\mathrm{.}701$

$42$

$15$

$142\mathrm{.}24$

$43$

$16$

$138$

$44$

$16$

$137\mathrm{.}157$

$45$

$16$

$138\mathrm{.}065$

$46$

$17$

$139\mathrm{.}104$

$47$

$17$

$138\mathrm{.}806$

$48$

$18$

$140\mathrm{.}107$

$49$

$18$

$137\mathrm{.}817$

$50$

$18$

$137\mathrm{.}511.$ Suppose you have width measurements from a part produced using an injection molding process that is difficult to adjust settings to correct mean deviations. Assume specification for critical width dimension is $\backslash (140\backslash$pm $3\backslash$mathrm$\{$~mm$\}\backslash ).$ For this process, individual parts are produced in four different cavities $($thus$,$ each cycle, you obtain four parts$).$ For each part, a critical width dimension is measured. They have $50$ different part sample measurements taken from $18$ batches. Currently, the production department is using an Individual$/$Moving Range chart to show that their process is in a state of statistical control. The design engineering group, however, feels strongly that the production operation team is not sufficiently controlling the process. They have asked you to examine the historical data to assess the current state. Given the available data $($see data below for Injection Mold$-$Width$),$

a$)$ First construct an I$/$MR Chart and assess the current state

b$)$ Next, create additional SPC charts$/$settings or other analysis as deemed appropriate. Note: May try different charts under different assumptions along with any related analysis necessary to make an assessment if the process is stable and$/$or capable $($Cpk $\backslash (\backslash$geq $1\mathrm{.}33\backslash )).$

c$)$ Given your analysis, recommend a chart for future process monitoring?

Control Chart $($provide I$-$MR control chart using Minitab and comment on stability$)\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|\}$

$\backslash$hline SampleNum & BatchiD & Width $\backslash \backslash$

$\backslash$hline $1$ & $1$ & $141\mathrm{.}136\backslash \backslash$

$\backslash$hline $2$ & $1$ & $139\mathrm{.}232\backslash \backslash$

$\backslash$hline $3$ & $2$ & $141\mathrm{.}474\backslash \backslash$

$\backslash$hline $4$ & $2$ & $141\mathrm{.}445\backslash \backslash$

$\backslash$hline $5$ & $2$ & $140\mathrm{.}493\backslash \backslash$

$\backslash$hline $6$ & $3$ & $140\mathrm{.}161\backslash \backslash$

$\backslash$hline $7$ & $3$ & $141\mathrm{.}051\backslash \backslash$

$\backslash$hline $8$ & $4$ & $137\mathrm{.}155\backslash \backslash$

$\backslash$hline $9$ & $4$ & $137\mathrm{.}289\backslash \backslash$

$\backslash$hline $10$ & $4$ & $140\mathrm{.}873\backslash \backslash$

$\backslash$hline $11$ & $4$ & $140\mathrm{.}616\backslash \backslash$

$\backslash$hline $12$ & $5$ & $139\mathrm{.}666\backslash \backslash$

$\backslash$hline $13$ & $5$ & $138\mathrm{.}904\backslash \backslash$

$\backslash$hline $14$ & $6$ & $138\mathrm{.}945\backslash \backslash$

$\backslash$hline $15$ & $6$ & $137\mathrm{.}313\backslash \backslash$

$\backslash$hline $16$ & $6$ & $138\mathrm{.}438\backslash \backslash$

$\backslash$hline $17$ & $7$ & $139\mathrm{.}635\backslash \backslash$

$\backslash$hline $18$ & $7$ & $140\mathrm{.}596\backslash \backslash$

$\backslash$hline $19$ & $8$ & $143\mathrm{.}073\backslash \backslash$

$\backslash$hline $20$ & $8$ & $142\mathrm{.}845\backslash \backslash$

$\backslash$hline $21$ & $9$ &