$1.(15$ points$)$ Solve problem $4\mathrm{.}1$ using Table $4\mathrm{.}2.$

$($a$)$ Show calculations of the center line, the UCL and the LCL by hand.

$($b$)$ Find the center line, the UCL and LCL by Excel and plot the R chart and the $$

$($c$)$ Did the process appear to be in control based on the charts created?

$2.(15$ points$)$ Solve problem $4\mathrm{.}3$ using Table $4\mathrm{.}2.$

$($a$)$ Show calculations of the center line, the UCL and the LCL by hand.

$($b$)$ Find the center line, the UCL and the LCL by Excel and plot the R chart and the $$

$($c$)$ Did the process appear to be in control based on the charts created?

$3.(20$ points$)$ A high voltage power supply should have a normal output voltage of $350$V$.$ A

sample of four units is selected each day and tested for process$-$control purposes. The data shown below give the difference between the observed reading on each unit and the normal voltage times the; that is$,$ xi $=($observed voltage on unit i $350)$ x $10.$

Note that an Excel datasheet can be found on Moodle.

$($a$)$ Show calculations of the center line, the UCL and the LCL by hand using Table $4\mathrm{.}2.$

$($b$)$ Find the center line, the UCL, the LCL by Excel and plot the R chart and the $$

$($c$)$ Did the process appear to be in control based on the charts created?

$($d$)$ If specifications are at $350$V $+/-5$V$,$ find the percentage of conforming power supplies

produced by this process. Assume the voltage is normally distributed.

$4.(40$ points$)$ The data shown in the table below are $$ and R values for $24$ samples of size n$=5$

taken from a process producing bearings. The measurements are made on the inside diameter of the bearing, with only the last three decimals recorded $($i$.$e$.,34\mathrm{.}5$ should be $0\mathrm{.}50345).$

$($a$)$ Do not use A$2,$ A$3,$ D$3$ and D$4$ values from Table $4\mathrm{.}2.$ Setup $$ chart and R charts on this

process. Note that the standard deviation for this production process can be estimated as $$

$($b$)$ Show the control charts using Excel. Does the process seem to be in statistical control?

$($c$)$ If necessary, revise the trial control limits by removing the points that fall out the first control

limits and recalculate the new control limits$.$ Have the changes made any difference?

$($d$)$ If specifications on this diameter are $0\mathrm{.}5030+0\mathrm{.}0020,$ find the percentage of nonconforming

bearings produced by this process. Assume that diameter is normally distributed.