Management Chapter 8 Homework Sample Number Precision Piston Company Continued Revised

Process

Capability

Calculations

Six

sigma

6.815

Upper specification

12.4

Cp

0.939

Capability indexes are Cpu = 0.948, Cpl = 0.930, Cpk = 0.930 and Cp = 0.939. All the

indexes are unsatisfactory, since they fall below 1. The Cpu = 0.948, indicates that the

d) Using the previously calculated control limits to monitor the last 20 samples, there is

one unusual occurrence, with seven out of the last eight samples below the centerline,

indicating a probable out-of-control condition. See spreadsheet Prob08-

Lower specification

6.0

Cpu

0.948

Cpl

0.930

Cpk

0.930

10. Construct x– and s-charts for the data for Babbage Chips from Prob. 8-09. Is the process

in control? What recommendation would you make to management concerning the

process, based on your analysis?

6.00

6.50

10.50

11.00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

Sample number

Prob. 8-09B X-bar Chart

Averages

Lower control limit

Upper control limit

Center line

0.00

4.00

5.00

6.00

1357911 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

Sample number

Prob. 8-09B R-Chart

Ranges

Lower control limit

Upper control limit

Center line

Answer

See data and control charts below and spreadsheet Prob08-10XS.xlsx for details. Using the

data from Babbage Chips, Prob08-09, we see:

For the center line, CL

x

:

x

= 9.170; CLs :

s

= 1.046

Control limits for the

x

– s charts are:

As in Prob. 08-09A, results from 30 samples of 5 show that both the

x

and s charts are

7

10

10.5

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Sample number

Prob. 8-10 X-bar Chart

Averages

Lower control limit

Upper control limit

Center line

x

s

11. Ricardo’s Widgets makes a critical part for a popular brand of cell phones. Consider the

data for 15 samples of size 4 of a key dimension for the part, shown in the worksheet

C08DataInsRsv.xlsx. Specifications are 0.110 ± 0.02.

a. Construct the

x

– and R charts and an x-MR chart for “individuals” using the data.

Interpret the results. Use a four period moving range for the x-MR chart.

b. Estimate the process capability by using the actual sample standard deviation.

Answer

a) See spreadsheet Prob08-11InsRsvXMR.xls (X and MR chart template) for details.

CLx :

x

= 0.1115; CLR :

R

= 0.0124 (with a 4-period moving average)

0

2

2.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Sample number

Prob. 8-10 s-Chart

Standard Deviations

Lower control limit

Upper control limit

Center line

b) The detailed comparisons of process capability using estimated  are shown in a table on

the spreadsheets Prob08-11InsRsvXbarR.xlsx.

Although individual values must be plotted on x-charts, as shown above, students need to

understand their relationship to

x

– chart and R-chart results for comparison with the

charts for individuals.

For the

x

– chart:

x

± A2

R

= 0.1115 ± 0. 0.729 (0.0119) = 0.1028 to 0.1202

0.080

0.130

0.140

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73

Observation number

Problem 8-11 Individuals (X) Chart

Individuals

Upper control limit

Center line

Lower control limit

0.000

0.025

0.030

14710 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73

Observation number

Problem 8-11 Moving Range Chart

Moving ranges

Lower control limit

Center line

Upper control limit

These limits apply to sample groups of 4 items each.

0.090

0.095

0.120

0.125

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Sample number

Problem 8-11 X-bar Chart

Averages

Lower control limit

Upper control limit

Center line

0.000

0.025

0.030

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Sample number

Problem 8-11 R-Chart

Ranges

Lower control limit

Upper control limit

Center line

c) The calculations of process capability using the estimated  value is shown on the

following table. (See spreadsheet Prob08-11InsRsvXbarR.xlsx for details.)

Nominal specification

0.110

Average

0.1115

Cp

1.115

Upper tolerance limit

0.130

Standard deviation

0.0119

Cpu

1.064

Lower tolerance limit

0.090

Cpl

1.237

Cpk

1.064

12. Thirty-five samples of 25 packages each at the Bakery Bread, Inc. were inspected, and 25

items were found to be defective. Compute control limits for a p-chart.

Answer

Control limits for Bakery Bread, Inc. orders can be calculated using:

Control limits:

UCLp =

p

+ 3 s

p

UCLp = 0.029 + 3(0.034) = 0.131

LCLp =

p

– 3 s

p

LCLp = 0.029 – 3(0.034) = – 0.073, use 0

13. The fraction defective of automotive pistons made by the Precision Piston Co. is given in

the worksheet the worksheet C08DataInsRsv.xlsx for Prob. 8-13 for 20 samples. One

hundred units are inspected each day. Construct a p-chart and interpret the results.

Answer

See data and control charts for Precision Piston Company, below. See spreadsheets

Prob08-13InsRsvP.xlsx, for details.

p

Control limits:

UCLp =

p

+ 3 s

p

UCLp = 0.096 + 3(0.029) = 0.183

LCLp =

p

– 3 s

p

b. Precision Piston Company (Continued)

Revised CL

p

= 0.092 (after sample 12, with a fraction defective

of 0.19, was removed).

p

0.140

0.160

0.180

0.200

Sample number

Problem 8-13A Attribute (p) Chart

Fraction nonconforming

Lower control limit

Center line

Upper control limit

p

Control limits:

UCLp =

p

+ 3 s

p

UCLp = 0.092 + 3(0.029) = 0.179

Prob8-13InsRsvP-B Final Revised Control Chart

14. Ellswater Hospital surveys all outgoing patients by means of a patient satisfaction

questionnaire. A random sample of 350 surveys is taken each month. Control charts that

monitor the proportion of unsatisfied patients for key questions are constructed and

studied. Construct an np-chart for the data in the C08DataInsRsv.xlsx for Prob. 8-14,

which represent responses to a question on satisfaction with hospital housekeeping

services.

0.160

0.180

0.200

Sample number

Problem 8-13B Attribute (p) Chart

Fraction nonconforming

Lower control limit

Center line

Upper control limit

p

Answer

See spreadsheet Prob08-14InsRsvNP.xls for details Using data for problem 14, found in the

worksheet C08DataInsRsv.xlsx, we get results for the np chart shown below:

Initial

So, CL n

p

= n

p

= 350 (0.024667) = 8.633

As was shown in the previous control charts for problem, the value for sample 21 is out of

limits. Therefore this sample had to be eliminated, leaving 29 usable data points. After

15.00

20.00

25.00

Problem 8-14 A – Number nonconforming (np) chart

Number nonconforming

Lower control limit

Upper control limit

Center line

p

Final

So, CL n

p

= n

p

= 350 (0.02345) = 8.207

Control limits:

p

15. Calculate the centerline and control limits for a c-chart involving 35 samples and having

a total of 300 defects and interpret the results.

Answer

Center line for the c-chart:

c

= 300/35 = 8.57

16.00

18.00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Sample number

Problem 8-14 B – Number nonconforming (np) chart

Number nonconforming

Lower control limit

Upper control limit

Center line

p

16. Consider the sample data for defects per pizza in a new store being opened by Rob’s

Pizza Palaces in the worksheet C08DataInsRsv.xlsx for Prob. 8-16. Construct a c-chart

for these data. What does the chart show?

Answer

See spreadsheet Prob08-16InsRsvCC.xls for details.

17. Find 3 s control limits for a small u-chart with the following errors per sample unit. What

do the limits show?

Errors Sample Unit

5 92

3 136

4 70

8

10

Sample number

Prob 8-16

Attribute (c) Chart

Number of defects

Lower control limit

Upper control limit

Center line

8 78

7 165

Answer

For the u-chart conditions: Number of samples = range from 70 to 165 and the number of

errors from 3 to 8.

Center Line for the u-chart:

u

= (5+3+4+8+7) / (92+136+70+78+165) = 27/541 = 0.05

Limits for the samples are, respectively:

Sample

Defects

Standard

Errors

Size

per unit

Deviation

LCLu

CL

UCLu

5

92

0.0543

0.0233

0

0.0499

0.1198

3

136

0.0221

0.0192

0

0.0499

0.1074

4

0.0571

0.0267

0

0.0499

0.1300

0.1258

7

165

0.0424

0.0174

0

0.0499

0.1021

Note: The following problems address sample size determination and refer to theory

covered in the Bonus Material for this chapter as contained on the Student Companion

Site.

18. Determine the appropriate sample size to estimate the proportion of sorting errors in an

apparel warehouse at a 95 percent confidence level. Historically, the sorting error rate is

0.01, and you wish to have an allowable statistical error of 0.02.

Answer

18. The sample size for the proportion of sorting errors at the apparel warehouse, using a

95% confidence level is:

0.0000

0.0200

0.1200

0.1400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Nonconformances per unit

Sample number

Prob. 8-17

Attribute (u) Chart

Defects per unit

Lower control limit

Upper control limit

Center line

Answer

First, we must find an estimated p for Localtel’s sample:

20. An engineer at Shoefactory, Inc. wants to calculate the cycle time (time to make one unit)

for a process based on work sampling observations of workers performing the job. The

standard deviation for the process times is 0.4 minutes, based on a small sample of

observations which she took. If the engineer want to be 95% confident of being in error

by only 0.125 minutes, what sample size should she take?

Answer