A process that is considered in control measures liquid in ounces. Below are the
ID: 361344 • Letter: A
Question
A process that is considered in control measures liquid in ounces. Below are the last 12 samples taken. The sample size =4.
Sample 1 2 3 4
1 20.2 20.1 19.8 19.9
2 19.9 19.9 19.7 20.1
3 19.8 19.9 20.0 20.1
4 20.1 19.9 19.9 19.8
5 19.8 20.1 20.1 19.9
6 19.7 19.6 19.7 19.4
7 20.2 20.1 19.9 19.9
8 20.1 20.1 19.8 19.7
9 20.0 19.9 20.1 19.8
10 19.9 19.8 20.1 19.9
11 20.1 20.1 19.8 19.9
12 19.9 19.4 19.7 19.8
a) Standard deviation of the sampling means = __ounces (round your response to three decimal places).
b) With z =3, the control limits for the mean chart are:
UCLx = ___ ounces (round your response to three decimal places).
LCLx= ____ ounces (round your response to three decimal places)
C) The control limits for the R-chart are:
UCLr = ____ ounces (round your response to three decimal places)
LCLr = ____ ounces (round your response to three decimal places)
d) Based on the x overbarx-chart, is one or more samples beyond the control limits? Yes/No
Based on the R-chart, is one or more samples beyond the control limits? Yes/No
Explanation / Answer
Below table highlights the calculated values of Sample mean and Range for each of 10 samples:
O B S E R V A T I O N ---------------- >>>>>
Sample
1
2
3
4
Sample mean
Range
1
20.2
20.1
19.8
19.9
20.00
0.4
2
19.9
19.9
19.7
20.1
19.90
0.4
3
19.8
19.9
20
20.1
19.95
0.3
4
20.1
19.9
19.9
19.8
19.93
0.3
5
19.8
20.1
20.1
19.9
19.98
0.3
6
19.7
19.6
19.7
19.4
19.60
0.3
7
20.2
20.1
19.9
19.9
20.03
0.3
8
20.1
20.1
19.8
19.7
19.93
0.4
9
20
19.9
20.1
19.8
19.95
0.3
10
19.9
19.8
20.1
19.9
19.93
0.3
11
20.1
20.1
19.8
19.9
19.98
0.3
12
19.9
19.4
19.7
19.8
19.70
0.5
TOTAL =
238.85
4.1
Also to be noted :
Sum of 12 sample means = 238.85
Sum of 12 range values = 4.1
The standard deviation of sample means
= STDDEV.S ( all values of sampling means ) in excel = 0.125605 ( 0.126 rounded to 3 decimal places )
Note: STDDEV.S returns standard deviation based on a sample
The other function STDDEV.P returns Standard deviation based on a population and its value in this case 0.120
STANDARD DEVIATION OF SAMPLING MEANS = 0.126 OUNCES
Following are the relevant values of constants as derived from standard table for Xbar and R chart :
D4 = 2.282
D3 = 0
Now,
Xbar-bar = Mean of sample means = 238.85 / 12 = 19.904
Rbar = Mean of range values = 4.1/12 = 0.341
Sample size = n = 4
Control limits for Xbar chart with Z = 3 :
Upper Control Limit = UCLx = Xbar-bar + 3 x ( Standard deviation/ Square root ( n))
Lower Control Limit = LCLx = Xbar-bar – 3 x ( Standard deviation / Square root ( n))
Therefore ,
UCLx = 19.904 +3 x 0.126/ Square root ( 4) = 19.904 + 0.189 = 20.093 Ounce
LCLx = 19.904 – 3 x 0.126/Square root ( 4) = 19.904 – 0.189 = 19.715 ounce
Control Limits for R chart :
Upper Control Limit = UCLr = D4. Rbar = 2.282 x 0.341 = 0.778
Lower Control Limit = 0 ( since D3 = 0)
Answer to question d :
For the process to be in control as per Xbar chart , all sample data must be within range : 19.715 – 20.093 Ounce
However there are data e.g 20.2, 20.1, 19.7, 19.6, 19.4 which are outside above control limits
BASED ON Xbar CHART, ONE OR MORE SAMPLES ARE BEYOND CONTROL LIMITS
As per control limits for R chart, all range values should be within range : 0 to 0.778
All the range values are either 0.3 or 0.4 or 0.5 and therefore are within stipulated control limits
BASED ON R CHART, NO SAMPLES ARE BEYOND CONTROL LIMIT
O B S E R V A T I O N ---------------- >>>>>
Sample
1
2
3
4
Sample mean
Range
1
20.2
20.1
19.8
19.9
20.00
0.4
2
19.9
19.9
19.7
20.1
19.90
0.4
3
19.8
19.9
20
20.1
19.95
0.3
4
20.1
19.9
19.9
19.8
19.93
0.3
5
19.8
20.1
20.1
19.9
19.98
0.3
6
19.7
19.6
19.7
19.4
19.60
0.3
7
20.2
20.1
19.9
19.9
20.03
0.3
8
20.1
20.1
19.8
19.7
19.93
0.4
9
20
19.9
20.1
19.8
19.95
0.3
10
19.9
19.8
20.1
19.9
19.93
0.3
11
20.1
20.1
19.8
19.9
19.98
0.3
12
19.9
19.4
19.7
19.8
19.70
0.5
TOTAL =
238.85
4.1
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