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Question
A machine is set to deliver packets of a given weight. Ten samples of size five each were recorded. Below are given relevant data:
| Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| `bar"X"` | 15 | 17 | 15 | 18 | 17 | 14 | 18 | 15 | 1 | 16 |
| R | 7 | 7 | 4 | 9 | 8 | 7 | 12 | 4 | 11 | 5 |
Calculate the control limits for mean chart and the range chart and then comment on the state of control, (conversion factors for n = 5, A2 = 0.58, D3 = 0 and D4 = 2.115)
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Solution
| Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Tota |
| Mean `(bar"X")` | 15 | 17 | 15 | 18 | 17 | 14 | 18 | 15 | 1 | 16 | `sum"X"` = 162 |
| Range (R) | 7 | 7 | 4 | 9 | 8 | 7 | 12 | 4 | 11 | 5 | `sum"R"` = 74 |
The control limits for `bar"X"` chart is
`\overset{==}{"X"} = (sum"X")/"No. of samples" = 162/10` = 16.2
`bar"R" = (sum"R")/"No. of samples" = 74/10` = 7.4
UCL = `\overset{==}{"X"} + "A"_2+ bar"R"`
= 16.2 + (0.58)(7.4)
= 16.2 + 4.292
= 20.492
= 20.49
CL = `\overset{=}{"X"}` = 16.2
LCL = `\overset{==}{"X"} - "A"_2 bar"R"`
= 16.2 – (0.58)(7.4)
= 16.2 – 4.292
= 11.908
= 11.91
The control limits for Range chart is
CL = `"D"_4bar"R"`
= (2.115)(7.4)
= 15.651
= 15.65
CL = `bar"R"` = 7.4
LCL = `"D"_3 bar"R"` = (0)(7.4) = 0
The above diagram shows all the three control lines with the data points plotted.
We see that all the points of the sample mean are within the control limits.
We now draw the R chart for the given data.
The above diagram shows all the three control lines with the sample range points plotted.
We observe that all the points are within the control limits.
Conclusion: From the above two plots of the sample mean `bar"X"` and sample range R, we conclude that the process is in control.
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Ten samples each of size five are drawn at regular intervals from a manufacturing process. The sample means `(bar"X")` and their ranges (R) are given below:
| Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| `bar"X"` | 49 | 45 | 48 | 53 | 39 | 47 | 46 | 39 | 51 | 45 |
| R | 7 | 5 | 7 | 9 | 5 | 8 | 8 | 6 | 7 | 6 |
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| Samples number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| `bar"X"` | 12 | 13 | 11 | 12 | 14 | 13 | 16 | 15 |
| R | 2 | 5 | 4 | 2 | 3 | 2 | 4 | 3 |
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| Sample Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Mean | 5.10 | 4.98 | 5.02 | 4.96 | 4.96 | 5.04 | 4.94 | 4.92 | 4.92 | 4.98 |
| Range | 0.3 | 0.4 | 0.2 | 0.4 | 0.1 | 0.1 | 0.8 | 0.5 | 0.3 | 0.5 |
