Problems
A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that the process standard deviation is two ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appears below.
-
Weight
Day
Package 1
Package 2
Package 3
Package 4
Monday
23
22
23
24
Tuesday
23
21
19
21
Wednesday
20
19
20
21
Thursday
18
19
20
19
Friday
18
20
22
20
(a) Calculate all sample means and the mean of all sample means.
(b) Calculate
upper and lower control limits that allow
for natural variations.
(c) Is this process in control?
(a) The five sample means are 23, 21, 20, 19, and 20. The mean of all sample means is 20.6
(b) UCL
=
=
22.6; LCL =
= 18.6
(c) Sample 1 is above the UCL; all others are within limits. The process is out of control.
(Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that when the process is operating as intended, packaging weight is normally distributed with a mean of twenty ounces, and a process standard deviation of two ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appears below.
-
Weight
Day
Package 1
Package 2
Package 3
Package 4
Monday
23
22
23
24
Tuesday
23
21
19
21
Wednesday
20
19
20
21
Thursday
18
19
20
19
Friday
18
20
22
20
(a) If he sets an upper control limit of 21 and a lower control limit of 19 around the target value of twenty ounces, what is the probability of concluding that this process is out of control when it is actually in control?
(b) With the UCL and LCL of part a, what do you conclude about this process—is it in control?
(a) These control limits are one standard error away from the centerline, and thus include 68.268 percent of the area under the normal distribution. There is therefore a 31.732 percent chance that, when the process is operating in control, a sample will indicate otherwise.
(b) The mean of sample 1 lies outside the control limits. All other points are on or within the limits. The process is not in control.
(Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
An operator trainee is attempting to monitor a filling process that has an overall average of 705 cc. The average range is 17 cc. If you use a sample size of 6, what are the upper and lower control limits for the X-bar and R chart?
From table, A2 = 0.483, D4 = 2.004, D3 = 0
UCL
=
+ A2
*
LCL
=
- A2
*
UCLR
= D4
*
LCLR
= D3
*
= 705 + 0.483 x 17 = 705 - 0.483 * 17 = 2.004 * 17 = 0 * 17
= 713.211 = 696.789 = 34.068 = 0
(Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
The defect rate for a product has historically been about 1.6%. What are the upper and lower control chart limits for a p-chart, if you wish to use a sample size of 100 and 3-sigma limits?
UCLp
=
=
0.016 + 3.
=
.0536
LCLp
=
=
0.016 - 3 .
=
-0.0216, or zero.
(Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
A small, independent amusement park collects data on the number of cars with out-of-state license plates. The sample size is fixed at n=25 each day. Data from the previous 10 days indicate the following number of out-of-state license plates:
-
Day
Out-of-state Plates
1
6
2
4
3
5
4
7
5
8
6
3
7
4
8
5
9
3
10
11
(a) Calculate the overall proportion of "tourists" (cars with out-of-state plates) and the standard deviation of proportions.
(b) Using
limits, calculate the LCL and UCL for these data.
(c) Is the process under control? Explain.
(a) p-bar is 56/250 = 0.224; the standard deviation of proportions is the square root of
.224 x .776 / 25 = 0.0834
(b) UCL = .224 + 3 x 0.834 = .4742; LCL = .224 -3 x .0834 which is negative, so the LCL = 0
(c) The largest percentage of tourists (day 10) is 11/25 = .44, which is still below the UCL. Thus, all the points are within the control limits, so the process is under control. (Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
Cartons of Plaster of Paris are supposed to weigh exactly 32 oz. Inspectors want to develop process control charts. They take ten samples of six boxes each and weigh them. Based on the following data, compute the lower and upper control limits and determine whether the process is in control.
-
Sample
Mean
Range
1
33.8
1.1
2
34.6
0.3
3
34.7
0.4
4
34.1
0.7
5
34.2
0.3
6
34.3
0.4
7
33.9
0.5
8
34.1
0.8
9
34.2
0.4
10
34.4
0.3
n
= 6; overall mean = 34.23;
= 0.52.
-
Upper control limit
34.48116
1.04208
Center line
34.23
0.52
Lower control limit
33.97884
0
The mean values for samples 1, 2, 3, and 7 fall outside the control limits on the X-bar chart and sample 1 falls outside the upper limit on the R-chart. Therefore, the process is out of control. (Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
McDaniel Shipyards wants to develop control charts to assess the quality of its steel plate. They take ten sheets of 1" steel plate and compute the number of cosmetic flaws on each roll. Each sheet is 20' by 100'. Based on the following data, develop limits for the control chart, plot the control chart, and determine whether the process is in control.
-
Sheet
Number of flaws
1
1
2
1
3
2
4
0
5
1
6
5
7
0
8
2
9
0
10
2
-
Total units sampled
10
Total defects
14
Defect rate, c-bar
1.4
Standard deviation
1.183216
z value
3
Upper Control Limit
4.949648
Center Line
1.4
Lower Control Limit
0
Sample six is above the control limits; therefore, the process is out of control.
(Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
The mean and standard deviations for a process are
= 90 and
= 9. For the variable control chart, a sample size of 16 will be
used. Calculate the standard deviation of the sampling distribution.
Sigma
x-bar =
(Statistical Process Control (SPC), moderate) {AACSB: Analytic Skills}
If
= 9 ounces,
= 0.5 ounces, and n = 9, calculate the 3-sigma control limits.