CHAPTER 6: PROCESS CAPABILITY ANALYSIS STATE OF CONTROL Process in Control When the assignable causes have been eliminated from the process to the extent the points plotted on the control chart remain within the control limits, the process is in a state of control. No higher degree of uniformity can be attained with the existing process. Greater uniformity can, however can, however, be attained through a change in the basic process through quality improvement ideas. When a process is in control, there occurs a normal pattern of variation which is illustrated in Figure 3-7. This natural pattern of variation has (1) about two thirds of the points near the central line, (2) a few points closer to the control limits, (3) points located back and forth across the central line, (4) points balanced on both sides of the central line, and (5) no points beyond the control limits. The natural pattern of the points or subgroup values forms its own frequency distribution, which follows or subgroup values forms its own frequency distribution, which follows a normal curve. As the number of plotted points increases, the frequency distribution will take on the appearance of a smooth polygon. The dashed normal curve at the left of Figure 3-7 represents the distribution of the points when a process is in control.
Figure 3-7 Natural pattern of variation of a control chart. Control limits are usually established at three standard deviations from the central line. They are used as a basis to judge whether there is evidence of lack of control. The choice of 3σ limits is an economic one with respect to two types of errors that can occur. One error, called Type I by statisticians, occurs when looking for an assignable cause of variation when in reality a chance cause is present. When the limits are set at three standard deviations, a Type I error will occur 0.27% (3 out of 1,000) of the time. In other words, when a point is outside the control limits, it is assumed to be due to an assignable cause even though it would be due to a chance cause 0.27% of the time. The other type error, called T ype II, occurs when assuming that a chance cause of variation is present when in reality there is an assignable cause. In other words, when a point is inside the control limits, it assumed to be due to a chance cause even though it might be an assignable cause. Abundant experience since 1930 in all types of industry indicates that 3σ limits provide an economic balance between the costs resulting from the two types of errors. Unless there are strong practical reasons for doing otherwise, the ±3 standard deviation limits should be used. When a process is in control, only chance causes of variation are present. Small variations in machine performance, operator performance, and material characteristics are expected and are considered to be part of a sta ble process. When a process is in control, certain practical advantages accrue to the producer and consumer. 1. Individual units of the product will be more uniform-or, stated another way, there will be less variation. 2. Since the product is more uniform, fewer samples are needed to judge the quality. Therefore, the cost of inspection can be reduced to a minimum. This advantage is extremely important when 100% conformance to specifications in not essential.
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3. The process capability or spread of the process is easily attained from 6σ. With a knowledge of the process capability, a number of reliable decisions relative to specifications can be made, such as: a. To decide the product specifications. b. To decide the amount of rework or scrap when there is insuffici ent tolerance. c. To decide whether to produce the product to tight specifications and permit interchangeability of components or to produce the product to loose specifications and use selective matching of components. 4. The percentage of product that falls within any pair of values may be predicted with the highest degree of assurance. For example, this advantage can be very important when adjusting filling machines to obtain different percentage of items below, between, or above particular values. 5. It permits the consumer as a check on the producer’s data and, therefore, to test only a few subgroups as a check on the producer’s records. 6. The operator is performing satisfactorily from a quality viewpoint. Process Out of Control When a point (subgroup value) falls outside its control limits, the process is out of control. This means that an assignable cause of variation is present. Another way of viewing the outof-control point is to think of the subgroup value as coming from a different population than the one from which the control limits were obtained. Figure 3-8 shows a frequency distribution of subgroup averages for cereal boxes which was developed from a large number of subgroups and, therefore, represents the population mean, μ = 450 g, and the population standard deviation for the averages, = 8 g. the frequency distribution for subgroup x
averages is shown by a dashed line, which represents a smooth polygon. For instructional purposes the individual dots represents the number of subgroup averages at particular values. Future explanations will use only the dashed line to represent the frequency distribution of averages and will use a solid line for the frequency distribution of individual values. The outof-control point has a value of 483 g. This point is so far away from the 3σ limits (99.73%) that it can only be considered to have come from another population. In other words, the process that produced the subgroup average of 483 g is a different process than the stable process from which the 3σ control limits were developed. Therefore, something has gone wrong with the process; some assignable cause of variation is present. This assignable cause must be found and corrected before a normal, stable process can continue. A process can also be considered out of control even the point’s fall inside the 3σ limits. This situation occurs when unnatural patterns of variation are present in the process. It is not normal for seven or more consecutive points to be above or below the central line. Also when 10 out of 11 points or 12 out of 14 points, etc., are located on one side of the central value, it is an unnatural pattern. These unnatural patterns are shown in Figure 3-9. The chance that these unnatural patterns or runs will occur is the same chance that a point will fall outside the 3σ control limits.
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Figure 3-8 Frequency distribution of subgroup averages with control limits.
Figure 3-9 Some unnatural patterns of variation-process out of control. There are many other unnatural patterns, and it is important to remember the conditions of normality described in the preceding section. One technique for recognizing unnatural patterns is to divide the control chart into six equal imaginary bands. Three equals bands are between the central line, and the lower control limit and three equal bands are between the central line and the upper control limit. A normal pattern of variation occurs when (1) about 34% of the points fall in each of the two bands adjacent to the center value, (2) about 13½ % of the points fall in each of the two middle bands, and (3) 2½ % of the points fall in each of the three outer bands. Any significant divergence from the normal pattern, such as 2 out of 3 consecutive points in the o uter band, would be an unnatural pattern and would be classified as an out-of-control condition. Analysis of Out-of-Control Condition When a process is out of control, the assignable cause responsible for the condition must be found. The detective work necessary to locate t he cause of the out-of-control condition can be minimized by a knowledge of the types of out-of-control patterns and their assignable causes.
Types of out-of-control X and R patterns are (1) change or jump in level, (2) trend or steady change in level, (3) recurring cycles, (4) two populations, and (5) mistakes. 1. Change or jump in level. This type is connected with a sudden change in level to the X chart, to the R chart, or to both charts. Figure 3-10 illustrates the change in level. For an X chart, the change in the process average can be due to: (a) An intentional or unintentional change in the process setting. (b) A new or inexperienced operator. (c) A different raw material.
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(d) A minor failure of a machine part. Some causes for a sudden change in the process spread or variability as shown on the R chart are: (a) Inexperienced operator. (b) Sudden increase in gear play. (c) Greater variation in incoming material. Sudden changes in level can occur on both the X and the R charts. This situation is common during the beginning of control chart activity prior to the attainment of a state of control. There may be more than one assignable cause, or it may be a cause that could affect both charts, such as an inexperienced operator.
Figure 3-10 Out-of-control pattern: change or jump in level. 2. Trend or steady change in level. Steady changes in control chart level are a very common industrial phenomena. Figure 3-11 illustrates a trend or steady change that is occurring in the upward direction; the trend could have been illustrated in the downward direction. Some causes of steady progressive changes on an X chart are: (a) Tool or die wear (b) Gradual deterioration of equipment (c) Gradual change in temperature or humidity (d) Viscosity in a chemical process (e) Buildup of chips in a work-holding device A steady change in level or trend on the R chart is not as common as the X chart. It does, however, occur and some possible causes are: (a) An improvement in worker skill (downward trend) (b) A decrease in worker skill due to fatigue, boredom, inattention, and so on (c) A gradual improvement in the homogeneity of incoming material.
Figure 3-11 Out-of-control pattern: trend or steady change in level. 3. Recurring cycles. When the plotted points on an X or R chart show a wave or periodic high and low points, it is called a cycle. A typical recurring out-of-control pattern is shown in Figure 3-12. For an X chart, some of the causes of recurring cycles are: (a) The seasonal effects of incoming material (b) Any daily or weekly chemical, mechanical, or psychological event (c) The periodic rotation of operators
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Periodic cycles on an R chart are not as common as for an X chart. Some affecting the R chart are due to: (a) Operator fatigue and rejuvenation (upgrading) resulting from morning, noon, and afternoon breaks (b) Lubrication cycles The out-of-control pattern of a recurring cycle sometimes goes unreported because of the inspection cycle. Thus, a cycle pattern of variation that occurs approximately every month 2 hours could coincide with the inspection frequency. Therefore, only the low points on the cycle are reported, and there is no evidence that a cyclic event is present.
Figure 3-12 Out-of-control pattern: recurring cycles. 4. Two populations. When there are a large number of points near or outside the control limits, a two-population situation may be present. This type of out-of-control pattern is illustrated in Figure 3-13. For an X chart the out-of-control pattern can be due to: (a) Large differences in material quality (b) Two or more machines on the same chart (c) Large differences in test method or equipment Some causes for an out-of-control pattern on an R chart are due to: (a) Different workers using the same chart (b) Materials from different suppliers.
Figure 3-13 Out-of-control pattern: two populations. 5. Mistakes. Mistakes can be very embarrassing to the quality assurance operation. Some causes of out-of-control patterns resulting from mistakes are: (a) Measuring equipment out of calibration (b) Errors in calculations (c) Errors in using test equipment (d) Taking samples from different populations Many of the out-of-control patterns that have been described can be attributed to some sort of inspection error. The causes given for the different types of out-of-control patterns are suggested possibilities and are not meant to be all-inclusive. These causes will give production and quality personnel ideas for the solution of particular industrial problems. They can be a start toward the development of an assignable cause checklist which is applicable to a particular manufacturing entity.
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When out-of-control patterns occur in relation to the lower control limits of the R chart, it is the result of outstanding performance. The cause should be determined so that the outstanding performance can continue. The preceding discussion has used the R chart as the measure of the dispersion. Information on patterns and causes also pertains to an s chart. SPECIFICATIONS Individual Values Compared to Averages Before discussing specifications and their relationship with control charts, it appears desirable, at this time, to obtain a better understanding of individual values and average values. Figure 3-14 shows a tally of the subgroup values or individual values ( X ’s) and a
tally of the subgroup averages ( X ’s) for the data on keyway widths given in Table 3-2. The four out-of-control subgroups were not used in the two tallys; therefore, there are 84 individual values and 21 averages. It is observed that the averages are grouped much closer to the center than the individual values. This is true because when we average four values, the affect of an extreme value is minimized, since the chance of four extremely high or four extremely low values in one subgroup is slight. Table 3-2 Data on the Depth of the Keyway (millimeters) Subgroup Measurements Average Range Date Time Comment Number R X 3 X 1 X 2 X 4 X 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Sum
23/12
27/12
28/12
29/12
30/12
8:50 11:30 1:45 3:45
6.35 6.46 6.34 6.69
6.40 6.37 6.40 6.64
6.32 6.36 6.34 6.68
6.33 6.41 6.36 6.59
6.35 6.40 6.36 6.65
0.08 0.10 0.06 0.10
4:20 8:35 9:00 9:40 1:30 2:50 8:30 1:35 2:25 2:35 3:55 8:25 9:25 11:00 2:35 3:15 9:35 10:20 11:35 2:00 4:25
6.38 6.42 6.44 6.33 6.48 6.47 6.38 6.37 6.40 6.38 6.50 6.33 6.41 6.38 6.33 6.56 6.38 6.39 6.42 6.43 6.39
6.34 6.41 6.41 6.41 6.52 6.43 6.41 6.37 6.38 6.39 6.42 6.35 6.40 6.44 6.32 6.55 6.40 6.42 6.39 6.36 6.38
6.44 6.43 6.41 6.38 6.49 6.36 6.39 6.41 6.47 6.45 6.43 6.29 6.29 6.28 6.37 6.45 6.45 6.35 6.39 6.35 6.43
6.40 6.34 6.46 6.36 6.51 6.42 6.38 6.37 6.35 6.42 6.45 6.39 6.34 6.58 6.38 6.48 6.37 6.40 6.36 6.38 6.44
6.39 6.40 6.43 6.37 6.50 6.42 6.39 6.38 6.40 6.41 6.45 6.34 6.36 6.42 6.35 6.51 6.40 6.39 6.39 6.38 6.44 160.25
0.10 0.09 0.05 0.08 0.04 0.11 0.03 0.04 0.12 0.07 0.08 0.10 0.12 0.30 0.06 0.11 0.08 0.07 0.06 0.08 0.06 2.19
New, temporary operator
Damaged oil line
Bad material
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Figure 3-14 Comparison of individual values and averages using the same data. Calculations of the average for both the individual values and for the subgroup
averages are the same, X = 38.9. However, the sample standard deviation of the individual values ( s ) is 4.16, while the sample standard deviation of the subgroup average ( s X ) is 2.77. If there are a large number of individual values and subgroup averages, the smooth polygons of Figure 3-14 would represent their frequency distribution if the distribution is normal. The curve for the frequency distribution of the averages has a dashed line while the curve for the frequency distribution of individual values has a solid line; this convention will be followed throughout the text. In comparing the two distributions it is observed that both distributions are normal in shape; in fact, even if the curve for individual values was not quite normal, the curve for averages would be close to a normal shape. The base of the curve for individual values is about twice as large as the base of the curve for averages. When population values are available for the standard deviation for individual values ( ) and for the standard deviation for averages ( X ), there is a definite relationship between them, as given by the formula X
n
, where
X
= population standard deviation of subgroup
averages ( X ’ s ), = population standard deviation of individual values, n = subgroup size. Thus, for a subgroup of size 5, X =0.45 , and for a subgroup of size 4, X =0.50 . If we assume normality (which may or may not be true), the population standard
deviation can be estimated from
s
c ,where 4
is the “estimate” of the population
standard deviation and =
4.16 0.996997 =4.17
is “appr oximately equal to ( ) 0.996997 for 4
c
and
X
n
= 4.17
4 =2.09.
n
=84. Thus,
s c
4
Note that s X , which was calculated
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from sample data, and
X
which was calculated above, are different. This difference is due
to sample variation or the small number of samples, which was only 21, or some combination thereof. The difference would not be caused by a non-normal population of X ’s. Since the height of the curve is a function of the frequency, the curve for individual values is higher. This is easily verified by comparing the tally sheet in Figure 3-14 and is a true relationship when making comparisons using frequencies from the sample data. However, if the curves represent relative or percentage frequency distributions, then the area under the curve must be equal to 100%. Therefore, the percentage frequency distribution curve for averages, with its smaller base, would need to be much higher to enclose the same area as the percentage frequency distribution curve for individual values. Central Limit Theorem Now that you are aware of the difference between the frequency distribution of individual
values, X ’s, and the frequency distribution of averages, be discussed. In simple terms it is:
X
’s, the central limit theorem can
If the population from which samples are taken is not normal, the distribution of sample averages will tend toward normality provided that the sample size, n , is at least 4. This tendency gets better and better as the sample size gets larger. Furthermore, the standardized normal can be used for the distribution of averages with the modification, Z
X
X
X
n
This theorem was illustrated by W.A. Shewhart for a uniform population distribution and a triangular population distribution of individual values as shown in Figure 3-15. Obviously, the distribution of X ’s is approximately normal. The central limit theorem is the reason the X chart works, in that we do not need to be concerned if the distribution of
X
’s is not normal provided that the sample size is 4 or more.
Figure 3-15 Illustration of central limit theorem. Control Limits and Specifications
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Control limits are established as a function of the averages; in other words, control limits are for averages. Specifications, on the other hand, are the permissible variation in the size of the part and are, therefore, for individual values. The specification or tolerance limits are established by product engineers to meet a particular function. Figure 3-16 shows that the location of the specification is optional and is not related to any other features in the figure. The control limits, process spread, distribution of averages, and distribution of individual values are interdependent, since X
n
.
Figure 3-16 Relation of limits, specifications, and distributions. Process Spread and Specifications While specifications can be established by the product engineer without regard for the spread of the process, serious situations can result when this type of action is adopted. There are three situations: (1) when the process spread is less than the difference between specifications, (2) when the process spread is equal to the difference between specifications, (3) and when the process spread is greater than the difference between specifications. Case I: 6 U L . This situation, where the spread of the process ( 6 ) is less than the difference between specifications ( U L ), is the most desirable case. Figure 3-17 illustrates this ideal relationship by the distribution of individual values labeled A. Since the specifications are appreciably greater than the process spread, no difficulty is encountered even when there is a substantial shift in the process average, as shown by the distributions at B. At C a shift in the dispersion is illustrated, and all the i ndividual values are between specifications. Case I is economically advantageous since an out-of-control condition, as illustrated at B and C, does not produce defective product. Therefore, frequent machine adjustments or searches for assignable causes are not necessary. In fact, this satisfactory state of affairs suggests that the control chart may be discontinued.
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Figure 3-17 Changes in the process average and dispersion when 6
U L
.
Case II: 6 U L . Figure 3-18 illustrates this case where the spread of the process, or process capability, is equal to the difference between specifications. The frequency distribution at A represents a natural pattern of variation. However, when there is a shift in the process average, as indicated at B, or a change in the dispersion, as indicated at C, the individual values exceed the specifications. As long as the process remains in control as indicated at A, no defective is produced; however, when the process is out of control as indicated at B and C, defective is being produced. Therefore, assignable causes of variation must be corrected as soon as they occur.
Figure 3-18 Changes in the process average and dispersion when 6 U L . Case III: 6 U L . When the spread of the process or process capability is greater than
the difference between specifications, an undesirable situation exists. Figure 3-19 illustrates this case. Even though a natural pattern of variation is occurring, as shown by the frequency distribution at A, some of the individual values are greater than the upper specification and are less than the lower specification. This case presents the unique situation where the process is in control, but defective product is produced. In other words, the process is not capable of manufacturing a product that will meet the specifications.
Figure 3-19 Changes in the process average and dispersion when 6
U L
.
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One solution is to discuss with the product engineer the possibility of increasing the difference between the upper and lower specifications. This solution may require reliability studies with mating parts to determine if the product can function with increased specification differences. Another solution is to leave the process and the specifications alone and perform 100% inspection to eliminate the defective parts. This is not an attractive solution, but it may be the most economical or only one. A third possibility to change the process dispersion so that a more peaked distribution occurs, as illustrated by frequency distribution B. to obtain such a substantial shift in the standard deviation might require new material, a more experienced operator or retraining, a new or overhauled machine, or possibly automatic in-process control. Another solution is to shift the process average so that all of the defective product occurs at one tail of the frequency distribution as indicated at C of Figure 3-19. To illustrate, assume that a shaft is being ground to tight specifications. If too much metal is removed, the part is scraped; if too little is removed, the part must be reworked. By shifting the process average the amount of scarp is eliminated and the amount of rework is increased. A similar situation exists for an internal member such as a hole or keyway except that scrap occurs above the upper specification and rework occurs below the lower specification. This type of solution is feasible when the cost of the part is sufficient economically to justify the reworking operation. Note that the crosshatched area at C is much more than that at A. Example Problem Location pins for work holding devices are ground to a diameter of 12.50 mm (approximately ½ in.), with a tolerance of 0.05 mm. if the process is centered at 12.50 mm ( ) and the dispersion is 0.02 mm ( ), what percent of the product must be scrapped and what percent can be reworked? How can be process center be changed to eliminate the scrap? What is the rework percentage? Solution
U L
Z
X t
0.05 0.05
12.50
12.50
12.45
0.05
0.05
mm 12.45 mm
12.55
12.50
0.02
2.50
From Table A of the Appendix for a Z value of -2.50: Area 1= 0.0062 or 0.62% scrap
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Since the process is centered between the specifications and a symmetrical distribution is assumed, the rework percentage will be equal to the scrap percentage of 0.62%. The second part of the problem is solved using the following sketch:
If the amount of scrap is to be zero, then Area 1=0. From Table A, the closest value to an Area1 value of zero is 0.00017, which has a Z value of -3.59. Thus, Z
X i
, 3.59
12.45
0.02
,
12.52 mm
The percentage of rework is obtained by first determining Area 3. Z
X i
12.55 12.52 0.02
1.50
The Table A, Area3 = 0.9332 and Area 2 = AreaTotal- Area3 =1.0000-0.9332=0.0668 or 6.68% The amount of rework is 6.68%, which, incidentally, is considerably more than the combined rework and scrap percentage (1.24%) when the process is centered. The preceding analysis of the process spread and the specifications was made utilizing an upper and a lower specification. Many times there is only one specification and it may be either upper or lower. A similar and much simpler analysis could be for a single specification limit.
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13
PROCESS CAPABILITY
The true process capability cannot be determined until the X and R charts have achieved the optimal quality improvement without a substantial investment for new equipment or equipment modification. Process capability or spread of the process is equal to 6 standard deviations, which is 6 0 or 6 if the population standard deviation is known In the example problem for the X and R charts, the quality improvement process began in January with 0 0.038 . The process capability is for 6 = (6) (0.038) = 0.228 mm or 0.038
mm or
0.114mm. By July,
0
=0.030, which gives a process capability of 0.180 mm or
0.090 mm. This is a 20% improvement in the process capability, which in most situations would be sufficient to solve a quality problem. It is frequently necessary to obtain the process capability by a quick method rather than by using the X and R charts. The procedure is:
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1. Take 20 subgroups of size 4 for a total of 80 measurements. These subgroups should be selected at random to minimize any bias. 2. Calculate the sample standard deviation, s , for the each subgroup. 3. Calculate the average sample standard deviation,
s
s / g s / 20 .
4. Calculate the estimate of the population standard deviation.
0
5. Process capability will be
s c
4
6 0 .
Remember that this technique does not give the true process capability and should be used only if circumstances require its use. Process capability and the tolerance are combined to form a capability index, defined as U L C p
where
6 0
= capability index
C p
U L = 6 0
upper specification – lower specification, or tolerance = process capability
If the capability index is 1.00, we have the case II situation discussed in the preceding section; if the ratio is greater than 1.00, we have the case I situation, which is desirable; and if the ratio is less than 1.00, we have the case III situation, which is undesirable. Example Problem Assume that the specifications are 6.50 and 6.30 in the depth of the keyway problem. Determine the capability index before and after im provement. U L 6.50 6.30 U L 6.50 6.30 0.88 1.11 C p C p 6 0 6(0.038) 6 0 6(0.030)
In the example problem the improvement in quality resulted in a desirable capability index. The minimum capability index is frequently established at 1.33. Below this value, design engineers have to seek approval from manufacturing before the product can be released for production. In Chapter 1, quality was defined as conformance to specifications. Using the capability index concept, we can measure quality provided the process is centered correctly. The larger the capability index, the better the quality. We should strive to make the capability index as large as possible. This is accomplished by having realistic specifications and continual striving to improve the process capability. Table A The equivalent Cp value corresponding to capability percentage. Equivalent Cp Capability in Equivalent Cp Capability in percentage percentage 0.50 86.64 0.86 99.00 0.62 93.50 0.91 99.35 0.68 96.00 1.00 99.73 0.75 97.50 1.33 99.994 0.81 98.50
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Table B Interpreting the process capability index. Cp < 1 Not capable Cp > 1 Capable at 3 Cp > 1.33 Capable at 4 Cp > 1.67 Capable at 5 Cp > 2 Capable at 6 Table C Six sigma value chart. Sigma
DPMO
COPQ
Capability
6 Sigma
3.4
<10% of sales
World class
5 Sigma 4 Sigma 3 Sigma 2 Sigma 1 Sigma
230 6200 67000 310,000 691,500
10 to 15% of sales 15 to 20% of sales 20 to 30% of sales 30 to 40% of sales -
Industry Noncompetitive -
DPMO=Defects Per Million Opportunities, COPQ= Cost of Poor Quality PROBLEMS 1. In filling bags of nitrogen fertilizer, it is desired to hold the average overfill to as low a value as possible. The lower specifications limits is 22.00 kg (48.50 lb), the population mean weight of the bags is 22.73 kg (50.11 lb), and the population standard deviation is 0.80 kg (1.76 lb). What percent of the bags contain less than 22 kg? If it is permissible for 5% of the bags to be below 22 kg, what would be the average weight? Assume a normal distribution. 2. Plastic strips that are used in a sensitive electronic device are manufactured to a maximum specification of 305.70 mm (approximately 12 in.) and a minimum specifications of 304.55 mm. If the strips are less than the minimum specification, they are scrapped; if greater than the maximum specification, they are reworked. The part dimensions are normally distributed with a population mean of 305.20 mm and a population standard deviations of 0.25 mm. What percentage of the product is scrap? What percentage is rework? How can the process be centered to eliminate all but 0.1% of the scrap? What is the rework percentage now? 3. A company that manufactures oil seals found the population mean to be 49.15 mm (1.935 in.), the population standard deviation to be 0.51 mm (0.020 in.), and the data to be normally distributed. If the ID of the seal is below the lower specification limit of 47.80 mm, the part is reworked. However, if above the upper specification limit of 49.80 mm, the seal is scrapped. (a) What percentage of the seals are reworked? What percentage are scrapped? (b) For various reasons the process average is changed to 48.50 mm. With this new mean or process center, what percentage of the seals is reworked? What percentage is scrapped? If rework is economically feasible, is the change in the process center a wise decision? 4. Determine the process capability of the data on Table 3-2. Use the first 20 subgroups. 5. Repeat problem 13 using the last 20 subgroups and compare the results. 6. Determine the capability ratio before and after improvement for the chapter example problem using specifications of 6.40 ± 0.15 mm. 7. A new process is started and the sum of the ample standard deviations for 20 subgroups of size 4 is 600. If the specifications are 700 ± 80, what is the process capability? What action would you recommend?
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