A company has been contracted to produce medicine X. The key ingredient of medic
ID: 448738 • Letter: A
Question
A company has been contracted to produce medicine X. The key ingredient of medicine X is chemical Y. To check if their production process is in control, the company collected five samples, where each sample consists of six bottles of medicine X. The company then determined the weight (in milligram) of chemical Y in each of the bottles as shown in the following table.
Sample
Bottle 1
Bottle 2
Bottle 3
Bottle 4
Bottle 5
Bottle 6
1
104.87
105.65
106.52
107.40
106.95
108.65
2
102.50
100.92
98.22
100.00
96.63
97.65
3
104.80
101.61
103.46
103.90
101.87
101.72
4
102.50
101.23
109.81
105.25
111.20
101.85
5
102.85
106.12
109.62
104.60
101.19
105.33
What is UCLR?
What is LCLR?
All points on the R-chart are within the control limits T/F
What is UCLX-bar?
What is LCLX-bar?
All points on the X-bar chart are within the control limits. T/F
Is this process considered in control?
Sample
Bottle 1
Bottle 2
Bottle 3
Bottle 4
Bottle 5
Bottle 6
1
104.87
105.65
106.52
107.40
106.95
108.65
2
102.50
100.92
98.22
100.00
96.63
97.65
3
104.80
101.61
103.46
103.90
101.87
101.72
4
102.50
101.23
109.81
105.25
111.20
101.85
5
102.85
106.12
109.62
104.60
101.19
105.33
Explanation / Answer
sample size = 5.
First we will calculate the r bar. R for a sample will be highest value - lowest value.
R for sample 1 = 108.65 - 104.87 = 3.78. R values for all the samples are shown below:
r bar = sum of all R's/sample size = (3.78+5.87+3.19+8.58+8.43)/5 = 29.85/5 = 5.97
UCLR = D4*r bar. When sample size = 5, D4 = 2.114
Thus, UCLR = 2.114*5.97 = 12.62
LCLR = D3*r bar. When sample size = 5, D3 = 0. Hence LCLR = 0*5.97 = 0.
As we can see that all the values of "R" as shown in the table above are below UCL and above LCL. So all points are within the control limits - True.
For UCL and LCL x-bar, we will calculate x bar and x double bar. x bar = sum of (x1, x2.....xn)/n
for sample 1 it will be 104.87+105.65+106.52......+108.65/6. The x bars have been calculated using this formula/method for all the samples and shown below:
x double bar = average of all x bars = (106.6733+99.32+102.8933+105.3067+104.9517)/5 = 519.145/5 = 103.829
UCLX-bar = x double bar+A2*R bar. For sample size = 5, A2 = 0.577
Thus UCLX-bar = 103.829+0.577*5.97 = 107.27369
LCL x-bar = x double bar - A2*R bar
= 103.829 - 0.577*5.97
= 100.38431
All x bars are below the UCL of 107.27. However, the mean for sample 2 is 99.32. This is lower than the LCL of 100.38. Thus all points on xbar chart are not within control limits. Hence the statement "All points on the X-bar chart are within the control limits" is False.
As one point in x-bar chart is outside the LCL, the process cannot be considered in control.
Sample R 1 3.78 2 5.87 3 3.19 4 8.58 5 8.43Related Questions
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