Gauge Repeatability and Reproducibility $($GR&R$)$ studies is a measure of the capability of a gauge or instrument to obtain the same measurement reading every time the measurement process is undertaken for the same characteristic or parameter. The following Table $1$ shows the measurement result of part diameter $($Tolerance $=+0\mathrm{.}003$mm $0\mathrm{.}002$mm$)$ by two appraisers $($observers$)$ through the use of an outside micrometer. Answer the following questions in order to perform a GR&R study. Refer Appendix $1$ for major equations and constant factors.

Table $1$: GR&R Study Worksheet

Appraiser: A B

Sample

$($n$)$ Trial $1$

$($r$1)$ Trial $2$

$($r$2)$ Mean

$($Xbar$.$A$)$ Range

$($RA$)$ Trial $1$

$($r$1)$ Trial $2$

$($r$2)$ Mean

$($Xbar$.$ B$)$ Range

$($RB$)$

$10\mathrm{.}4730\mathrm{.}472$ Xbar.n$1$A Rn$1$A $0\mathrm{.}4720\mathrm{.}469$ Xbar.n$1$B Rn$1$B

$20\mathrm{.}4680\mathrm{.}471$ Xbar.n$2$A Rn$2$A $0\mathrm{.}4710\mathrm{.}469$ Xbar.n$2$B Rn$2$B

$30\mathrm{.}4720\mathrm{.}470$ Xbar.n$3$A Rn$3$A $0\mathrm{.}4700\mathrm{.}471$ Xbar.n$3$B Rn$3$B

$40\mathrm{.}4710\mathrm{.}473$ Xbar.n$4$A Rn$4$A $0\mathrm{.}4700\mathrm{.}471$ Xbar.n$4$B Rn$4$B

$50\mathrm{.}4680\mathrm{.}470$ Xbar.n$5$A Rn$5$A $0\mathrm{.}4670\mathrm{.}469$ Xbar.n$5$B Rn$5$B

$\backslash$Sigma $--\backslash$Sigma Xbar.A $\backslash$Sigma RA $--\backslash$Sigma Xbar.B $\backslash$Sigma RB

$($a$)$ Calculate:

i$.$ The mean $($Xbar$.$n$1$A ~ Xbar.n$5$A ; Xbar.n$1$B ~ Xbar.n$5$B$),$ range $($RnA$1$ ~ RnA$5$ ; Rn$1$B ~ Rn$5$B$)$ and summation $(\backslash$Sigma Xbar.A ; $\backslash$Sigma Xbar.B ; $\backslash$Sigma RA ; $\backslash$Sigma RB$)$ for each appraiser.

ii$.$ The mean of each appraiser$$s means $($Xdoublebar$.$A and Xdoublebar.B$)$

iii. The different $($Xdoublebar$.$diff.$)$ between the maximum appraiser mean value and the minimum appraiser mean value.

iv$.$ The mean of range for each appraiser $($Rbar$.$A and Rbar.B$)$ and the grand range $($Rbar$).$

v$.$ The control limit $($LCL and UCL$)$ for R$-$chart.

vi$.$ The Equipment Variation $($EV$),$ Appraiser Variation $($AV$)$ and GR&R$.$

vii. The Equipment Variation $\%($EV$\%),$ Appraiser Variation $\%($AV$\%)$ and GR&R$\%.$

$($b$)$ Briefly explain the conclusion on the Measurement System Analysis $($MSA$)$ based on the result you have calculated.