Statistical Process Control at Taco Bear $($Salsa Packets$)$

Step $1$: Establish Your Upper and Lower Control Limits

Everyone loves eating at Taco Bear especially because they have such great salsa. Their famous salsas come in small, single serve packets $($much like a ketchup packet$).$ These packets are filled in a state$-$of$-$the$-$art food packing facility with an long$-$standing quality management program based on Statistical Process Control. The following data comes from the Mega$-$Hot salsa package line and displays the weight of the salsa packet in grams. An important part of the quality control process is collecting baseline data for centerline and range of values against which process control measurements can be taken. This baseline data will be used to calculate control limits to determine if the Mega$-$Hot salsa packet line is ever out of control. As everyone knows, too little salsa in the packet means a dissatisfying customer taste experience, In contrast, too much salsa in the packet could result in salsa squirting in someone's eye when they tear it open.

In this facility, quality control associates collected baseline data for ten working days from July $20$ until July $31$ by weighing six randomly selected Mega$-$Hot packets each day. The resulting data is displayed in the table below:

Packet Weight $($grams$)$

$\backslash$table$[[$Day:$,$Sample $1$:$,$Sample $2$:$,$Sample $3$:$,$Sample $4$:$,$Sample $5$:$,$Sample $6$:$],[20$ July,$2\mathrm{.}798,2\mathrm{.}814,2\mathrm{.}79,2\mathrm{.}848,2\mathrm{.}739,2\mathrm{.}807],[21$ July,$2\mathrm{.}796,2\mathrm{.}799,2\mathrm{.}773,2\mathrm{.}829,2\mathrm{.}748,2\mathrm{.}834],[22$ July,$2\mathrm{.}802,2\mathrm{.}803,2\mathrm{.}844,2\mathrm{.}839,2\mathrm{.}865,2\mathrm{.}754],[23$ July,$2\mathrm{.}741,2\mathrm{.}774,2\mathrm{.}788,2\mathrm{.}776,2\mathrm{.}776,2\mathrm{.}75],[24$ July,$2\mathrm{.}83,2\mathrm{.}74,2\mathrm{.}804,2\mathrm{.}794,2\mathrm{.}805,2\mathrm{.}842],[27$ July,$2\mathrm{.}804,2\mathrm{.}776,2\mathrm{.}867,2\mathrm{.}742,2\mathrm{.}73,2\mathrm{.}833],[28$ July,$2\mathrm{.}776,2\mathrm{.}741,2\mathrm{.}781,2\mathrm{.}811,2\mathrm{.}757,2\mathrm{.}855],[29$ July,$2\mathrm{.}827,2\mathrm{.}845,2\mathrm{.}79,2\mathrm{.}796,2\mathrm{.}838,2\mathrm{.}762],[30$ July,$2\mathrm{.}818,2\mathrm{.}743,2\mathrm{.}845,2\mathrm{.}79,2\mathrm{.}737,2\mathrm{.}777],[31$ July,$2\mathrm{.}832,2\mathrm{.}805,2\mathrm{.}853,2\mathrm{.}825,2\mathrm{.}824,2\mathrm{.}746]]$

What is the Lower Control Limit for the X$-$bar Control Chart? $($Please input your answer to three decimal places$)$