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As a quality analyst you are also responsible for controlling the weight of a box of cereal. The Operations Manager asks you to identify the ways in which statistical quality control methods can be applied to the weights of the boxes. Provide your recommendations to the Operations Manager in a two-three page report. Using the data provided in the Doc Sharing area labeled M4A2Data, create Xbar and R charts.
Your report should indicate the following along with valid justifications of your answers:
- The control limits of the weights of the boxes.
- Nonrandom patterns or trends, if any.
- If the process is in control.
- The appropriate action if the process is not in control.
>2
Xb
rR
]
©
0
Vertex
2 LLC
ness (mm),
Size, n
0
.
40
.8
4
0.861
%
X-bar
40
0
X-bar
.
5
X-bar
4
0.000
.8
5 34.840 37.355 32.325 4.360 9.219 0.000
6 34.840 37.355 32.325 4.360 9.219 0.000
5 34.840 37.355 32.325 4.360 9.219 0.000
34 3 34.840 37.355 32.325 4.360 9.219 0.000
35 4 34.840 37.355 32.325 4.360 9.219 0.000
35.6 4 34.840 37.355 32.325 4.360 9.219 0.000
33.4 5 34.840 37.355 32.325 4.360 9.219 0.000
3 34.840 37.355 32.325 4.360 9.219 0.000
5 34.840 37.355 32.325 4.360 9.219 0.000
35.2 5 34.840 37.355 32.325 4.360 9.219 0.000
3 34.840 37.355 32.325 4.360 9.219 0.000
5 34.840 37.355 32.325 4.360 9.219 0.000
5 34.840 37.355 32.325 4.360 9.219 0.000
4 34.840 37.355 32.325 4.360 9.219 0.000
5 34.840 37.355 32.325 4.360 9.219 0.000
37 4 34.840 37.355 32.325 4.360 9.219 0.000
34.8 3 34.840 37.355 32.325 4.360 9.219 0.000
4 34.840 37.355 32.325 4.360 9.219 0.000
&L&8
&R&8Templates by Vertex42.com
Probability of a Type I Error (a):
If a sample value falls outside the control limits, we would conclude that the process is out of control. A Type I error is made when the process is concluded to be out of control when it is really in control. This probability is calculated assuming a normal distribution for the process.
The estimated process mean is calculated as the mean of the X-bar values from the data table. It is used as the Center Line for the X-bar Chart.
The estimated process standard deviation.
The number of measurements within each sample. For this chart, all samples are assumed to be the same size. In this spreadsheet, the sample size must be between 2 and 25.
The k-value is number of standard deviations (typically 3) that the upper and lower control limits are placed away from the center line.
In-Control Average Run Length:
If the process is in-control, the ARL is the number of samples, on average, you would observe before getting an out-of-control signal. In other words, you expect to get a false alarm (a point outside the control limits) every N samples, where N is the ARL.
R-bar is the mean of the Ranges in the data table and is used as the center line for the R-Chart.
The range for each sample is calculated as the Max value minus the Min value.
X-bar is the sample mean calculated as the sum of the observations divided by the number of observations in the sample (n).
This is the standard deviation of the sample mean, calculated as the process standard deviation divided by the square root of the sample size.
The Cp index is calculated as (USL-LSL)/(6*sigma) where sigma is the process standard deviation. You want Cp to be greater than 1.
The CPU index is the upper capability index for when you are only given an upper spec limit, USL. You want CPU > 1
The CPL index is the lower capability index for when you are only given a lower spec limit, LSL. You want a CPL > 1.
The Cpk index is used when the process mean is shifted away from the target value, or the point half way between the spec limits. It is the minimum of the CPU and CPL. You want a Cpk>1.
Percent Yield measures the proportion of the output that is within the spec limits, assuming a Normal population distribution.
Instructions:
This Control Chart template creates an X-bar Chart and R Chart with control limits calculated from values contained in the data table. All samples are assumed to be the same size.
– Enter the label and the sample size for the quality characteristic that you are monitoring.
– Choose a k-value (typically 3) for setting the control limits.
– Replace the X-bar and Range values in the Data Table with your own data set (use Paste Special – Values).
– You can delete unused rows in the data table.
– If you need to insert additional rows in the data table, insert rows above the gray line below the table so that series in the chart expand accordingly. Copy the formulas for CL, UCL, and LCL to fill in the blank spaces.
– The labels for CL, UCL, and LCL within the chart are created by selecting the last Data Point and formatting it so that the Data Labels include both the Series name and the Value.
XbarR
X-bar
CL
UCL
LCL
Sample #
XbarS
Range
CL
UCL
LCL
Sample #
Range
TermsOfUse
| Control Chart for Mean and Standard Deviation | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Sample Size, n | Statistics from Data Table | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| s-bar | 45 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| c4 | 0.9213 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.732 | 0.704 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 0.716 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.366 | 0.693 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.840 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 96.53% | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Control Limits for X-bar Chart | Control Limits for S Chart | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| CLX-bar | CLS | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| UCLX-bar | 41.939 | UCLS | 9.880 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| LCLX-bar | 27.741 | LCLS | |||||||||||||||||||||||||||||||||||||||||||||||||||
| St. Dev., s | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
&L&8© 2009 Vertex42 LLC&R&8Templates by Vertex42.com
Probability of a Type I Error (a):
If a sample value falls outside the control limits, we would conclude that the process is out of control. A Type I error is made when the process is concluded to be out of control when it is really in control. This probability is calculated assuming a normal distribution for the process.
The estimated process mean is calculated as the mean of the X-bar values from the data table. It is used as the Center Line for the X-bar Chart.
The estimated population standard deviation. This is calculated by dividing s-bar by c4.
The number of measurements within each sample. For this chart, all samples are assumed to be the same size. In this spreadsheet, the sample size must be between 2 and 25.
The k-value is number of standard deviations (typically 3) that the upper and lower control limits are placed away from the center line.
In-Control Average Run Length:
If the process is in-control, the ARL is the number of samples, on average, you would observe before getting an out-of-control signal. In other words, you expect to get a false alarm (a point outside the control limits) every N samples, where N is the ARL.
s-bar is the mean of the sample standard deviations from the data table. It is the center line for the S Chart.
The sample standard deviation for each sample.
X-bar is the sample mean calculated as the sum of the observations divided by the number of observations in the sample (n).
This is the standard deviation of the sample mean, calculated as the process standard deviation divided by the square root of the sample size.
This is the standard deviation of the sample mean, calculated from the estimated population standard deviation and c4.
c4 is a factor that depends on the sample size and can be found tabulated in most control chart factor tables. Assuming the population distribution is Normal, c4 is used to find the mean and standard deviation of the sample standard deviation.
The Cp index is calculated as (USL-LSL)/(6*sigma) where sigma is the process standard deviation. You want Cp to be greater than 1.
The CPU index is the upper capability index for when you are only given an upper spec limit, USL. You want CPU > 1
The CPL index is the lower capability index for when you are only given a lower spec limit, LSL. You want a CPL > 1.
The Cpk index is used when the process mean is shifted away from the target value, or the point half way between the spec limits. It is the minimum of the CPU and CPL. You want a Cpk>1.
Percent Yield measures the proportion of the output that is within the spec limits, assuming a Normal population distribution.
Instructions:
This Control Chart template creates an X-bar Chart and Standard Deviation Chart (s Chart) with control limits calculated from values contained in the data table. All samples are assumed to be the same size.
– Enter the label and the sample size for the quality characteristic that you are monitoring.
– Choose a k-value (typically 3) for setting the control limits.
– Replace the X-bar and St.Dev. values in the Data Table with your own data set (use Paste Special – Values).
– You can delete unused rows in the data table.
– If you need to insert additional rows in the data table, insert rows above the gray line below the table so that series in the chart expand accordingly. Copy the formulas for CL, UCL, and LCL to fill in the blank spaces.
– The labels for CL, UCL, and LCL within the chart are created by selecting the last Data Point and formatting it so that the Data Labels include both the Series name and the Value.
TermsOfUse
X-bar
CL
UCL
LCL
Sample #
Average Thickness (mm), X-bar
©
St. Dev., s
CL
UCL
LCL
Sample #
Standard Deviation
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http://www.vertex42.com/ExcelTemplates/control-chart.html
| Control Chart Template |
http://www.vertex42.com/ExcelTemplates/control-chart.html
>Sheet ote: The following consists of sets of three box weights in ounces
.3
6.32 735027
3333333
6.33 88 58
6.33 6.323 6.28 6.39 6.4 6.339 Control weight of Cereal Boxes1
N
12
N
#1
#2
#
3
StdDev
Xbar
1
6
6.2
8
6.26
0.02
6.28
2
6.32
6.33
0.00
5
7
6.323
3
6.2
9
6.36
0.035
11
4
6.3266666667
4 6.3 6.29
6.34
0.0264575131
6.31
5
6.295
6.315
6.39
0.050083264
6.3333333333
6
6.292
6.319
0.0195533458
6.3136666667
7
6.289
6.4
0.0568711995
6.3373333333
8
6.286
6.327
6.471
0.0971613778
6.3613333333
9
6.283
6.331
6.498
0.1128553647
6.3706666667
10
6.335
6.525
0.1285496013
6.38
11
6.277
6.339
0.0565891627
6.3353333333
12
6.274
6.343
0.0630951662
