Problem $6-4$: Diners Delight $($ and R Charts$)$

The management of the Diners Delight franchised restaurant chain is in the process of establishing quality$-$control charts for the time that its service people give to each customer. Management thinks the length of time that each customer is given should remain within certain limits to enhance service quality.

A sample of six service people was selected, and the customer service they provided was observed four times. The activities that the service people were performing were identified, and the time to service one customer was recorded as noted below:

Service Person Sample $1$ Sample $2$ Sample $3$ Sample $4$

$120015017590$

$21208510575$

$38393130150$

$468150145175$

$51109075105$

$611565115125$

a$)$ Copy and paste the data from the table above into Excel. Do the calculations in Excel and develop and R Charts showing the LCL and UCL. Copy and paste the spreadsheet calculations and the and R Charts below. Hint: The sample size is six since each sample contains six observations.

b$)$ After the control chart was established, a sample of six service personnel was observed, and the following customer service times in seconds were recorded: $180,125,110,98,156,$ and $190.$ Is corrective action called for?