$($S$6\mathrm{.}3)$

$3.$ Sampling of four pieces of precision$-$cut wire $($to be used in computer assembly$)$ every hour for the past $24$ hours has produced the following results. Develop appropriate control charts and determine whether there is any cause of concern in the cutting process. Observe the patterns.

$($S$6\mathrm{.}6)$

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline HOUR & $\backslash (\backslash$bar$\{\backslash$chi$\}\backslash )$ & $\backslash (\backslash$mathbf$\{$R$\}\backslash )$ & HOUR & $\backslash (\backslash$bar$\{$X$\}\backslash )$ & $\backslash (\backslash$mathbf$\{$R$\}\backslash )\backslash \backslash$

$\backslash$hline $1$ & $\backslash (3\mathrm{.}25^\{\backslash$prime $\backslash$prime$\}\backslash )$ & $\backslash (\mathrm{.}71^\{\backslash$prime $\backslash$prime$\}\backslash )$ & $13$ & $\backslash (3\mathrm{.}11^\{\backslash$prime $\backslash$prime$\}\backslash )$ & $\backslash (\mathrm{.}85^\{\backslash$prime $\backslash$prime$\}\backslash )\backslash \backslash$

$\backslash$hline $2$ & $3\mathrm{.}10$ & $1\mathrm{.}18$ & $14$ & $2\mathrm{.}83$ & $1\mathrm{.}31\backslash \backslash$

$\backslash$hline $3$ & $3\mathrm{.}22$ & $1\mathrm{.}43$ & $15$ & $3\mathrm{.}12$ & $1\mathrm{.}06\backslash \backslash$

$\backslash$hline $4$ & $3\mathrm{.}39$ & $1\mathrm{.}26$ & $16$ & $2\mathrm{.}84$ & $\mathrm{.}50\backslash \backslash$

$\backslash$hline $5$ & $3\mathrm{.}07$ & $1\mathrm{.}17$ & $17$ & $2\mathrm{.}86$ & $1\mathrm{.}43\backslash \backslash$

$\backslash$hline $6$ & $2\mathrm{.}86$ & $\mathrm{.}32$ & $18$ & $2\mathrm{.}74$ & $1\mathrm{.}29\backslash \backslash$

$\backslash$hline $7$ & $3\mathrm{.}05$ & $\mathrm{.}53$ & $19$ & $3\mathrm{.}41$ & $1\mathrm{.}61\backslash \backslash$

$\backslash$hline $8$ & $2\mathrm{.}65$ & $1\mathrm{.}13$ & $20$ & $2\mathrm{.}89$ & $1\mathrm{.}09\backslash \backslash$

$\backslash$hline $9$ & $3\mathrm{.}02$ & $\mathrm{.}71$ & $21$ & $2\mathrm{.}65$ & $1\mathrm{.}08\backslash \backslash$

$\backslash$hline $10$ & $2\mathrm{.}85$ & $1\mathrm{.}33$ & $22$ & $3\mathrm{.}28$ & $\mathrm{.}46\backslash \backslash$

$\backslash$hline $11$ & $2\mathrm{.}83$ & $1\mathrm{.}17$ & $23$ & $2\mathrm{.}94$ & $1\mathrm{.}58\backslash \backslash$

$\backslash$hline $12$ & $2\mathrm{.}97$ & $\mathrm{.}40$ & $24$ & $2\mathrm{.}64$ & $\mathrm{.}97\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$