Question

Happy Oil Processing needs to monitor the amount of oil they put in 1-liter containers on their filling line. They have had problems with underfilled containers.

Samples of 250 containers were collected 4 times a day for 12 days.
They’ve asked you to create an *np*-chart for
the filling process and interpret the results.

a) What values will be used for the centerline, upper control limit, and lower control limit? (You only need to make a table of the values to answer this question and others like it in this assignment.)

b) Create the control chart.

c) Does the process appear to be in statistical control? Why or why not?

d) What percentage of containers are underfilled?

Number of Underfilled | |

Sample | Cans (n = 250) |

1 | 8 |

2 | 7 |

3 | 5 |

4 | 7 |

5 | 7 |

6 | 6 |

7 | 6 |

8 | 7 |

9 | 12 |

10 | 7 |

11 | 7 |

12 | 0 |

13 | 11 |

14 | 6 |

15 | 10 |

16 | 5 |

17 | 6 |

18 | 7 |

19 | 6 |

20 | 1 |

21 | 6 |

22 | 4 |

23 | 9 |

24 | 5 |

25 | 6 |

26 | 10 |

27 | 6 |

28 | 0 |

29 | 6 |

30 | 7 |

31 | 4 |

32 | 6 |

33 | 3 |

34 | 2 |

35 | 7 |

36 | 4 |

37 | 13 |

38 | 8 |

39 | 6 |

40 | 0 |

41 | 9 |

42 | 6 |

43 | 9 |

44 | 10 |

45 | 10 |

46 | 8 |

47 | 7 |

48 | 11 |

Answer #1

a) First we calculate the average proportions p_bar. This is done by

P_bar = total defects/total samples = 0.026083

Now we know that n = 250.

The formula for center line (CL) is n*p_bar = 250*0.026083 = 6.52

The formula for UCL and LCL are

UCL = n*p_bar + 3*sqrt(n*p_bar*(1-p_bar)) = 14.08104

LCL = n*p_bar - 3*sqrt(n*p_bar*(1-p_bar)) = -1.03837, however since the defects cannot be in negative, we will take this as 0.

b) The control chart is shown below

c) The process appears to be in statistical control. There is no anomaly in the data or the pattern. Also all the points are within the control limit.

d) P_bar provides the percentage. It is 0.026 or 2.6% of the containers are under filled.

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1 2
2 3
3 5
4 1
5 2
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2 10
3 14
4 8
5 9
6 11
7 9
8 9
9 14
10 11
11 7
12 8
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Units Inspected
Units Defective
1
200
6
2
150
2
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250
8
4
150
0
5
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3
6
200
4
7
250
9
8
150
1
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7
10
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13
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16
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17
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18
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21
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2
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4
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6
7
8
9
10
11
12
13
14
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Observ. 2
Observ. 3
Observ. 4
Observ. 5
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333.6226
339.3906
361.9761
339.1192
346.4578
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365.5820
347.4967
349.5748
352.6524
363.7096
3
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367.4003
335.0422
328.8487
355.8509
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338.4916
338.6541
346.3491
366.9538
343.1767
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345.7635
356.5218
347.2718
334.5434
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345.6990
326.0756
328.9903
362.4881
352.8718
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334.7083
359.4960
333.1609
352.2697
360.8256
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2
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4
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