Question

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 three samples are displayed below. In each sample, four service people were randomly selected, and the customer service they provided was observed. The time (in seconds) to service each customer was recorded as below.

Service person	Sample 1 (morning)	Sample 2 (afternoon)	Sample 3 (evening)
#1	201	196	201
#2	198	206	210
#3	206	208	211
#4	195	202	206

• Keep two decimals if not exact, do not round. For example, 3.24923... will be kept as 3.24, but the exact value of 0.625 will be kept as 0.625

1. What is the sample size of this study?  
2. What is the X-bar value of sample 1?  
3. What is the R-value of sample 2?  
4. What is 99.73% (3-sigma) the upper control limit of the X-bar chart   and R-chart   ?
5. Assume the established control chart are as follows: X-bar chart is [120, 170], and R-chart is [5, 20]. A sample of six service personnel was observed, and the following customer service times in seconds were recorded: 180, 125, 110, 145, 156 and 190. What is the sample mean here?   Based on the information of sample mean and sample range, and the assumed established control chart, please determine whether the sample process is in control or not. (answer Y or N here)

Several complaints recently have been sent to the LA city office regarding the increasing incidence of congestion on the city’s streets. The complaints attribute the cause of these traffic tie-ups to a lack of synchronization of the traffic lights. The lights are controlled by a main computer system, and adjusting this program is costly. Therefore, the city is reluctant to change the situation unless a clear need is shown. During the past three months, the office initiated a study and collected 150 data each month at major intersections and identify the # of congestion incidents. The congestion data is shown below.

Month	Congestion Incidents
1	12
2	18
3	15

• Keep 3-decimal if not exact, do not round up. For example, 0.24364....will be kept as 0.243
• Both formats, 0.243 or .243, are acceptable

1. What is the sample size of the study?  
2. What is the p-value of sample 1 (month 1)?  
3. What is the p-bar value of the study?  
4. What is the upper control limit, for a 3-sigma quality standard, based on the above data?
5. Assume the p-chart for an acceptable traffic congestion rate is [0.02, 0.09], and last month's # of congestions is 203 from a 2300 traffic-observation, what is last month's sample p-value?  Does last month's congestion situation acceptable or not, comparing to the assumed p-chart? (answer Y or N here)  

A process fills baby formula into bottles with a target of 3 ounces ± 0.12 ounce. Two hundred bottles of baby formula from the process were sampled. The results showed the average amount of baby formula placed in the bottles to be 3.022 ounces. The standard deviation of the amounts was 0.035 ounce.

• Keep 3-decimal if not exact, do not round up. For example, 0.24364....will be kept as 0.243
• Both formats, 0.243 or .243, are acceptable

1. Determine the value of C_pk for the process.  
2. Does it meet the 3-sigma quality control standard? (answer Y or N here)

Answer 1

1st question

Part questions:

What is the sample size of this study?

Answer is 4.

What is the X-bar value of sample 1?

x-bar of sample 1 = (196+206+208+202) = 812

What is the R-value of sample 2?

R-value of sample 2 = 208-196 = 12

What is 99.73% (3-sigma) the upper control limit of the X-bar chart and R-chart ?

For sample size = 4, control chart constants A2 = 0.729, D3 = 0 and D4 = 2.282

Mean of sample 1 = (201+198+206+195)/4 =200; Range = 206-195 = 11

Mean of sample 2 = (196+206+208+202)/4 =203; Range = 208-196 = 12

Mean of sample 3 = (201+210+211+206)/4 =207; Range = 211-201 = 10

Mean of sample means = x-bar-bar = (200+203+207)/3 = 203.3333333
Mean of all sample ranges = (11+12+10)/3 = 11 = R-bar

Upper control limit of x-bar chart = x-bar-bar+A2*R-bar = 203.333333+0.729*11 = 211.35

Upper control limit of R chart = D4*R-bar = 2.282*11 = 25.1

Assume the established control chart are as follows: X-bar chart is [120, 170], and R-chart is [5, 20]. A sample of six service personnel was observed, and the following customer service times in seconds were recorded: 180, 125, 110, 145, 156 and 190. What is the sample mean here? Based on the information of sample mean and sample range, and the assumed established control chart, please determine whether the sample process is in control or not. (answer Y or N here)

Sample mean = (180+125+110+145+156+190)/6 = 151
Sample range = 190-110 = 80

The sample process is not in control as the sample process range is not within control chart limits