TriolaE S CH14pp732 757

14

14-1 Overview

14-2 Control Charts for Variation and Mean

14-3 Control Charts for Attributes

Statistical ProcessControl

5014_TriolaE/S_CH14pp732-757 11/22/05 6:17 AM Page 732

C H A P T E R P R O B L E M

Is the production of aircraft altimetersdangerous for those who fly?The Altigauge Manufacturing Company produces aircraftaltimeters, which provide pilots with readings of theirheights above sea level. The accuracy of altimeters is im-portant because pilots rely on them to maintain altitudeswith safe vertical clearance above mountains, towers, andtall buildings, as well as vertical separation from other air-craft. The accuracy of altimeters is especially importantwhen pilots fly approaches to landing while not being ableto see the ground. In the past, pilots and passengers havebeen killed in crashes caused by wrong altimeter readingsthat led pilots to believe that they were safely above theground when they were actually flying dangerously low.

Because aircraft altimeters are so critically importantto aviation safety, their accuracy is carefully controlledby government regulations. According to Federal Avia-tion Administration Regulation Part 43, Appendix E, analtimeter must give a reading with an error of no morethan 20 ft when tested for an altitude of 1000 ft.

At the Altigauge Manufacturing Company, four al-timeters are randomly selected from production on eachof 20 consecutive business days, and Table 14-1 lists theerrors (in feet) when they are tested in a pressure cham-ber that simulates an altitude of 1000 ft. On Day 1, forexample, the actual altitude readings for the four se-lected altimeters are 1002 ft, 992 ft, 1005 ft, and 1011 ft,so the corresponding errors (in feet) are 2, 28, 5, and 11.

In this chapter we will evaluate this altimeter manu-facturing process by analyzing the behavior of the er-rors over time. We will see how methods of statisticscan be used to monitor a manufacturing process withthe goal of identifying and correcting any serious prob-lems. In addition to helping companies stay in business,methods of statistics can positively affect our safety invery significant ways.

Table 14-1 Aircraft Altimeter Errors (in feet)

Day Error Mean Median Range Standard Deviation

1 2 28 5 11 2.50 3.5 19 7.942 25 2 6 8 2.75 4.0 13 5.743 6 7 21 28 1.00 2.5 15 6.984 25 5 25 6 0.25 0.0 11 6.085 9 3 22 22 2.00 0.5 11 5.236 16 210 21 28 20.75 24.5 26 11.817 13 28 27 2 0.00 22.5 21 9.768 25 24 2 8 0.25 21.0 13 6.029 7 13 22 213 1.25 2.5 26 11.32

10 15 7 19 1 10.50 11.0 18 8.0611 12 12 10 9 10.75 11.0 3 1.5012 11 9 11 20 12.75 11.0 11 4.9213 18 15 23 28 21.00 20.5 13 5.7214 6 32 4 10 13.00 8.0 28 12.9115 16 213 29 19 3.25 3.5 32 16.5816 8 17 0 13 9.50 10.5 17 7.3317 13 3 6 13 8.75 9.5 10 5.0618 38 25 25 5 8.25 0.0 43 20.3919 18 12 25 26 12.25 15.0 31 13.2820 227 23 7 36 9.75 15.0 63 27.22


734 Chapter 14 Statistical Process Control

14-1 OverviewIn Chapter 2 we noted that when describing, exploring, or comparing data sets,it is usually important to consider center, variation, distribution, outliers, andchanging characteristics over time. The main objective of this chapter is to ad-dress the last item: changing characteristics of data over time. When investigat-ing characteristics such as center and variation, it is important to know whetherwe are dealing with a stable population or one that is changing with the passageof time.

There is currently a strong trend toward trying to improve the quality ofAmerican goods and services, and the methods presented in this chapter are beingused by growing numbers of businesses. Evidence of the increasing importance ofquality is found in its greater role in advertising and the growing number of booksand articles that focus on the issue of quality. In many cases, job applicants (you?)have a definite advantage when they can tell employers that they have studiedstatistics and methods of quality control. This chapter will present some of the ba-sic tools commonly used to monitor quality.

Minitab, Excel, and other software packages include programs for automati-cally generating charts of the type discussed in this chapter, and we will includeseveral examples of such displays. Control charts are good examples of wonderfulgraphical devices that allow us to see and understand some property of data thatwould be more difficult or impossible to understand without graphs. The worldneeds more people who can construct and interpret important graphs, such as thecontrol charts described in this chapter.

Control Charts for Variation14-2 and MeanKey Concept The main objective of this section is to construct run charts, Rcharts, and charts so that we can monitor important features of data over time.We will use such charts to determine whether some process is statistically stable(or within statistical control).

The following definition formally describes the type of data that will be con-sidered in this chapter.

x

DefinitionProcess data are data arranged according to some time sequence. They aremeasurements of a characteristic of goods or services that result from somecombination of equipment, people, materials, methods, and conditions.

For example, Table 14-1 includes process data consisting of the measured error(in feet) in altimeter readings over 20 consecutive days of production. Each day,four altimeters were randomly selected and tested. Because the data in Table 14-1


14-2 Control Charts for Variation and Mean 735

are arranged according to the time at which they were selected, they are processdata. It is very important to recognize this point:

Important characteristics of process data can change over time.

In making altimeters, a manufacturer might use competent and well-trained per-sonnel along with good machines that are correctly calibrated, but if the personnelare replaced or the machines wear with use, the altimeters might begin to becomedefective. Companies have gone bankrupt because they unknowingly allowedmanufacturing processes to deteriorate without constant monitoring.

Run ChartsThere are various methods that can be used to monitor a process to ensure that theimportant desired characteristics don’t change—analysis of a run chart is onesuch method.

DefinitionA run chart is a sequential plot of individual data values over time. One axis(usually the vertical axis) is used for the data values, and the other axis (usu-ally the horizontal axis) is used for the time sequence.

Figure 14-1 Run Chart of Individual AltimeterErrors in Table 14-1

EXAMPLE Manufacturing Aircraft Altimeters Treatingthe 80 altimeter errors in Table 14-1 as a string of consecutive mea-surements, construct a run chart by using a vertical axis for the errors

and a horizontal axis to identify the order of the sample data.

SOLUTION Figure 14-1 is the Minitab-generated run chart for the data inTable 14-1. The vertical scale is designed to be suitable for altimeter errors

Minitab

continued



Interpreting Run Charts Only when a process is statistically stable can itsdata be treated as if they came from a population with a constant mean, standarddeviation, distribution, and other characteristics.

ranging from 227 ft to 38 ft, which are the minimum and maximum values inTable 14-1. The horizontal scale is designed to include the 80 values arrangedin sequence. The first point represents the first value of 2 ft, the second pointrepresents the second value of 28 ft, and so on.

In Figure 14-1, the horizontal scale identifies the sample number, so thenumber 20 indicates the 20th sample item. The vertical scale represents thealtimeter error (in feet). Now examine Figure 14-1 and try to identify anypatterns that jump out begging for attention. Figure 14-1 does reveal thisproblem: As time progresses from left to right, the heights of the pointsappear to show a pattern of increasing variation. See how the points at theleft fluctuate considerably less than the points farther to the right. The Fed-eral Aviation Administration regulations require errors less than 20 ft (orbetween 20 ft and 220 ft), so the altimeters represented by the points at theleft are okay, whereas several of the points farther to the right correspond toaltimeters not meeting the required specifications. It appears that themanufacturing process started out well, but deteriorated as time passed. Ifleft alone, this manufacturing process will cause the company to go out ofbusiness.

DefinitionA process is statistically stable (or within statistical control) if it has onlynatural variation, with no patterns, cycles, or unusual points.

Figure 14-2 consists of run charts illustrating typical patterns showing ways inwhich the process of filling 12-oz cola cans may not be statistically stable.

● Figure 14-2(a): There is an obvious upward trend that corresponds tovalues that are increasing over time. If the filling process were to followthis type of pattern, the cans would be filled with more and more colauntil they began to overflow, eventually leaving the employees swim-ming in cola.

● Figure 14-2(b): There is an obvious downward trend that corresponds tosteadily decreasing values. The cans would be filled with less and less colauntil they were extremely underfilled. Such a process would require a com-plete reworking of the cans in order to get them full enough for distributionto consumers.

● Figure 14-2(c): There is an upward shift. A run chart such as this one mightresult from an adjustment to the filling process, making all subsequent val-ues higher.

The Flynn Effect:Upward Trend inIQ Scores

A run chart or control chart of

IQ scores would reveal that

they exhibit an upward trend,

because IQ scores have been

steadily increasing since they

began to be used about 70 years

ago. The trend is worldwide,

and it is the same for different

types of IQ tests, even those

that rely heavily on abstract and

nonverbal reasoning with mini-

mal cultural influence. This up-

ward trend has been named the

Flynn effect, because political

scientist James R. Flynn dis-

covered the trend in his studies

of U.S. military recruits. The

amount of the increase is quite

substantial: Based on a current

mean IQ score of 100, it is esti-

mated that the mean IQ in 1920

would be about 77. The typical

student of today is therefore

brilliant when compared to his

or her great-grandparents. So

far, there is no generally ac-

cepted explanation for the

Flynn effect.



● Figure 14-2(d): There is a downward shift—the first few values are rela-tively stable, and then something happened so that the last several valuesare relatively stable, but at a much lower level.

● Figure 14-2(e): The process is stable except for one exceptionally highvalue. The cause of that unusual value should be investigated. Perhaps thecans became temporarily stuck and one particular can was filled twice in-stead of once.

● Figure 14-2(f): There is an exceptionally low value.

Figure 14-2

Processes That Are Not Statistically Stable

Minitab

(a) (b)

(c) (d )

(e) (f )

(g) (h)



DefinitionsRandom variation is due to chance; it is the type of variation inherent in anyprocess that is not capable of producing every good or service exactly thesame way every time.

Assignable variation results from causes that can be identified (such factorsas defective machinery, untrained employees, and so on).

Later in the chapter we will consider ways to distinguish between assignable vari-ation and random variation.

The run chart is one tool for monitoring the stability of a process. We willnow consider control charts, which are also extremely useful for that samepurpose.

Control Chart for Monitoring Variation: The R ChartIn the article “The State of Statistical Process Control as We Proceed into the21st Century” (Stoumbos, Reynolds, Ryan, and Woodall, Journal of the Ameri-can Statistical Association, Vol. 95, No. 451), the authors state that “controlcharts are among the most important and widely used tools in statistics. Theirapplications have now moved far beyond manufacturing into engineering, en-vironmental science, biology, genetics, epidemiology, medicine, finance, andeven law enforcement and athletics.” We begin with the definition of a controlchart.

● Figure 14-2(g): There is a cyclical pattern (or repeating cycle). This pat-tern is clearly nonrandom and therefore reveals a statistically unstable pro-cess. Perhaps periodic overadjustments are being made to the machinery,with the effect that some desired value is continually being chased butnever quite captured.

● Figure 14-2(h): The variation is increasing over time. This is a commonproblem in quality control. The net effect is that products vary more andmore until almost all of them are worthless. For example, some cola canswill be overflowing with wasted cola, and some will be underfilled and un-suitable for distribution to consumers.

A common goal of many different methods of quality control is this:reduce variation in the product or service. For example, Ford became con-cerned with variation when it found that its transmissions required signifi-cantly more warranty repairs than the same type of transmissions made byMazda in Japan. A study showed that the Mazda transmissions had substan-tially less variation in the gearboxes; that is, crucial gearbox measurementsvaried much less in the Mazda transmissions. Although the Ford transmissionswere built within the allowable limits, the Mazda transmissions were more reli-able because of their lower variation. Variation in a process can result from twotypes of causes.

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We will assume that the population standard deviation s is not known as weconsider only two of several different types of control charts: (1) R charts (orrange charts) used to monitor variation and (2) charts used to monitor means.When using control charts to monitor a process, it is common to consider R chartsand charts together, because a statistically unstable process may be the result ofincreasing variation or changing means or both.

An R chart (or range chart) is a plot of the sample ranges instead of individ-ual sample values, and it is used to monitor the variation in a process. (It mightmake more sense to use standard deviations, but range charts are used more oftenin practice. This is a carryover from times when calculators and computers werenot available. See Exercise 17 for a control chart based on standard deviations.) Inaddition to plotting the range values, we include a centerline located at , whichdenotes the mean of all sample ranges, as well as another line for the lower controllimit and a third line for the upper control limit. Following is a summary of nota-tion for the components of the R chart.

R

x

x

DefinitionA control chart of a process characteristic (such as mean or variation) con-sists of values plotted sequentially over time, and it includes a centerline aswell as a lower control limit (LCL) and an upper control limit (UCL). Thecenterline represents a central value of the characteristic measurements,whereas the control limits are boundaries used to separate and identify anypoints considered to be unusual.

NotationGiven: Process data consisting of a sequence of samples all of the same size n,and the distribution of the process data is essentially normal.n 5 size of each sample, or subgroup

5 mean of the sample ranges (that is, the sum of the sample ranges divided bythe number of samples)

R

Monitoring Process Variation: Control Chart for RPoints plotted: Sample ranges

Centerline:

Upper control limit (UCL): D4 (where D4 is found in Table 14-2)

Lower control limit (LCL): D3 (where D3 is found in Table 14-2)R

R

R

The values of D4 and D3 were computed by quality-control experts, and they areintended to simplify calculations. The upper and lower control limits of D4 andD3 are values that are roughly equivalent to 99.7% confidence interval limits. Itis therefore highly unlikely that values from a statistically stable process wouldfall beyond those limits. If a value does fall beyond the control limits, it’s verylikely that the process is not statistically stable.

RR

Costly AssignableVariation

The Mars Climate Orbiter was

launched by NASA and sent

to Mars, but it was destroyed

when it flew too close to its

destination planet. The loss

was estimated at $125 million.

The cause of the crash was

found to be confusion be-

tween the use of units used for

calculations. Acceleration data

were provided in the English

units of pounds of force, but

the Jet Propulsion Laboratory

assumed that those units were

in metric “newtons” instead of

pounds. The thrusters of the

spacecraft subsequently pro-

vided wrong amounts of force

in adjusting the position of the

spacecraft. The errors caused

by the discrepancy were fairly

small at first, but the cumula-

tive error over months of the

spacecraft’s journey proved to

be fatal to its success.

In 1962, the rocket carry-

ing the Mariner 1 satellite

was destroyed by ground con-

trollers when it went off

course due to a missing minus

sign in a computer program.



Table 14-2 Control Chart Constants

n: Number of s RObservationsin Subgroup A2 A3 B3 B4 D3 D4

2 1.880 2.659 0.000 3.267 0.000 3.2673 1.023 1.954 0.000 2.568 0.000 2.5744 0.729 1.628 0.000 2.266 0.000 2.2825 0.577 1.427 0.000 2.089 0.000 2.1146 0.483 1.287 0.030 1.970 0.000 2.0047 0.419 1.182 0.118 1.882 0.076 1.9248 0.373 1.099 0.185 1.815 0.136 1.8649 0.337 1.032 0.239 1.761 0.184 1.816

10 0.308 0.975 0.284 1.716 0.223 1.77711 0.285 0.927 0.321 1.679 0.256 1.74412 0.266 0.886 0.354 1.646 0.283 1.71713 0.249 0.850 0.382 1.618 0.307 1.69314 0.235 0.817 0.406 1.594 0.328 1.67215 0.223 0.789 0.428 1.572 0.347 1.65316 0.212 0.763 0.448 1.552 0.363 1.63717 0.203 0.739 0.466 1.534 0.378 1.62218 0.194 0.718 0.482 1.518 0.391 1.60819 0.187 0.698 0.497 1.503 0.403 1.59720 0.180 0.680 0.510 1.490 0.415 1.58521 0.173 0.663 0.523 1.477 0.425 1.57522 0.167 0.647 0.534 1.466 0.434 1.56623 0.162 0.633 0.545 1.455 0.443 1.55724 0.157 0.619 0.555 1.445 0.451 1.54825 0.153 0.606 0.565 1.435 0.459 1.541

Source: Adapted from ASTM Manual on the Presentation of Data and Control Chart Analysis,© 1976 ASTM, pp. 134–136. Reprinted with permission of American Society for Testing andMaterials.

x

Don’t Tamper!

Nashua Corp. had trouble with

its paper-coating machine and

considered spending a million

dollars to replace it. The ma-

chine was working well with a

stable process, but samples

were taken every so often and,

based on the results, adjust-

ments were made. These over-

adjustments, called tampering,

caused shifts away from the

distribution that had been

good. The effect was an in-

crease in defects. When statis-

tician and quality expert W.

Edwards Deming studied the

process, he recommended that

no adjustments be made unless

warranted by a signal that the

process had shifted or had be-

come unstable. The company

was better off with no adjust-

ments than with the tampering

that took place.EXAMPLE Manufacturing Aircraft Altimeters Refer tothe altimeter errors in Table 14-1. Using the samples of size n 5 4collected each day of manufacturing, construct a control chart for R.

SOLUTION We begin by finding the value of , the mean of the sample ranges.

The centerline for our R chart is therefore located at 5 21.2. To find the upperand lower control limits, we must first find the values of D3 and D4. Referring to

R

R 519 1 13 1 c 1 63

205 21.2

R



Interpreting Control ChartsWhen interpreting control charts, the following point is extremely important:

Upper and lower control limits of a control chart are based on theactual behavior of the process, not the desired behavior. Upper andlower control limits are totally unrelated to any process specificationsthat may have been decreed by the manufacturer.

When investigating the quality of some process, there are typically two key ques-tions that need to be addressed:

1. Based on the current behavior of the process, can we conclude that the pro-cess is within statistical control?

2. Do the process goods or services meet design specifications?

The methods of this chapter are intended to address the first question, but not thesecond. That is, we are focusing on the behavior of the process with the objectiveof determining whether the process is within statistical control. Whether the pro-cess results in goods or services that meet some stated specifications is anotherissue not addressed by the methods of this chapter. For example, the precedingMinitab R chart includes upper and lower control limits of 48.36 and 0, whichresult from the sample values listed in Table 14-1. Government regulations

Table 14-2 for n 5 4, we get D3 5 0.000 and D4 5 2.282, so the control limitsare as follows:

Upper control limit: D4 5 (2.282)(21.2) 5 48.4Lower control limit: D3 5 (0.000)(21.2) 5 0.0

Using a centerline value of 5 21.2 and control limits of 48.4 and 0.0, wenow proceed to plot the sample ranges. The result is shown in the Minitabdisplay.

R

RR

Minitab



require that altimeters have errors between 220 ft and 20 ft, but those desired (orrequired) specifications are not included in the control chart for R.

Also, we should clearly understand the specific criteria for determiningwhether a process is in statistical control (that is, whether it is statistically stable).So far, we have noted that a process is not statistically stable if its pattern resemblesany of the patterns shown in Figure 14-2. This criterion is included with some oth-ers in the following list.

Criteria for Determining When a Process Is Not Statistically Stable(Out of Statistical Control)

1. There is a pattern, trend, or cycle that is obviously not random (such as thosedepicted in Figure 14-2).

2. There is a point lying outside of the region between the upper and lower con-trol limits. (That is, there is a point above the upper control limit or below thelower control limit.)

3. Run of 8 Rule: There are eight consecutive points all above or all below thecenterline. (With a statistically stable process, there is a 0.5 probability that apoint will be above or below the centerline, so it is very unlikely that eightconsecutive points will all be above the centerline or all below it.)

We will use only the three out-of-control criteria listed above, but some businessesuse additional criteria such as these:

● There are six consecutive points all increasing or all decreasing.

● There are 14 consecutive points all alternating between up and down (suchas up, down, up, down, and so on).

● Two out of three consecutive points are beyond control limits that are 2standard deviations away from the centerline.

● Four out of five consecutive points are beyond control limits that are 1 stan-dard deviation away from the centerline.

EXAMPLE Statistical Process Control Examine the R chartshown in the Minitab display for the preceding example and deter-mine whether the process variation is within statistical control.

SOLUTION We can interpret control charts for R by applying the three out-of-control criteria just listed. Applying the three criteria to the Minitab displayof the R chart, we conclude that variation in this process is out of statisticalcontrol. There are not eight consecutive points all above or all below the cen-terline, so the third condition is not violated, but the first two conditions areviolated.

1. There is a pattern, trend, or cycle that is obviously not random: Going fromleft to right, there is a pattern of upward trend, as in Figure 14-2(a).

2. There is a point (the rightmost point) that lies above the upper control limit.



Control Chart for Monitoring Means: The ChartAn chart is a plot of the sample means, and it is used to monitor the center in aprocess. In addition to plotting the sample means, we include a centerline locatedat , which denotes the mean of all sample means (equal to the mean of all samplevalues combined), as well as another line for the lower control limit and a thirdline for the upper control limit. Using the approach common in business and in-dustry, the centerline and control limits are based on ranges instead of standarddeviations. See Exercise 18 for an chart based on standard deviations.x

x

x

x

Monitoring Process Mean: Control Chart for Points plotted: Sample means

Centerline: 5 mean of all sample means

Upper control limit (UCL): 1 A2 (where A2 is found in Table 14-2)

Lower control limit (LCL): 2 A2 (where A2 is found in Table 14-2)Rx

Rx

x

x

continued

Bribery Detectedwith Control Charts

Control charts were used to

help convict a person who

bribed Florida jai alai players

to lose. (See “Using Control

Charts to Corroborate

Bribery in Jai Alai,” by

Charnes and Gitlow, The

American Statistician, Vol.

49, No. 4.) An auditor for one

jai alai facility noticed that

abnormally large sums of

money were wagered for cer-

tain types of bets, and some

contestants didn’t win as

much as expected when those

bets were made. R charts and

charts were used in court as

evidence of highly unusual

patterns of betting. Examina-

tion of the control charts

clearly shows points well be-

yond the upper control limit,

indicating that the process of

betting was out of statistical

control. The statistician was

able to identify a date at

which assignable variation

appeared to stop, and prose-

cutors knew that it was the

date of the suspect’s arrest.

x

INTERPRETATION We conclude that the variation (not necessarily themean) of the process is out of statistical control. Because the variation appearsto be increasing with time, immediate corrective action must be taken to fix thevariation among the altimeter errors.

EXAMPLE Manufacturing Aircraft Altimeters Refer tothe altimeter errors in Table 14-1. Using samples of size n 5 4 col-lected each working day, construct a control chart for . Based on the

control chart for only, determine whether the process mean is within statisti-cal control.

SOLUTION Before plotting the 20 points corresponding to the 20 values of ,we must first find the value for the centerline and the values for the controllimits. We get

Referring to Table 14-2, we find that for n 5 4, A2 5 0.729. Knowing the val-ues of , A2, and , we can now evaluate the control limits.

Upper control limit: 1 A2 5 6.45 1 (0.729)(21.2) 5 21.9Lower control limit: 2 A2 5 6.45 2 (0.729)(21.2) 5 29.0

INTERPRETATION The resulting control chart for will be as shown inthe accompanying Excel display. Examination of the control chart showsthat the process mean is out of statistical control because at least one of thethree out-of-control criteria is not satisfied. Specifically, the third criterion

x

RxRx

Rx

R 519 1 13 1 . . . 1 63

205 21.2

x 52.50 1 2.75 1 . . . 1 9.75

205 6.45

x

xx



is not satisfied because there are eight (or more) consecutive points all be-low the centerline. Also, there does appear to be a pattern of an upwardtrend. Again, immediate corrective action is required to fix the productionprocess.

Using Technology

STATDISK See the STATDISK StudentLaboratory Manual and Workbook that is asupplement to this book.

MINITAB Run Chart: To construct arun chart, such as the one shown in Figure14-1, begin by entering all of the sample datain column C1. Select the option Stat, thenQuality Tools, then Run Chart. In the indi-cated boxes, enter C1 for the single column

variable, enter 1 for the subgroup size, andthen click on OK.

R Chart: First enter the individual samplevalues sequentially in column C1. Next, se-lect the options Stat, Control Charts, Vari-ables Charts for Subgroups, and R. EnterC1 in the data entry box, enter the samplesize in the box for the subgroup size, andclick on R Options, then estimate. SelectRbar. (Selection of the R bar estimatecauses the variation of the population dis-tribution to be estimated with the sampleranges instead of the sample standard devi-ations, which is the default.) Click OKtwice.

Chart: First enter the individual samplevalues sequentially in column C1. Next, se-lect the options Stat, Control Charts, Vari-ables Charts for Subgroups, and Xbar.Enter C1 in the data entry box, enter the size

of each of the samples in the “subgroupsizes” box. Click on Xbar Options, then se-lect estimate and choose the option ofRbar. Click OK twice.

EXCEL To use the Data Desk XL add-in, click on DDXL and select Process Con-trol. Proceed to select the type of chart youwant. (You must first enter the data in col-umn A with sample identifying codes enteredin column B. For the data of Table 14-1, for example, enter a 1 in column B ad-jacent to each value from day 1, enter a 2 foreach value from day 2, and so on.)

To use Excel’s built-in graphics featuresinstead of Data Desk XL, see the following:

Run Chart: Enter all of the sample data incolumn A. On the main menu bar, click onthe Chart Wizard icon, which looks like abar graph. For the chart type, select Line.For the chart subtype, select the first graph

x

Excel

continued



14-2 BASIC SKILLS AND CONCEPTSStatistical Literacy and Critical Thinking

1. Process Data What are process data?

2. Statistical Control What does it mean for a process to be out of statistical control?

3. Control Charts What is a control chart? What is an R chart? What is an chart? Whatis the difference between an R chart and an chart?

4. Variation What is the difference between random variation and assignable variation?

Interpreting Run Charts. In Exercises 5–8, examine the run chart from a process of fill-ing 12-oz cans of cola and do the following: (a) Determine whether the process is withinstatistical control; (b) if the process is not within statistical control, identify reasons whyit is not; (c) apart from being within statistical control, does the process appear to be be-having as it should?

5. 6.

xx

in the second row, then click Next. Continueto click Next, then Finish. The graph can beedited to include labels, delete grid lines,and so on.

R Chart: Step 1: Enter the sample data inrows and columns corresponding to the dataset. For example, enter the data in Table 14-1in four columns (A, B, C, D) and 20 rows asshown in the table.

Step 2: Next, create a column of therange values using the following procedure.Position the cursor in the first empty cell tothe right of the block of sample data, then en-ter this expression in the formula box: 5

MAX(A1:D1) 2 MIN(A1:D1), where therange A1:D1 should be modified to de-scribe the first row of your data set. Afterpressing the Enter key, the range for thefirst row should appear. Use the mouse toclick and drag the lower right corner of this

cell, so that the whole column fills up withthe ranges for the different rows.

Step 3: Next, produce a graph by follow-ing the same procedure described for therun charts, but be sure to refer to the columnof ranges when entering the input range.You can insert the required centerline andupper and lower control limits by editingthe graph. Click on the line on the bottom ofthe screen, then click and drag to position theline correctly.

Chart: Step 1: Enter the sample data inrows and columns corresponding to the dataset. For example, enter the data in Table 14-1in four columns (A, B, C, D) and 20 rows asshown in the table.

Step 2: Next, create a column of the sam-ple means using the following procedure.Position the cursor in the first empty cell tothe right of the block of sample data, then

enter this expression in the formula box: 5AVERAGE(A1:D1), where the range A1:D1should be modified to describe the first rowof your data set. After pressing the Enterkey, the mean for the first row should ap-pear. Use the mouse to click and drag thelower right corner of this cell, so that thewhole column fills up with the means forthe different rows.

Step 3: Next, produce a graph by follow-ing the same procedure described for the runchart, but be sure to refer to the column ofmeans when entering the input range. Youcan insert the required centerline and upperand lower control limits by editing the graph.Click on the line on the bottom of the screen,then click and drag to position the line cor-rectly. It’s not easy.

x



7. 8.

Constructing Control Charts for Aluminum Cans. Exercises 9 and 10 are based on theaxial loads (in pounds) of aluminum cans that are 0.0109 in. thick, as listed in Data Set15 in Appendix B. An axial load of a can is the maximum weight supported by its side,and it is important to have an axial load high enough so that the can isn’t crushed whenthe top lid is pressed into place. The data are from a real manufacturing process, andthey were provided by a student who used an earlier edition of this book.

9. R Chart In each day of production, seven aluminum cans with thickness 0.0109 in.were randomly selected and the axial loads were measured. The ranges for the differ-ent days are listed below, but they can also be found from the values given in Data Set15 in Appendix B. Construct an R chart and determine whether the process variationis within statistical control. If it is not, identify which of the three out-of-control crite-ria lead to rejection of statistically stable variation.

78 77 31 50 33 38 84 21 38 77 26 78 78

17 83 66 72 79 61 74 64 51 26 41 31

10. Chart In each day of production, seven aluminum cans with thickness 0.0109 in.were randomly selected and the axial loads were measured. The means for the differ-ent days are listed below, but they can also be found from the values given in Data Set15 in Appendix B. Construct an chart and determine whether the process mean iswithin statistical control. If it is not, identify which of the three out-of-control criterialead to rejection of statistically stable variation.

252.7 247.9 270.3 267.0 281.6 269.9 257.7 272.9 273.7 259.1 275.6 262.4 256.0

277.6 264.3 260.1 254.7 278.1 259.7 269.4 266.6 270.9 281.0 271.4 277.3

Monitoring the Minting of Quarters. In Exercises 11–13, use the following informa-tion: The U.S. Mint has a goal of making quarters with a weight of 5.670 g, but anyweight between 5.443 g and 5.897 g is considered acceptable. A new minting machineis placed into service and the weights are recorded for a quarter randomly selectedevery 12 min for 20 consecutive hours. The results are listed in the accompanyingtable.

11. Minting Quarters: Run Chart Construct a run chart for the 100 values. Does there ap-pear to be a pattern suggesting that the process is not within statistical control? Whatare the practical implications of the run chart?

12. Minting Quarters: R Chart Construct an R chart and determine whether the processvariation is within statistical control. If it is not, identify which of the three out-of-control criteria lead to rejection of statistically stable variation.

13. Minting Quarters: Chart Construct an chart and determine whether the processmean is within statistical control. If it is not, identify which of the three out-of-controlcriteria lead to rejection of a statistically stable mean. Does this process need correc-tive action?

xx

x

x



Weights (in grams) of Minted Quarters

Hour Weight (g) s Range

1 5.639 5.636 5.679 5.637 5.691 5.6564 0.0265 0.0552 5.655 5.641 5.626 5.668 5.679 5.6538 0.0211 0.0533 5.682 5.704 5.725 5.661 5.721 5.6986 0.0270 0.0644 5.675 5.648 5.622 5.669 5.585 5.6398 0.0370 0.0905 5.690 5.636 5.715 5.694 5.709 5.6888 0.0313 0.0796 5.641 5.571 5.600 5.665 5.676 5.6306 0.0443 0.1057 5.503 5.601 5.706 5.624 5.620 5.6108 0.0725 0.2038 5.669 5.589 5.606 5.685 5.556 5.6210 0.0545 0.1299 5.668 5.749 5.762 5.778 5.672 5.7258 0.0520 0.110

10 5.693 5.690 5.666 5.563 5.668 5.6560 0.0534 0.13011 5.449 5.464 5.732 5.619 5.673 5.5874 0.1261 0.28312 5.763 5.704 5.656 5.778 5.703 5.7208 0.0496 0.12213 5.679 5.810 5.608 5.635 5.577 5.6618 0.0909 0.23314 5.389 5.916 5.985 5.580 5.935 5.7610 0.2625 0.59615 5.747 6.188 5.615 5.622 5.510 5.7364 0.2661 0.67816 5.768 5.153 5.528 5.700 6.131 5.6560 0.3569 0.97817 5.688 5.481 6.058 5.940 5.059 5.6452 0.3968 0.99918 6.065 6.282 6.097 5.948 5.624 6.0032 0.2435 0.65819 5.463 5.876 5.905 5.801 5.847 5.7784 0.1804 0.44220 5.682 5.475 6.144 6.260 6.760 6.0642 0.5055 1.285

x

Appendix B Data Set: Constructing Control Charts for Boston Rainfall. In Exercises14–16, refer to the daily amounts of rainfall in Boston for one year, as listed in Data Set10 in Appendix B. Omit the last entry for Wednesday so that each day of the week has ex-actly 52 values.

14. Boston Rainfall: Constructing a Run Chart Using only the 52 rainfall amounts forMonday, construct a run chart. Does the process appear to be within statistical control?

15. Boston Rainfall: Constructing an R Chart Using the 52 samples of seven values each,construct an R chart and determine whether the process variation is within statisticalcontrol. If it is not, identify which of the three out-of-control criteria lead to rejectionof statistically stable variation.

16. Boston Rainfall: Constructing an Chart Using the 52 samples of seven values each,construct an chart and determine whether the process mean is within statistical con-trol. If it is not, identify which of the three out-of-control criteria lead to rejection of astatistically stable mean. If not, what can be done to bring the process within statisti-cal control?

14-2 BEYOND THE BASICS

17. Constructing an s Chart In this section we described control charts for R and basedon ranges. Control charts for monitoring variation and center (mean) can also bebased on standard deviations. An s chart for monitoring variation is made by plottingsample standard deviations with a centerline at (the mean of the sample standard de-viations) and control limits at B4 and B3 , where B4 and B3 are found in Table 14-2.Construct an s chart for the data of Table 14-1. Compare the result to the R chart givenin this section.

sss

x

xx



18. Constructing an Chart Based on Standard Deviations An chart based on standarddeviations (instead of ranges) is made by plotting sample means with a centerline at and control limits at 1 A3 and 2 A3 , where A3 is found in Table 14-2 and is themean of the sample standard deviations. Use the data in Table 14-1 to construct an chart based on standard deviations. Compare the result to the chart based on sampleranges (shown in this section).

14-3 Control Charts for AttributesKey Concept This section presents a method for constructing a control chart tomonitor the proportion p for some attribute, such as whether a service or manu-factured item is defective or nonconforming. (A good or a service is nonconform-ing if it doesn’t meet specifications or requirements; nonconforming goods aresometimes discarded, repaired, or called “seconds” and sold at reduced prices.)The control chart is interpreted by using the same three criteria from Section 14-2to determine whether the process is statistically stable.

Section 14-2 discussed control charts for quantitative data, but this section de-scribes the construction of control charts for qualitative data. As in Section 14-2,we select samples of size n at regular time intervals and plot points in a sequentialgraph with a centerline and control limits. (There are ways to deal with samples ofdifferent sizes, but we don’t consider them here.)

xx

ssxsxx

xx

DefinitionA control chart for p (or p chart) is a graph of proportions plotted sequen-tially over time, and it includes a centerline, a lower control limit (LCL), andan upper control limit (UCL).

The notation and control chart values are as follows (where the attribute of“defective” can be replaced by any other relevant attribute).

Notation5 pooled estimate of the proportion of defective items in the process

5 pooled estimate of the proportion of process items that are not defective

n 5 size of each sample (not the number of samples)

5 1 2 p

q

5total number of defects found among all items sampled

total number of items sampled

p

Quality Controlat Perstorp

Perstorp Components, Inc. uses

a computer that automatically

generates control charts to

monitor the thicknesses of the

floor insulation the company

makes for Ford Rangers and

Jeep Grand Cherokees. The

$20,000 cost of the computer

was offset by a first-year

savings of $40,000 in labor,

which had been used to manu-

ally generate control charts to

ensure that insulation thick-

nesses were between the

specifications of 2.912 mm

and 2.988 mm. Through the

use of control charts and other

quality-control methods,

Perstorp reduced its waste by

more than two-thirds.


14-3 Control Charts for Attributes 749

We use for the centerline because it is the best estimate of the proportion ofdefects from the process. The expressions for the control limits correspond to99.7% confidence interval limits as described in Section 7-2.

p

Control Chart for pCenterline:

Upper control limit:

Lower control limit:

(If the calculation for the lower control limit results in a negative value, use 0instead. If the calculation for the upper control limit exceeds 1, use 1 instead.)

p 2 3Åp q

n

p 1 3Åp q

n

p

EXAMPLE Defective Aircraft Altimeters The ChapterProblem describes the process of manufacturing aircraft altimeters.Section 14-2 includes examples of control charts for monitoring the

errors in altimeter readings. An altimeter is considered to be defective if it can-not be calibrated or corrected to give accurate readings (within 20 ft of the truealtitude). The Altigauge Manufacturing Company produces altimeters inbatches of 100, and each altimeter is tested and determined to be acceptable ordefective. Listed below are the numbers of defective altimeters in successivebatches of 100. Construct a control chart for the proportion p of defectivealtimeters and determine whether the process is within statistical control. Ifnot, identify which of the three out-of-control criteria apply.

Defects: 2 0 1 3 1 2 2 4 3 5 3 7

SOLUTION The centerline for the control chart is located by the value of

Because 5 0.0275, it follows that 5 12 5 0.9725. Using 5 0.0275,5 0.9725, and n 5 100, we find the control limits as follows:

Upper control limit:

Lower control limit:

p 2 3Åp q

n5 0.0275 2 3Å s0.0275ds0.9725d

1005 20.0216

p 1 3Åp q

n5 0.0275 1 3Å s0.0275ds0.9725d

1005 0.0766

qppqp

52 1 0 1 1 1 # # # 1 7

12 ? 1005

33

12005 0.0275

p 5total number of defects from all samples combined

total number of altimeters sampled

p:

continued

Six Sigma in Industry

Six Sigma is the term used in

industry to describe a process

that results in a rate of no

more than 3.4 defects out of a

million. The reference to Six

Sigma suggests six standard

deviations away from the

center of a normal distribu-

tion, but the assumption of a

perfectly stable process is

replaced with the assumption

of a process that drifts

slightly, so the defect rate is

no more than 3 or 4 defects

per million.

Started around 1985 at

Motorola, Six Sigma pro-

grams now attempt to

improve quality and increase

profits by reducing variation

in processes. Motorola saved

more than $940 million in

three years. Allied Signal

reported a savings of $1.5

billion. GE, Polaroid, Ford,

Honeywell, Sony, and Texas

Instruments are other major

companies that have adopted

the Six Sigma goal.



Because the lower control limit is less than 0, we use 0 instead. Having foundthe values for the centerline and control limits, we can proceed to plot the pro-portions of defective altimeters. The Minitab control chart for p is shown in theaccompanying display.

INTERPRETATION We can interpret the control chart for p by consideringthe three out-of-control criteria listed in Section 14-2. Using those criteria, weconclude that this process is out of statistical control for this reason: There ap-pears to be an upward trend. The company should take immediate action tocorrect the increasing proportion of defects.

Using Technology

MINITAB Enter the numbers of defects(or items with any particular attribute) incolumn C1. Select the option Stat, thenControl Charts, Attributes Charts, thenP. Enter C1 in the box identified as variable,and enter the size of the samples in the boxidentified as subgroup size, then click OK.

EXCEL Using DDXL: To use theDDXL add-in, begin by entering the num-bers of defects or successes in column A,and enter the sample sizes in column B. Forthe example of this section, the first threeitems would be entered in the Excel spread-sheet as shown below.

A B

1 2 1002 0 1003 1 100

Click on DDXL, select Process Control,then select Summ Prop Control Chart (forsummary proportions control chart). A dia-log box should appear. Click on the pencilicon for “Success Variable” and enter therange of values for column A, such as

A1:A12. Click on the pencil icon for “TotalsVariable” and enter the range of values forcolumn B, such as B1:B12. Click OK. Nextclick on the Open Control Chart bar andthe control chart will be displayed.

Using Excel’s Chart Wizard: Enter the sam-ple proportions in column A. Click on theChart Wizard icon, which looks like a bargraph. For the chart type, select Line. Forthe chart subtype, select the first graph in thesecond row, then click Next. Continue toclick Next, then Finish. The graph can beedited to include labels, delete grid lines,and so on. You can insert the required cen-terline and upper and lower control limits byediting the graph. Click on the line on thebottom of the screen, then click and drag toposition the line correctly.

STATISTICSIN THE NEWS

High Cost of LowQuality

The Federal Drug Adminis-tration recently reached anagreement whereby a phar-maceutical company, theSchering-Plough Corpora-tion, would pay a record$500 million for failure tocorrect problems in manu-facturing drugs. Accordingto a New York Times articleby Melody Petersen, “Someof the problems relate to thelack of controls that wouldidentify faulty medicines,while others stem from out-dated equipment. Theyinvolve some 200 medicines,including Claritin, the al-lergy medicine that is Scher-ing’s top-selling product.”

Minitab


14-3 Control Charts for Attributes 751

14-3 BASIC SKILLS AND CONCEPTSStatistical Literacy and Critical Thinking

1. p Chart What is a p chart, and what is its purpose?

2. Contract Specification The Paper Chase office supply company requires a supplier ofpens to maintain a production process with a defect rate less than 2%. If the supplieruses a p chart to determine that the manufacturing process is within statistical control,does that indicate that the defect rate is less than 2%?

3. Control Limits Using the methods of this section, a process is analyzed and the upperand lower control limits are found to be 0.250 and 20.050 respectively. What upperand lower control limits are used to construct the control chart?

4. Interpreting a Control Chart When monitoring the process of producing altimeters, acompany finds that the process is out of statistical control because there is a down-ward pattern of defects that is not random. Should the downward pattern be cor-rected? What should the company do?

Determining Whether a Process Is in Control. In Exercises 5–8, examine the given con-trol chart for p and determine whether the process is within statistical control. If it is not,identify which of the three out-of-control criteria apply.

5. 6.

7. 8.

Constructing Control Charts for p. In Exercises 9–12, use the given process data to con-struct a control chart for p. In each case, use the three out-of-control criteria listed inSection 14-2 and determine whether the process is within statistical control. If it is not,identify which of the three out-of-control criteria apply.

9. p Chart for Birth Rate In each of 10 consecutive and recent years, 10,000 peoplewere randomly selected and the numbers of births they generated were found, withthe results given below (based on data from the National Center for Health Statistics).How might the results be explained?

Births: 155 152 148 147 145 146 145 144 141 139

10. p Chart for Divorce Rate In each of 10 consecutive and recent years, 10,000 peoplewere randomly selected and the numbers of divorces were found, with the resultsgiven below (based on data from the National Center for Health Statistics). Howmight the results be explained?

Divorces: 48 46 46 44 43 43 42 41 42 40



11. p Chart for Defective Car Batteries Defective car batteries are a nuisance becausethey can strand and inconvenience drivers. A car battery is considered to be defec-tive if it fails before its warranty expires. Defects are identified when the batteriesare returned under the warranty program. The Powerco Battery corporation manu-factures car batteries in batches of 1000 and the numbers of defects are listedbelow for each of 12 consecutive batches. Does the manufacturing process requirecorrection?

Defects: 8 6 5 9 10 7 7 4 6 11 5 8

12. Polling When the Infopop polling organization conducts a telephone survey, a call isconsidered to be a defect if the respondent is unavailable or refuses to answer ques-tions. For one particular poll about consumer preferences, 200 people are called eachday, and the numbers of defects are listed below. Does the calling process require cor-rective action?

Defects: 92 83 85 87 98 108 96 115 121 125 112 127 109 131 130

14-3 BEYOND THE BASICS

13. p Chart for Boston Rainfall Refer to the Boston rainfall amounts in Data Set 10 inAppendix B. For each of the 52 weeks, let the sample proportion be the proportion ofdays that it rained. (Delete the 53rd value for Wednesday.) In the first week, for exam-ple, the sample proportion is 3 7 5 0.429. Do the data represent a statistically stableprocess?

14. Constructing an np Chart A variation of the control chart for p is the np chart inwhich the actual numbers of defects are plotted instead of the proportions of defects.The np chart will have a centerline value of , and the control limits will have valuesof 1 3 and 2 3 . The p chart and the np chart differ only in thescale of values used for the vertical axis. Construct the np chart for the example givenin this section. Compare the result with the control chart for p given in this section.

Review

In Chapter 2 we identified important characteristics of data: center, variation, distribution,outliers, and changing pattern of data over time. The focus of this chapter is the changingpattern of data over time. Process data were defined to be data arranged according to sometime sequence, and such data can be analyzed with run charts and control charts. Controlcharts have a centerline, an upper control limit, and a lower control limit. A process is sta-tistically stable (or within statistical control) if it has only natural variation with no pat-terns, cycles, or unusual points. Decisions about statistical stability are based on how aprocess is actually behaving, not how we might like it to behave because of such factors asmanufacturer specifications. The following graphs were described:

● Run chart: a sequential plot of individual data values over time● R chart: a control chart that uses ranges in an attempt to monitor the variation in a

process● chart: a control chart used to determine whether the process mean is within sta-

tistical control● p chart: a control chart used to monitor the proportion of some process attribute,

such as whether items are defective

x

!np qnp!np qnpnp

>


Review Exercises 753

Statistical Literacy and Critical Thinking

1. Pattern over Time Why is it important to monitor a changing pattern of data over time?

2. Statistical Process Control The title of this chapter is “Statistical Process Control.”What does that mean?

3. Manufacturing Run Amok What would be a possible adverse consequence of a Pepsibottling plant running a process that is not monitored?

4. Control Charts When monitoring the times it takes technicians to repair computers,why is it important to use an chart and an R chart together?

Review Exercises

Constructing Control Charts for Consumption of Electricity. The following table listsamounts of electrical consumption (in kWh) for the author’s home, as given in Data Set 9in Appendix B. Use the data for Exercises 1–3.

Electrical Consumption (kWh)

Year 1: 1st Half 3375 2661 2073Year 1: 2nd Half 2579 2858 2296Year 2: 1st Half 2812 2433 2266Year 2: 2nd Half 3128 3286 2749Year 3: 1st Half 3427 578 3792

1. Run Chart Construct a run chart for the 15 values. Does there appear to be a patternsuggesting that the process is not within statistical control?

2. R Chart Using subgroups of size n 5 3 corresponding to the rows of the table, con-struct an R chart and determine whether the process variation is within statistical con-trol. If it is not, identify which of the three out-of-control criteria lead to rejection ofstatistically stable variation.

3. Chart Using subgroups of size n 5 3 corresponding to the rows of the table, con-struct an chart and determine whether the process mean is within statistical control.Does the process appear to be statistically stable? If it is not, identify which of thethree out-of-control criteria lead to rejection of statistically stable variation.

4. Constructing a Control Chart for Infectious Diseases In each of 13 consecutive andrecent years, 100,000 adults 65 years of age or older were randomly selected andthe number who died from infectious diseases is recorded, with the results givenbelow (based on data from “Trends in Infectious Diseases Mortality in the UnitedStates,” by Pinner et al., Journal of the American Medical Association, Vol. 275,No. 3). Construct an appropriate control chart and determine whether the process iswithin statistical control. If not, identify which criteria lead to rejection of statisticalstability.

Number who died: 270 264 250 278 302 334 348 347 377 357 362 351 343

5. Control Chart for Defects The Medassist Pharmaceutical Company manufacturesaspirin tablets. Each day, 100 tablets are randomly selected and tested. A tablet isconsidered defective if it has obvious physical deformities or the aspirin content isless than 490 mg or greater than 510 mg. The numbers of defects are listed below for

xx

x


Cooperative Group Activities


consecutive days. Construct an appropriate control chart and determine whether theprocess is within statistical control. If not, identify which criteria lead to rejection ofstatistical stability.

Defects: 4 2 2 3 5 2 9 12 1 11 3 2 12 14

Cumulative Review Exercises

1. Control Chart for Defective Seat Belts The Flint Accessory Corporation manufac-tures seat belts for cars. Federal specifications require that the webbing must have abreaking strength of at least 5000 lb. During each week of production, 200 belts arerandomly selected and tested for breaking strength, and a belt is considered defectiveif it breaks before reaching the force of 5000 lb. The numbers of defects are listed be-low for a sequence of 10 weeks. Use a control chart for p to verify that the process iswithin statistical control. If it is not in control, explain why it is not.

6 4 12 3 7 2 3 5 4 2

2. Confidence Interval for Defective Seat Belts Refer to the data in Exercise 1 and, us-ing all of the data from the 2000 seat belts that were tested, construct a 95% confi-dence interval for the proportion of defects.

3. Hypothesis Test for Defective Seat Belts Refer to the data in Exercise 1 and, using allof the data from the 2000 seat belts that were tested, use a 0.05 significance level totest the claim that the rate of defects is greater than 1%.

4. Using Probability in Control Charts When interpreting control charts, one of thethree out-of-control criteria is that there are eight consecutive points all above or allbelow the centerline. For a statistically stable process, there is a 0.5 probability that apoint will be above the centerline and there is a 0.5 probability that a point will be be-low the centerline. In each of the following, assume that sample values are indepen-dent and the process is statistically stable.a. Find the probability that when eight consecutive points are randomly selected, they

are all above the centerline.b. Find the probability that when eight consecutive points are randomly selected, they

are all below the centerline.c. Find the probability that when eight consecutive points are randomly selected, they

are all above or all below the centerline.

1. Out-of-class activity Collect your own process dataand analyze them using the methods of this section. Itwould be ideal to collect data from a real manufactur-ing process, but that may be difficult to accomplish. Ifso, consider using a simulation or referring to publisheddata, such as those found in an almanac. Here are somesuggestions:● Shoot five basketball foul shots (or shoot five crum-

pled sheets of paper into a wastebasket) and recordthe number of shots made; then repeat this proce-

dure 20 times, and use a p chart to test for statisticalstability in the proportion of shots made.

● Your pulse rate can be measured by counting thenumber of times your heart beats in 1 min. Measureyour pulse rate four times each hour for severalhours, then construct appropriate control charts.What factors contribute to random variation?Assignable variation?

● Go through newspapers for the past 12 weeks andrecord the closing of the Dow Jones Industrial


Technology Project 755

Technology Project

a. Simulate the following process for 20 days: Eachday, 200 calculators are manufactured with a 5%rate of defects, and the proportion of defects isrecorded for each of the 20 days. The calculators forone day are simulated by randomly generating 200numbers, where each number is between 1 and 100.Consider an outcome of 1, 2, 3, 4, or 5 to be a defect,with 6 through 100 being acceptable. This corre-sponds to a 5% rate of defects. (See the technologyinstructions below.)

b. Construct a p chart for the proportion of defectivecalculators, and determine whether the process iswithin statistical control. Since we know the processis actually stable with p 5 0.05, the conclusion thatit is not stable would be a type I error; that is, wewould have a false positive signal, causing us to be-lieve that the process needed to be adjusted when infact it should be left alone.

c. The result from part(a) is a simulation of 20 days.Now simulate another 10 days of manufacturing cal-culators, but modify these last 10 days so that the de-fect rate is 10% instead of 5%.

d. Combine the data generated from parts(a) and (c) torepresent a total of 30 days of sample results. Con-struct a p chart for this combined data set. Is the pro-cess out of control? If we concluded that the processwas not out of control, we would be making a type IIerror; that is, we would believe that the process wasokay when in fact it should be repaired or adjustedto correct the shift to the 10% rate of defects.

Technology Instructions for Part(a):

STATDISK Select Data, Uniform Generator, and pro-ceed to generate 200 values with a minimum of 1 and amaximum of 100. Copy the data to the data window, thensort the values using the Data Tools button. Repeat this pro-cedure until results for 20 days have been simulated.

MINITAB Select Calc, Random Data, then Integer.Enter 200 for the number of rows of data, enter C1 as thecolumn to be used for storing the data, enter 1 for the mini-mum value, and enter 100 for the maximum value. Repeatthis procedure until results for 20 days have been simulated.

EXCEL Click on the fx icon on the main menu bar, thenselect the function category Math & Trig, followed byRANDBETWEEN. In the dialog box, enter 1 for bottomand 100 for top. A random value should appear in the firstrow of column A. Use the mouse to click and drag the lowerright corner of that cell, then pull down the cell to cover thefirst 200 rows of column A. When you release the mousebutton, column A should contain 200 random numbers. Youcan also click drag the lower right corner of the bottom cellby moving the mouse to the right so that you get 20 columnsof 200 numbers each. The different columns represent thedifferent days of manufacturing.

TI-83/84 PLUS Press the MATH key. Select PRB,then select the 5th menu item, randInt(, and proceed to en-ter 1, 100, 200; then press the ENTER key. Press STO andL1 to store the data in list L1. After recording the number ofdefects, repeat this procedure until results for 20 days havebeen simulated.

>

Average (DJIA) for each business day. Use run andcontrol charts to explore the statistical stability ofthe DJIA. Identify at least one practical consequenceof having this process statistically stable, and iden-tify at least one practical consequence of having thisprocess out of statistical control.

● Find the marriage rate per 10,000 population forseveral years. (See the Information Please Almanacor the Statistical Abstract of the United States.) As-sume that in each year 10,000 people were randomlyselected and surveyed to determine whether theywere married. Use a p chart to test for statistical sta-bility of the marriage rate. (Other possible rates:death, accident fatality, crime.)

Obtain a printed copy of computer results, and write areport summarizing your conclusions.

2. In-class activity If the instructor can distribute thenumbers of absences for each class meeting, groups ofthree or four students can analyze them for statisticalstability and make recommendations based on the con-clusions.

3. Out-of-class activity Conduct research to identifyDeming’s funnel experiment, then use a funnel and mar-bles to collect data for the different rules for adjustingthe funnel location. Construct appropriate controlcharts for the different rules of funnel adjustment. Whatdoes the funnel experiment illustrate? What do youconclude?



From Data to DecisionCritical Thinking: Are the axialloads within statistical control?

Is the process of manufacturingcans proceeding as it should?Exercises 9 and 10 in Section 14-2 used pro-cess data from a New York company thatmanufactures 0.0109-in. thick aluminumcans for a major beverage supplier. Refer toData Set 15 in Appendix B and conduct ananalysis of the process data for the cans thatare 0.0111 in. thick. The values in the data

set are the measured axial loads of cans, andthe top lids are pressed into place with pres-sures that vary between 158 lb and 165 lb.

Analyzing the ResultsBased on the given process data, should thecompany take any corrective action? Writea report summarizing your conclusions.Address not only the issue of statistical sta-bility, but also the ability of the cans to with-stand the pressures applied when the top lidsare pressed into place. Also compare thebehavior of the 0.0111-in. cans to the behav-

ior of the 0.0109-in. cans and recommendwhich thickness should be used.

Control Charts

This chapter introduces different charting tech-

niques used to summarize and study data asso-

ciated with a process along with methods for

analyzing the stability of that process. With the

exception of the run chart, individual data

points are not needed to construct a chart. For

example, the R chart is constructed from sample

ranges while the p chart is based on sample

proportions. This is an important point, as data

collected from third-party sources are often

given in terms of summarizing statistics. Go to

the Elementary Statistics Web site:

http://www.aw.com/Triola.

Locate the Internet Project dealing with control

charts. There you will be directed to data sets

and sources of data for use in constructing con-

trol charts. From the resulting charts you will be

asked to interpret and discuss trends in the

underlying processes.

Internet Project


Statistics @ Work 757

“There is a certainamount of respect thatis given to someone whoknows statistics andcan explain it tosomeone who doesn’tknow it.”

Dan O’Toole

Account Executive: A. C. Nielsen

In his work in the Advanced Ana-lytics Group at A. C. Nielsen, Dandevelops statistical solutions tohelp clients like Polaroid, OceanSpray, and Gillette understandwhich of their marketing vehiclesdrives sales most profitably. Danhas a Masters Degree in BusinessEconomics from Bentley College.

Statistics @ WorkWhat concepts of statistics do

you use?

I have worked with analyses as simple ascorrelation and general significancetests, all the way to multiple regression,factor analysis, correspondence analysis,and cluster analysis.

How do you use statistics

on the job?

My job is to discover or uncover client is-sues, and then find out if we can applyone of our statistical techniques to theirspecific issue. If a technique won’t help aclient, then you need to know that. Anexample of how I use stats: A client maysay, I sell product “X,” whether it is juice,bread, or a camera. Right now, they maycontrol 20% of the market. They maycome to us to see if they can increasemarket share by lowering their price. Myjob would be to design a study to ana-lyze this question. To do this I have todesign a study that will take into accounteverything that affects the sales of aproduct. Using techniques like regres-sion, if I am able to create a model withgood significance, I will be able to isolatespecific influences on the sales and offerrecommendations. Things like seasonal-ity distribution as well as any marketingefforts that may have taken place, mustbe included. In addition, I will have totake into account complementary prod-ucts’ price (butter is complementary tobread; while film is for a camera) andalso competitive products. For instance,

bread may compete with English muffins(I know it does for me).

Do you feel job applicants are viewed

more favorably if they have studied

some statistics?

By far. There is a certain amount of re-spect that is given to someone whoknows statistics and can explain it tosomeone who doesn’t know it (because itmeans you really know it and aren’t recit-ing from a textbook). Almost every jobuses statistics (particularly correlationsand regressions). People will say thingslike, “Oh, check if they’re correlated.”

Is your use of probability and statis-

tics increasing, decreasing, or re-

maining stable?

It definitely is increasing. In this busi-ness (consulting), you are constantlychallenged to learn a new technique orlook at an old technique in order toimprove it. In addition, since we areconstantly coming out with new prod-ucts, our understanding of statistics hasto increase to use these techniqueseffectively.

How beneficial do you find your

knowledge of statistics for perform-

ing your responsibilities?

It is not a question of beneficial, butrather it is a necessity. In fact, we findthat we have to know it so well, so thatwe can explain it in “layman’s” terms toour clients.


TriolaE S CH14pp732 757

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