Day Time 1 Time 2 Time 3 Time 4 1 7.1 8.6 9.9 8.7 2 5.8 8.8 3.3 4.1 3 5.5 8.7 3.
ID: 3296599 • Letter: D
Question
Day Time 1 Time 2 Time 3 Time 4
1 7.1 8.6 9.9 8.7
2 5.8 8.8 3.3 4.1
3 5.5 8.7 3.3 6.1
4 3.4 8.4 5.3 7.7
5 9.9 4.5 4.9 5.9
6 8.9 9.2 9.4 6.7
7 5.9 6.4 5.5 5.7
8 8.8 5.3 8.6 7.2
9 5.6 4.8 4.3 4.6
10 3.8 4.3 3.4 4.4
11 2.5 3.9 1.5 4.8
12 3.8 1.2 7.6 2.5
13 5.7 6.7 8.7 9.5
14 6.9 6.3 8.4 5.1
15 7.1 5.7 6.9 7.2
16 6.5 7.1 6.9 9.2
17 5.5 6.3 3.2 4.8
18 4.7 5.2 3.1 9.6
19 7.3 8.1 4.2 5.9
20 6.1 4.3 7.3 5.7
Control Chart Factors
A local bank wants to study waiting times of customers. A subgroup of four customers is selected, and the time is measured from the point each customer enters the line to when he or she reaches the teller window. The results are shown in the data table (a) Construct control charts for the range and the mean (b) Is the process in control? Click the icon to view the data Click the icon to view a table of control chart factors (a) Construct a control chart for the range. Choose the correct graph below OA. C. OD. -11*^-UCL UCL ELCL 20 20 20 20 Construct a control chart for the mean. Choose the correct graph below OA. O B. OC. O D. Fo---UCL F---UCL 20 20 (b) Is the process in control? o A. O B. Yes, because no points lie outside the control limits and there are no obvious patterns on either chart No, because there is an obvious pattern in the R chartExplanation / Answer
Solution
Back-up Theory
Let xij be the jth observation in the ith sub-group; i = 1,2,…..,k; j = 1,2,….n
Average of the ith sub-group, xiBar = (sum of xij over j)/ni
Range of ith sub-group, Ri = Max observation - min onservation of the ith sub-group
Grand Average, xdouble bar = (sum of xi bar over i)/k
Average Range, R bar = (sum of Ri over i)/k
LCL (Lower Control Limit) for x bar chart: x double bar - (A2).(R bar)
UCL (Upper Control Limit) for x bar chart: x double bar + (A2).(R bar)
LCL (Lower Control Limit for R chart): (D3).(R bar)
UCL (Upper Control Limit) for R chart: (D4).(R bar)
Part (a)
ANSWER option Graph (B)
Summary of Excel calculations:
Rbar = 3.555; LCL = 0; UCL = 8.113. [Detailed calculation is presented later after Part (c)]
Viewing the graphs w.r.t., LCL and UCL, only graphs A and B qualify. Further, looking at the values of range plotted, Graph A has a point at 7 which is not in the table given at the end.
So, graph A is ruled out leaving graph B as the correct choice.
Part (b)
ANSWER option Graph (C)
Summary of Excel calculations:
X double bar = 6.07; LCL = 3.48; UCL = 8.66. [Detailed calculation is presented later after Part (c)]
Viewing the graphs w.r.t., LCL and UCL, only graphs B and C qualify. Further, looking at the values of averages plotted, the 6th average value as per calculations is 8.55 which is depicted correctly only in Graph C
So, graph B is ruled out leaving graph C as the correct choice.
Part (c) ANSWER option (B)
While all points on R-chart are well within the control limits, one point (11th day) of average lies below the LCL of Xbar chart. Hence, process is not under control.
Details of Calculations
Day
Time1
Time2
Time3
Time 4
Xbari
Ri
1
7.1
8.6
9.9
8.7
8.575
2.8
2
5.8
8.8
3.3
4.1
5.5
5.5
3
5.5
8.7
3.3
6.1
5.9
5.4
4
3.4
8.4
5.3
7.7
6.2
5
5
9.9
4.5
4.9
5.9
6.3
5.4
6
8.9
9.2
9.4
6.7
8.55
2.7
7
5.9
6.4
5.5
5.7
5.875
0.9
8
8.8
5.3
8.6
7.2
7.475
3.5
9
5.6
4.8
4.3
4.6
4.825
1.3
10
3.8
4.3
3.4
4.4
3.975
1
11
2.5
3.9
1.5
4.8
3.175
3.3
12
3.8
1.2
7.6
2.5
3.775
6.4
13
5.7
6.7
8.7
9.6
7.675
3.9
14
6.9
6.3
8.4
5.1
6.675
3.3
15
7.1
5.7
6.9
7.2
6.725
1.5
16
6.5
7.1
6.9
9.2
7.425
2.7
17
5.5
6.3
3.2
4.8
4.95
3.1
18
4.7
5.2
3.1
9.6
5.65
6.5
19
7.3
8.1
4.2
5.9
6.375
3.9
20
6.1
4.3
7.3
5.7
5.85
3
Xdoublebar =
6.0725
Rbar
3.555
A2
0.729
D3
0
D4
2.282
Xbar LCL
3.480905
Xbar UCL
8.664095
RLCL
0
RUCL
8.11251
Let xij be the jth observation in the ith sub-group; i = 1,2,…..,k; j = 1,2,….n
Average of the ith sub-group, xiBar = (sum of xij over j)/ni
Range of ith sub-group, Ri = Max observation - min onservation of the ith sub-group
Grand Average, xdouble bar = (sum of xi bar over i)/k
Average Range, R bar = (sum of Ri over i)/k
LCL (Lower Control Limit) for x bar chart: x double bar - (A2).(R bar)
UCL (Upper Control Limit) for x bar chart: x double bar + (A2).(R bar)
LCL (Lower Control Limit for R chart): (D3).(R bar)
UCL (Upper Control Limit) for R chart: (D4).(R bar)
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