Problem 1: The management of the Diners Delight 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. For simplicity, the data from four samples are displayed below. In each sample, six service people were randomly selected , and the customer service they provided was observed . The activities that the service people were performing were identified , and the time to service each customer was recorded as below. Service time (in seconds Service person Sample 1 Sample 2 Sample 3 Sample 200 150 175 90 120 85 105 75 83 93 130 150 UI & W N 68 150 145 175 110 90 75 105 6 115 65 115 125 X 116 105.5 124.17 120 200 -68 = 132 150 -65 =85 175 -75 =100 175 -75 = 100 Use the given information , please (a) Find the 99.73 9% (3-sigma ) upper and lower control limits for X-bar chart and R-chart . (b) After the control c 1 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 the process in control ? Any corrective action needed ? Solution : (a) X = (116 + 105 .5 + 124 .17 + 120 )/ 4 =116.42 R = (132 + 85 + 100 + 100)/4 =104 .25 Sample size n = 6 - A2 = 0.48 , D3 =0, and D4 =2.0 Control limits for mean