Problems/Solutions

Problems/Solutions

Problems/Solutions
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Problems

1. The Perfect Circle Company manufactures bushings. Once each hour a sample of 125 finished bushings is drawn from the output; each bushing is examined by a technician. Those which fail are classified as defective; the rest are satisfactory. Here are data on ten consecutive samples taken in one week:

	Sample no.	1	2	3	4	5	6	7	8	9	10
	Defective	15	13	16	11	13	14	20	25	30	45

What type of control chart should be used here?
What is the centerline of the chart?
What is the lower control limit (LCL)? The upper control I?
What statistic should be plotted on the control chart for?
Draw the control chart on a piece of graph paper.
Is this system under control?
What should the quality control engineer do?

2. Use the data in Problem #1 in this problem. Assume that an assignable cause has been found for Sample #10 and has been corrected.

What is the value of the centerline of the revised control chart?
What is the lower control limit of the revised chart?
What is the upper control limit of the revised chart?
Compare the revised control chart to the original control chart.

3. The Take-Charge Company produces batteries. From time to time a random sample of six batteries is selected from the output and the voltage of each battery is measured, to be sure that the system is under control. Here are statistics on 16 such samples.

Sample

Mean

Range

Sample

Mean

Range

1

2

3

4

5

6

7

8

4.99

4.87

4.85

5.26

5.09

5.02

5.13

5.09

0.41

0.57

0.59

0.74

0.74

0.21

0.56

0.92

9

10

11

12

13

14

15

16

5.01

5.19

5.40

5.15

5.00

4.89

4.99

5.05

0.49

0.56

0.44

0.63

0.35

0.45

0.54

0.33

What type of control chart should be used here? Why?
What is the centerline of the chart?
What is the lower control limit? The upper control limit?
What statistic should be plotted on the control chart for each sample?
Draw the control chart on a piece of graph paper.
s this system under control?
What should the quality control engineer do?

4. Use the data in Problem 3 to draw an R chart.

What is the lower control limit? The upper control limit?
What statistic should be plotted on the control chart for each sample?
Draw the control chart on a piece of graph paper.
Is this system under control?
What should the quality control engineer do?

5. Assume that an assignable cause has been found and corrected for sample # 11 in Problem 3.

What is the value of the centerline of the revised control chart?
What is the lower control limit of the revised chart? The upper control Limit?
Compare the revised control chart to the original control chart.

6. Tinker Belle Peanut Butter is sold in .50 kilograms jars. The plant produces thousands of jars of peanut butter per working day; the process is rather simple and quite standardized, and is thought to be highly stable, with a standard deviation of .016 kg. Management has specified that the jars should fall between .446 kg and .554 kg.

What is the process capability index?
Is this process capable?

7. Occasionally, a random sample of five jars of Tinker Belle Peanut Butter (see problem #6) is selected from the output and weighed, to be sure that the system is under control. Here are data on ten such samples. Measurements are in kilograms.

	Sample	1	2	3	4	5	6	7	8	9	10
		.50	.50	.50	.51	.51	.51	.50	.50	.51	.50
		.47	.48	.49	.51	.50	.50	.51	.52	.48	.51
		.50	.48	.51	.52	.49	.52	.49	.47	.50	.49
		.49	.48	.47	.51	.52	.51	.50	.49	.49	.50
		.51	.47	.49	.51	.50	.51	.48	.49	.50	.47
	Total	2.47	2.41	2.46	2.56	2.52	2.55	2.48	2.47	2.48	2.47

What type of control chart should be used here? Why?
What is the centerline of the chart?
What is the lower control limit? The upper control limit?
What statistic should be plotted on the control chart for each sample?
Draw the control chart on a piece of graph paper.
Is this system under control?
What should the quality control engineer do?

8. (One step beyond. Use Table D in this study guide.) Use the data in Problem 6 to construct an R chart.

What is the lower control limit? The upper control limit?
What statistic should be plotted on the control chart for each sample?
Draw the control chart on a sheet of graph paper.
Is this system under control?
What should the quality control engineer do?

9. The Poseidon Fabric Co. produces large beach towels (among other things): they are supposed to be brightly colored and have a fringe on each end. From time to time, a towel is selected from the finished goods and subjected to an intense inspection in search of any and all defects. A defect is a stain, a badly dyed spot, a hole, a missing fringe, etc., each occurrence counts as a distinct defect. Here are data on 12 sample towels.

	Towel	1	2	3	4	5	6	7	8	9	10	11	12
	Number of defects	2	1	3	0	1	4	0	1	3	2	3	1

What type of control chart should be used here? Why?
What is the centerline of the chart?
What is the lower control limit? The upper control limit?
What statistic should be plotted on the control chart for each sample?
Draw the control chart on a piece of graph paper.
Is this system under control?
What should the quality control engineer do?

10. Here is an x-bar chart with 20 sample means plotted on it. Use this chart to perform a test for runs above and below the centerline. See next page.

What is the expected number of runs?
What is the standard deviation of the number of runs?
What is the actual number of runs?
What is the z-statistic?
What are the critical values of z for 95 percent?
Is this process under control? Explain.

30 UCL

·

·

·

·

--

20

·

·

·

·

·

·

·

·

·

·

·

·

·

·

·

10

·

LCL


		0		5		10		15		20

Solutions

1. a. Use the p chart. This is attribute data with the categories of satisfactory and defective bushings.
b. Image248 .
c. Image249 .
Image250
Image251
d.

Sample No.

Statistic (p)

Sample No.

Statistic (p)

1

16/125 = 12%

6

11.2%

2

3

4

5

10.4

12.8

8.8

10.4

7

8

9

10

16.0

20.0

24.0

36.0

e.

·

26.03 UCL

·

·

--

16.16

·

·

·

·

·

·

·

6.29

LCL


		0		5		10

f. No. Sample #10 falls above the UCL.
g. He should look for an assignable cause for sample #10.

2. a. The revised centerline will be obtained by deleting sample #10 and recalculating the value of Image252 on the basis of the remaining nine samples.
revised Image253
b. Revised Image254
Revised Image255
Revised Image256
c. The revised control limits have lower values than the original control limits, and they are slightly closer to the centerline.

3. a. Use the x-bar chart; this is measurement data.
b. Image257 v.
(k = the number of samples)
c. Because the value of the population standard deviation is unknown, use Image258.gif and the A₂ control chart factor.
Image259 v.
Image260 v.
Image261 v.
d. The sample mean.
e.

·

UCL

·

·

·

·

·

--

·

·

·

·

·

·

·

·

·

·

LCL


		0		5		10		15

f. No, sample #11 is out of control.
g. He should look for an assignable cause for sample #11.

4. a. Because s is unknown, use

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image262::/sites/dl/free/0072443901/24520/Image262.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image262 (0.0K)</a>Image262 and the factors, D₃ and D₄ from the Table of Control Chart
Factors.

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image263::/sites/dl/free/0072443901/24520/Image263.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image263 (1.0K)</a>Image263 v.

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image264::/sites/dl/free/0072443901/24520/Image264.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image264 (1.0K)</a>Image264 v.
b. The sample ranges.
c. UCL

·

·

·

·

·

--

·

·

·

·

·

·

·

·

·

·

·

LCL


		0		5		10		15

d. Yes.
e. No action is needed.

5. a. Image265 v.

b. LCL = 5.03 - .48(.53) = 4.78 v.
UCL = 5.03 + .48(.53) = 5.28 v.
c. The new control limits are lower but are still the same distance from the centerline.

6.a. Image266 (upper specification - lower specification/ Image267 = (.554 - .446)/6(.016) = 1.125.

b. Yes, but not by much.

7. a. Use the x-bar chart; this is measurement data.

b. Image268 kg.

c. Because the value of the population standard deviation is known, find the standard error of the sample means and use z = ±3s. Image269 kg.
Image270 kg.
Image271 kg.

d. The sample mean. Here are the first few.
Sample No. Mean
1 Image272 kg.
2 Image273 kg.

e. UCL

·

·

·

--
--

·

·

·

·

·

·

·

LCL


		0		5		10		15

f. Yes.
g. No action is necessary.

8. a. Because Image274 is known, use the factors, D₁ and D₂, from the Table of Control Chart Factors in this Study Guide. Image275 kg.
Image276 kg.

b. Plot the ranges. Here are the first few of them.
Sample No. Range (R)
1 Image277 kg.
2 Image278 kg.

c. UCL

·

·

·

·

·

·

·

·

·

·

LCL


		0		5		10		15

d. Yes.
e. No action is necessary.

9. a. Use a c-chart, because the data consists of the number of defects per unit; a unit is a towel.
b. Image279 defects per towel
c. Image280
Image281 .
d. The number of defects per towel.

e. UCL

·

--

·

·

·

·

·

·

·

·

·

·

·

LCL


		0		5		10		15

f. Yes.
g. No action is necessary.

10. a. Image282 runs
(N = the number of samples)
b. Image283 runs
c. r = 5 runs.
d. Image284
e. z = ±1.96.
f. No; -2.75 < - 1.96.

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Home > Chapter 10 > Problems/Solutions

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	Course-wide Content
		Data Sets Excel Templates PowerWeb Web Links Operations Management ... Interactive OM Updates and Errata

	Instructor Resources
		Instructors Manual PowerPoint Presentations

	News, Articles & Links
		News Feeds PowerWeb Articles Weekly Update Archive
	Quizzes
		Multiple Choice Quiz True or False Self Quiz from CD
	Interactive Exercises
		Interactive OM
	More Resources
		Problems/Solutions Glossary Supplement to Chapter S-Multiple Choice Quiz S-True or False S-Problems/Soutions S-Glossary S-Self Quiz from CD