$\backslash$table$[[$Number$,$Product$1,$Product$2,$Product$3,$Product$4,$Product$5,$Product$6,$Product$7,$Mean,Range$],[1,105\mathrm{.}94,104\mathrm{.}60,102\mathrm{.}41,98\mathrm{.}91,102\mathrm{.}29,97\mathrm{.}55,106\mathrm{.}14,102\mathrm{.}55,8\mathrm{.}59],[2,88\mathrm{.}56,99\mathrm{.}06,87\mathrm{.}28,100\mathrm{.}77,99\mathrm{.}16,93\mathrm{.}39,87\mathrm{.}08,93\mathrm{.}61,13\mathrm{.}68],[3,100\mathrm{.}39,100\mathrm{.}21,90\mathrm{.}28,107\mathrm{.}75,105\mathrm{.}06,92\mathrm{.}99,98\mathrm{.}25,99\mathrm{.}28,17\mathrm{.}47],[4,95\mathrm{.}60,93\mathrm{.}15,96\mathrm{.}32,89\mathrm{.}47,92\mathrm{.}34,96\mathrm{.}96,113\mathrm{.}32,96\mathrm{.}74,23\mathrm{.}85],[5,112\mathrm{.}28,103\mathrm{.}29,97\mathrm{.}76,103\mathrm{.}45,106\mathrm{.}58,105\mathrm{.}30,72\mathrm{.}82,100\mathrm{.}21,39\mathrm{.}45],[6,113\mathrm{.}82,103\mathrm{.}56,107\mathrm{.}82,107\mathrm{.}98,111\mathrm{.}93,99\mathrm{.}50,102\mathrm{.}26,106\mathrm{.}70,14\mathrm{.}32],[7,105\mathrm{.}98,111\mathrm{.}99,103\mathrm{.}99,96\mathrm{.}32,87\mathrm{.}87,97\mathrm{.}34,92\mathrm{.}06,99\mathrm{.}36,24\mathrm{.}12],[8,107\mathrm{.}04,103\mathrm{.}59,105\mathrm{.}44,95\mathrm{.}06,95\mathrm{.}40,102\mathrm{.}07,105\mathrm{.}98,102\mathrm{.}08,11\mathrm{.}98],[9,92\mathrm{.}92,101\mathrm{.}02,91\mathrm{.}82,98\mathrm{.}86,107\mathrm{.}51,110\mathrm{.}39,100\mathrm{.}39,100\mathrm{.}41,18\mathrm{.}57],[10,91\mathrm{.}94,103\mathrm{.}23,109\mathrm{.}51,103\mathrm{.}01,93\mathrm{.}90,118\mathrm{.}77,91\mathrm{.}17,101\mathrm{.}65,27\mathrm{.}60],[11,106\mathrm{.}70,94\mathrm{.}75,118\mathrm{.}19,96\mathrm{.}48,98\mathrm{.}78,106\mathrm{.}22,100\mathrm{.}02,103\mathrm{.}02,23\mathrm{.}44],[12,95\mathrm{.}93,109\mathrm{.}80,109\mathrm{.}69,101\mathrm{.}12,101\mathrm{.}37,102\mathrm{.}22,105\mathrm{.}88,103\mathrm{.}72,13\mathrm{.}88],[13,91\mathrm{.}20,107\mathrm{.}05,98\mathrm{.}58,88\mathrm{.}18,111\mathrm{.}11,101\mathrm{.}50,97\mathrm{.}72,99\mathrm{.}33,22\mathrm{.}93],[14,104\mathrm{.}52,87\mathrm{.}55,111\mathrm{.}88,93\mathrm{.}42,111\mathrm{.}28,107\mathrm{.}99,104\mathrm{.}67,103\mathrm{.}04,24\mathrm{.}33],[15,102\mathrm{.}90,101\mathrm{.}68,96\mathrm{.}27,99\mathrm{.}09,100\mathrm{.}91,99\mathrm{.}18,92\mathrm{.}73,98\mathrm{.}97,10\mathrm{.}16],[$x$-$bar$-$bar $=100\mathrm{.}71117$ and R$-$bar $=19\mathrm{.}62451,,,,,,,,,]]$

A new week just started and you were given a new task to manage the quality function that monitors the thickness of one of the company's major products.

As you are about to call Sam to collect data, you get an email from the HR department saying Sam quit. He said he was overworked. However, HR has found you another engineer, Edith Clarke, who$$s been working on operations projects all over the world. She$$ll be a perfect fit for your task.

$2/4$

Edith finds some historical data which was collected when the process was in control. In that dataset $($Dataset $4),$ there were $15$ samples and in each sample $7$ products were measured in the unit of millimeter $($mm$).$ In this case n $=\_\_\_\_\_\_\_\_\_\_$ and N $=\_\_\_\_\_\_\_\_\_\_.$

Go to Tab $"$P$4"$ and click "Show me the data". The collected data is shown in this table. The first column is the sample number while the next few columns show the product thickness of the measured products in each sample. Edith has also calculated sample means and sample ranges which are shown in the last two columns $($Mean and Range$),$ as well as x$-$bar$-$bar and R$-$bar values, which are shown below the table.

Edith knows the next step is to determine the "constants" in the table and comes up with: A$2=0\mathrm{.}22,$ D$3=0\mathrm{.}35,$ and D$4=1\mathrm{.}65.$ You need to verify whether these values are correct. Are all three constants chosen correctly? $($Yes or No$)\_\_\_\_\_\_\_\_\_\_\_$

If the values are correct, please proceed to the next step. If the values are not correct, please write down the correct values here: $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Use the x$-$bar$-$bar and R$-$bar values computed by Edith, and appropriate "constants" to calculate the center line and control limits $($rounding to $3$ decimal places$)$ for x$-$bar chart and R chart.

x$-$bar chart: CL $=\_\_\_\_\_\_\_\_\_\_\_\_,$ UCL $=\_\_\_\_\_\_\_\_\_\_\_\_\_,$ and LCL $=\_\_\_\_\_\_\_\_\_\_\_\_\_$; R chart: CL $=\_\_\_\_\_\_\_\_\_\_\_\_,$ UCL $=\_\_\_\_\_\_\_\_\_\_\_\_\_,$ and LCL $=\_\_\_\_\_\_\_\_\_\_\_\_$