A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Xbar/R Chart for C6-C9 25 UCL=24.04 24 Mean-23.93 23 LCL=2291 Subgroup 10 UCL-3,17 R=1.302 LCL=0 The revised charts do not show any out-of-control phenomenon. Based on all the given data and information, 4.1) Identify the average length of the beams Best-Wood produced when the process is in statistical control 4.2) Estimate the process standard deviation of the beam production process. 4.3) Calculate Process Capability Ratio if the design specifications of the beam length is 24±2.2cm. 4.4) Based on the given data and your calculation, briefly explain to Best-Wood management that the following steps should be taken or should not be taken in improving the process: 4.4.1) Use a more accurate measuring device to make the centering of the process closer to the targeted value of 24.00 cm. 4.4.2) Eliminate the impact of certain factors such as machine vibration to reduce process standard deviation. 4.4.3) Increase the range of the design specifications in order to improve Process Capability Ratio in achieving a 6-sigma process You may need the following equations and table in answering this question. ô = R/d2, C, = (USL – LSL )/(6 x ô), where USL and LSL are upper and lower specification limits. Factors for Factors for Factors for Observations Control Limits Center Line Factors for Control Limits Center Line Factors for Control Limits in Sample, a B, B. B, B. /d d, D, D D, D. 3.267 3.267 0 2.568 0 2.121 1.880 2.659 0.7979 1.2533 2.606 L.128 0.8865 0.853 0 0.888 0 3.686 1.732 1.500 1.342 1.023 1.954 0.729 1.628 2.276 2.088 1.964 3 0.8862 1.1284 0,9213 1.0854 1.693 0.5907 2.059 0.4857 4.358 4.698 2.574 2.282 2.266 2.089 0.030 1.970 4 0,880 0 0.577 0.483 1.427 1.287 0.9400 1.0638 0.9515 1.0510 2.326 0.4299 2.534 0.3946 2.704 0.3698 0.864 0 4.918 5.078 2.114 2.004 0.076 1.924 1.225 0.029 1.874 0.848 0 1.134 0.419 1.182 0.9594 1.0423 0.118 1.882 0.113 1.806 0.833 0.204 5.204 Sample Range Sample Mean
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Transcribed Image Text:Xbar/R Chart for C6-C9 25 UCL=24.04 24 Mean-23.93 23 LCL=2291 Subgroup 10 UCL-3,17 R=1.302 LCL=0 The revised charts do not show any out-of-control phenomenon. Based on all the given data and information, 4.1) Identify the average length of the beams Best-Wood produced when the process is in statistical control 4.2) Estimate the process standard deviation of the beam production process. 4.3) Calculate Process Capability Ratio if the design specifications of the beam length is 24±2.2cm. 4.4) Based on the given data and your calculation, briefly explain to Best-Wood management that the following steps should be taken or should not be taken in improving the process: 4.4.1) Use a more accurate measuring device to make the centering of the process closer to the targeted value of 24.00 cm. 4.4.2) Eliminate the impact of certain factors such as machine vibration to reduce process standard deviation. 4.4.3) Increase the range of the design specifications in order to improve Process Capability Ratio in achieving a 6-sigma process You may need the following equations and table in answering this question. ô = R/d2, C, = (USL – LSL )/(6 x ô), where USL and LSL are upper and lower specification limits. Factors for Factors for Factors for Observations Control Limits Center Line Factors for Control Limits Center Line Factors for Control Limits in Sample, a B, B. B, B. /d d, D, D D, D. 3.267 3.267 0 2.568 0 2.121 1.880 2.659 0.7979 1.2533 2.606 L.128 0.8865 0.853 0 0.888 0 3.686 1.732 1.500 1.342 1.023 1.954 0.729 1.628 2.276 2.088 1.964 3 0.8862 1.1284 0,9213 1.0854 1.693 0.5907 2.059 0.4857 4.358 4.698 2.574 2.282 2.266 2.089 0.030 1.970 4 0,880 0 0.577 0.483 1.427 1.287 0.9400 1.0638 0.9515 1.0510 2.326 0.4299 2.534 0.3946 2.704 0.3698 0.864 0 4.918 5.078 2.114 2.004 0.076 1.924 1.225 0.029 1.874 0.848 0 1.134 0.419 1.182 0.9594 1.0423 0.118 1.882 0.113 1.806 0.833 0.204 5.204 Sample Range Sample Mean
Best-Wood is a manufacturing company producing large number of short wood beams. The length of all beams should be close to or at 24.00cm. Quality engineers at Best-Wood have collected 15 groups (samples) of data with each group containing the measurements of the lengths (in centimeters) of 4 beams. The data are shown in the following table (next pate). As shown in the table, the 4 measurements of the first sample (second row of the table) are 23.50, 24.40, 23.60 and 23.90, respectively. The average length, I, of the 4 beams in the first sample is 23.85cm. The range, R, of the sample is max(x,,x,, x3,x,)- min(x,,x,, X3,x,)= 24.40 – 23.50 = 0.90 as shown in the last number of the first row of the table. The end of the table shows that the average of the 15 x values is 24.00 and the average of the 15 R values is 1.413. Sample X2 X4 1 23.50 24.40 23.60 23.90 23.85 0.90 25.70 24.20 23.90 24.80 24.65 1.80 22.60 23.90 23.20 23.20 23.23 1.30 4 26.50 24.80 26.40 26.90 26.15 2.10 23.80 24.00 24.20 22.20 23.55 2.00 6. 24.20 24.40 24.20 23.40 24.05 1.00 7 24.40 23.60 24.50 25.00 24.38 1.40 8 23.30 24.10 23.80 23.60 23.70 0.80 9 23.20 24.70 24.70 25.50 24.53 2.30 10 23.20 23.10 22.20 22.50 22.75 1.00 11 23.20 23.40 23.50 24.80 23.73 1.60 12 23.70 23.70 24.10 23.70 23.80 0.40 13 23.50 23.20 24.90 22.50 23.53 2.40 14 22.80 23.90 23.50 24.20 23.60 1.40 15 25.00 24.20 24.30 24.50 24.50 0.80 Average 24.00 1.413 The quality engineers at Best-Wood also constructed F-R charts using the above data. The charts are shown below. Xbar/R Chart for C1-C4 26.5 25.5 UCL=25.03 24.5 Mean=24.0 23.5 LCL-22.97 22.5 Subgroup 10 15 UCL=3.224 R=1,413 LCL-0 As can be seen from these two charts, Sample #4 and Sample #10 are out of control limits. A team of manufacturing engineers and technicians quickly identified the causes of these 2 out-of-control points. The team followed standard procedure to remove these 2 out-of-control samples from the data set and generated the revised X -R charts with the remaining 13 groups (samples) of data. The revised charts are shown next. Sample Range
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Transcribed Image Text:Best-Wood is a manufacturing company producing large number of short wood beams. The length of all beams should be close to or at 24.00cm. Quality engineers at Best-Wood have collected 15 groups (samples) of data with each group containing the measurements of the lengths (in centimeters) of 4 beams. The data are shown in the following table (next pate). As shown in the table, the 4 measurements of the first sample (second row of the table) are 23.50, 24.40, 23.60 and 23.90, respectively. The average length, I, of the 4 beams in the first sample is 23.85cm. The range, R, of the sample is max(x,,x,, x3,x,)- min(x,,x,, X3,x,)= 24.40 – 23.50 = 0.90 as shown in the last number of the first row of the table. The end of the table shows that the average of the 15 x values is 24.00 and the average of the 15 R values is 1.413. Sample X2 X4 1 23.50 24.40 23.60 23.90 23.85 0.90 25.70 24.20 23.90 24.80 24.65 1.80 22.60 23.90 23.20 23.20 23.23 1.30 4 26.50 24.80 26.40 26.90 26.15 2.10 23.80 24.00 24.20 22.20 23.55 2.00 6. 24.20 24.40 24.20 23.40 24.05 1.00 7 24.40 23.60 24.50 25.00 24.38 1.40 8 23.30 24.10 23.80 23.60 23.70 0.80 9 23.20 24.70 24.70 25.50 24.53 2.30 10 23.20 23.10 22.20 22.50 22.75 1.00 11 23.20 23.40 23.50 24.80 23.73 1.60 12 23.70 23.70 24.10 23.70 23.80 0.40 13 23.50 23.20 24.90 22.50 23.53 2.40 14 22.80 23.90 23.50 24.20 23.60 1.40 15 25.00 24.20 24.30 24.50 24.50 0.80 Average 24.00 1.413 The quality engineers at Best-Wood also constructed F-R charts using the above data. The charts are shown below. Xbar/R Chart for C1-C4 26.5 25.5 UCL=25.03 24.5 Mean=24.0 23.5 LCL-22.97 22.5 Subgroup 10 15 UCL=3.224 R=1,413 LCL-0 As can be seen from these two charts, Sample #4 and Sample #10 are out of control limits. A team of manufacturing engineers and technicians quickly identified the causes of these 2 out-of-control points. The team followed standard procedure to remove these 2 out-of-control samples from the data set and generated the revised X -R charts with the remaining 13 groups (samples) of data. The revised charts are shown next. Sample Range
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