You are in charge of quality control on a line which fills bottles of soda. The desired gross weight for a 2 liter bottle of soda is 2,160 grams. At the end of each shift you select 12 bottles at random and weigh them. After one month, the overall average weight of all bottles sampled is 2,162 grams. The average range of weights within a sample for all the samples of 12 is 5 grams. You are preparing x-bar and R control charts for the filling line. Give the target value, UCL and LCL for the x-bar and R charts.
To calculate the control limits for the x-bar and R charts, we need to use the following formulas:
For the x-bar chart:
- Target Value (Centerline): Equals to the average of the individual sample averages.
- UCL (Upper Control Limit): Equals to the target value plus three times the average range of weights (ARW).
- LCL (Lower Control Limit): Equals to the target value minus three times the average range of weights (ARW).
For the R chart:
- UCL: Equals to the average range of weights (ARW) multiplied by the D4 factor.
- LCL: Equals to zero, as there is no negative range of weights.
Now, let's calculate the control limits for the x-bar and R charts:
1. Target Value (Centerline) for x-bar chart:
The given information states that the overall average weight of all bottles sampled is 2,162 grams. Thus, the target value for the x-bar chart is 2,162 grams.
2. UCL and LCL for x-bar chart:
The average range of weights within a sample for all the samples of 12 bottles is given as 5 grams. Therefore, the average range of weights (ARW) is 5 grams. Now, we can calculate the UCL and LCL using the formulas mentioned above:
UCL = Target Value + (3 * ARW)
= 2,162 + (3 * 5)
= 2,162 + 15
= 2,177 grams
LCL = Target Value - (3 * ARW)
= 2,162 - (3 * 5)
= 2,162 - 15
= 2,147 grams
So, the UCL for the x-bar chart is 2,177 grams and the LCL is 2,147 grams.
3. UCL and LCL for R chart:
The given information states that the average range of weights within a sample for all the samples of 12 bottles is 5 grams. Therefore, the average range of weights (ARW) is 5 grams. The D4 factor, which is used to calculate the UCL for the R chart, can be obtained from statistical tables. For a sample size of 12, the D4 factor is 2.282.
UCL = ARW * D4
= 5 * 2.282
≈ 11.41 grams
LCL = 0 (as there is no negative range of weights)
So, the UCL for the R chart is approximately 11.41 grams, and the LCL is 0 grams.
In summary, the control limits for the x-bar chart are:
- Target Value: 2,162 grams
- UCL: 2,177 grams
- LCL: 2,147 grams
And the control limits for the R chart are:
- UCL: Approximately 11.41 grams
- LCL: 0 grams