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Design specifications require that a key dimension on a product measure 102 ± 15 units. A process being considered for producing this product has a standard deviation of 6 units.
(Format answers rounded to 4 decimal places when entering answers in the quiz, Leave the probabilty as decimal numbers, not %.)
a) Find the process capability index, Cpk, and the probability of defective output. Assume that the process is centered with respect to specifications, i.e., process mean is 102.
) Suppose the process mean shifts to 96. Calculate the new process capability index.
c) What is the probability of defective output after the process shift? Did it increase or decrease?
(For this question and the next, a relevant reference is Example 13.1 in the textbook, pp XXXXXXXXXXThe Excel calculation of that example is shown in sheet Ex 13.1 of Excel SPC workbook. )
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Thickness (mm)
1.9
1.8
2.1
2.1
2
2.2
2.4
2
1.9
2.1
2.2
2.4
1.8
2.2
2.1
1.7
2.2
1.9
2.1
2.2
1.7
1.8
2
2
2
2.1
1.8
1.6
1.9
1.6
1.7
2
1.7
1.8
1.9
2.1
1.8
1.6
1.9
2.2
To the right is a picture of a washer that is supposed to be 1.9 mm thick.
The tolerances on the thickness are 0.5 mm, so the thickness should be between 1.4 mm and 2.4 mm.
You are given here the thickness in mm for a sample of 40 washers. Assume the thickness is distributed normally.
Format your answers with 4 decimal places except where otherwise specified.
a) Verify that the mean = XXXXXXXXXXand the standard deviation from this data is XXXXXXXXXXRecall STDEV.S is the function for computing the sample standard deviation.)
) What is the Cpk for the process?
c) What % of the output is expected to be out of tolerance (outside the specification limits = defective)? (Express the probability as % with 2 decimal digits, e.g., 12.34%.)
d) If the process were centered, i.e., sample mean were equal to 1.9 mm (with the standard deviation unchanged), what would be the Cpk?
e) If the process were centered as in (d), what percentage of output would be expected to be out of tolerance?
f) If the process were centered AND the standard deviation was only about .10 millimeter, what would be the Cpk and percent defective?
(This percent defective is very close to 0. Increase decimal digits until you can see 2 significant figures (e.g., XXXXXXXXXX%))
g) Out of situations in (b), (d), and (f) which had the process considered capable?
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Sample n Number of Defective Items
1 15 1
2 15 2
3 15 3
4 15 2
5 15 1
6 15 2
7 15 1
8 15 3
9 15 2
10 15 1
Ten samples of 15 parts each were taken from an ongoing process to establish a p-chart. The samples and the number of defectives in each are shown in the following table.
a) Determine the p-bar, Sp, UCL and LCL for a p-chart with z = 3 (Format your answers with 4 decimal places.)
) Plot the p-chart.
c) Based on the plotted data points, what comments can you make? (i.e., is the process in control or out of control?) Decide based on whether at least one point is outside the control limits. Recall a point is outside the control limits if it is < LCL or > UCL.
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Sample # Defects
1 6
2 5
3 0
4 1
5 4
6 2
7 5
8 3
9 3
10 2
11 6
12 1
13 8
14 7
15 5
16 4
17 11
18 3
19 0
20 4
At Data Systems Services company owned by Donna, she wants to see if the data entry process is in control. She collected 100 records entered by each of the 20 clerks. She counted the number of inco
ectly entered records out of each sample. Below, you are given this data. Determine whether this process is in control. (Remember the sample size is not number of rows, but the number of items in each sample.)
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Area Number of Crimes Sample Size
1 10 910
2 1 910
3 15 910
4 14 910
5 10 910
6 22 910
7 16 910
8 15 910
9 8 910
10 5 910
11 12 910
12 11 910
13 14 910
14 10 910
15 13 910
16 18 910
17 4 910
18 12 910
19 6 910
20 18 910
A city has 20 police precincts (geographical areas in the city served by different units). Some citizens in the city complained to city council members that there should be equal protection under the law against the occu
ence of crimes. The citizens argued that this equal protection should be interpreted as indicating that high-crime areas should have more police protection than low-crime areas. Therefore, police patrols and other methods for preventing crime (such as street lighting or cleaning up abandoned areas and buildings) should be used proportionately to crime occu
ence.
The police recognize that not all crimes and offenses are reported. Because of this, the police have contacted by phone a random sample of 910 residences for data on crime in each area. (Respondents are guaranteed anonymity.) The 910 sampled from each area showed the following incidence of crime during the past month.
a) Determine the p-bar, Sp, UCL and LCL for a p-chart of 99.7 percent confidence (at Z = 3). (Round your answers to 4 decimal places in the quiz.)
) Is the process in control? If not, are there certain areas that wa
ant further investigation into their relative high or low crime rates?
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Factors for X-bar LCL UCL
Sample Size, n
Janice Winch: Same as Exhibit 13.7 on p. 377 of textbook
A2 D3 D4
2 1.88 0 3.267
3 1.023 0 2.574
4 0.729 0 2.282
5 0.577 0 2.114
6 0.483 0 2.004
7 0.419 0.076 1.924
8 0.373 0.136 1.864
9 0.337 0.184 1.816
Sample Mean, X-bar Range, R 10 0.308 0.223 1.777
1 10.002 0.011 11 0.285 0.256 1.744
2 10.002 0.014 12 0.266 0.283 1.717
3 9.991 0.007 13 0.249 0.307 1.693
4 10.006 0.018 14 0.235 0.328 1.672
5 9.997 0.013 15 0.223 0.347 1.653
6 9.999 0.012
7 10.001 0.008
8 10.005 0.013
9 9.995 0.004
10 10.001 0.011
11 10.001 0.014
12 10.006 0.009
Twelve samples, each containing five parts, were taken from a process that produces steel rods at a factory. The length of each rod (in inches) in the samples was determined. The results were tabulated and sample means and ranges were computed. The results were are shown below. Notice X-bar and R values are already given, so you don't need to compute them. To the right is the table containing A2, D3, and D4.
Do not round the intermediate results, but when entering answers, round them to 3 decimal places.
In problems involving X-bar and R charts, remember sample size n should be the number of items in each sample, not number of rows.
a) Determine the UCL and LCL for X-bar chart, and plot the X-bar chart.
) Determine the UCL and LCL for R chart, and plot the R chart.
c) Both X-bar and R should be in control to conclude the process is in control. Based on your results, is the process in control?
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READINGS (IN OHMS)
SAMPLE 1 2 3 4
1 1013 986 994 977
2 977 1027 982 1024
3 1027 990 998 1017
4 1023 1025 1016 1006
5 1005 1026 975 991
6 983 998 990 988
7 991 999 1001 985
8 1025 1023 1014 1023
9 1019 1004 982 979
10 999 995 991 1010
11 970 992 1006 1012
12 1010 985 983 1030
13 1030 1002 1016 982
14 979 986 1016 988
15 1028 1006 1019 1002
Resistors for electronic circuits are manufactured on a high-speed automated machine. The machine is set up to produce a large run of resistors of 1,000 ohms each.
To set up the machine and to create a control chart to be used throughout the run, 15 samples were taken with four resistors in each sample. The complete list of samples and their measured values are as follows. Use three-sigma control limits.
a) Calculate the mean and range for the samples. (Round "Mean" to 2 decimal places and "Range" to the nearest whole number in the quiz.)
) Determine the UCL and LCL for an X-bar chart. (Round your answers to 2 decimal places.)
c) Determine the UCL and LCL for an R chart. (Round your answers to 2 decimal places.)
d) Create an X-bar chart and an R chart. Is the process in control?
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Observations
Sample 1 2 3 4 5
1 20.2 20.1 19.8 19.9 19.7
2 19.9 19.8 19.7 20.1 20.1
3 19.8 19.9 20 20.1 19.7
4 20.1 19.9 19.5 19.8 19.7
5 19.8 20.1 20.1 19.9 19.4
6 20.1 19.6 19.7 19.4 20.2
7 20.2 20.1 19.9 19.9 19.8
8 20.1 20.1 20.3 19.7 19.9
9 19.7 19.9 20.1 19.8 19.8
10 19.9 19.8 20.1 19.9 20.2
11 20.1 20.1 19.9 19.9 19.9
12 19.9 19.5 19.7 19.8 20
Each bottle of iced tea is supposed to have 20 fluid ounces, but it is normal for the amount to vary slightly from bottle to bottle. Twelve samples were taken from the bottle-filling process in order to determine whether it is in control. Each sample consists of 5 bottles, so the sample size is 5. Round your answers to 3 decimal places.
a) What are the control limits for the mean chart?
) What are the control limits for the range chart?
c) Create an X-bar chart and an R chart. Is the process in control?
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Sample I
egularities
1 3
2 5
3 6
4 3
5 4
6 6
7 5
8 4
9 3
10 5
A shirt manufacturer buys cloth by the 100-yard roll from a supplier. For setting up a control chart to manage the i
egularities (e.g., loose threads and tears), the following data were collected from a sample provided by the supplier.
a) Determine the c_bar, UCL and LCL for a c -chart with z = 3. (format example: 12.34)
) Suppose the next five rolls from the supplier had two, four, four, three, and seven i
egularities. Is the supplier process under control? (Hint: Use the UCL and LCL already established and check if these new values are still between them.)
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For each situation below, determine which chart is appropriate and find the control chart limits. For variable measurements, you must find control limits for both the X-bar chart and R chart.
(Round your answers to 3 decimal places when entering them in the quiz.)
a) An inspector looked at 20 automobiles being prepared for shipment and found an average of 3.9 scratches per unit in the exterior paint.
) An Accounts Receivable department decided to implement SPC in its billing process due to complaints from customers that the bills are inaccurate. Ten samples of 50 bills each were taken over a month's time and checked. The mean proportion of inco
ect bills was XXXXXXXXXXi.e., 9.4%).
c) To make sure the filling process for M&M bags are in control, each hour, a sample of 5 M&M bags were weighed. After 20 samples of 5 weights were collected, averaging the 20 sample means yielded 51.2 grams, and averaging the 20 sample ranges resulted in 0.78 grams.