Engineering Statistics and Manufacturing
Note Please Attempt each question as asked using the software where ever mentioned. Full Report generated by STATGRAPHICS Must be included in Problems 1 .All Excel sheets be included in PROBLEMS where Excel has been used such as in Q
Each problem should be started on separate page after pasting the the problem statement at the top of the page. Your Solution will be typewritten.
A hard copy as (WORD +EXCELL SHEETS) be provided along with a Hard copy in PDF format. Plus Software generated reports
A soft copy as a CD (WORD +EXCELL SHEETS) be provided along with a soft copy copy in PDF format. Plus Software generated reports. CD Must have mentioned that it is STSTISTICS PROJECT and it shoulf have All Project students Name and ID mentioned on CD and a on a text file inside the cD.
Project copies both word and PDF will be uploaded on WebCT as will be explained later.
PROBLEM # 1 (1.5 point)
For alpha distribution
Find
a) Plot F(t) versus t/C for =0.05 , 0.1.0.15,0.2 ,0.25 and 0.3 (all curves should be in one Figure for range of t/C varying from (0+=0.05) to 4). Hint use Normal Distribution Function(select True) of Excel subroutine for Z=-
b) Find pdf for t varying from -∞ to +∞ ,(note at t=0 function is undefined). Note
where
Z=-
c) Plot f(t) versus t/C for =0.05 , 0.1.0.15,0.2 ,0.25 and 0.3 .all curves in one Figure for range of t/C varying from (0+=0.05) to 4 . Hint use Normal Distribution Function (select False) of Excel subroutine for Z=-
d) Median value of T, t0.5
e) Quantile, tp which is the solution of F(tp)=p .
f) Mode of T, (value of t where
PROBLEM # 2 (1.0 point) Use EXCEL and attach all spreadsheet analysis with solution)
Fifty measurements of the ultimate tensile strength of wire are given in the accompanying table.
a) Group the data and make an appropriate normalized histogram (with total area of histogram be 1 ) to approximate the PDF.
b) Calculate and for the distribution from the ungrouped data.
c) Using and from part b, draw a normal distribution through the normalized histogram .histogram.
Ultimate Tensile Strength
103,779
102,325
102,325
103,799
102,906
104,651
105,377
100,145
104,796
105,087
104,796
103,799
103,197
106,395
106,831
103,488
100,872
100,872
105,087
102,906
97,383
104,360
103,633
101,017
101,162
101,453
107,848
104,651
98,110
103,779
99,563
103,197
104,651
101,162
105,813
105,337
102,906
102,470
108,430
101,744
103,633
105,232
106,540
106,104
102,616
106,831
101,744
100,726
103,924
101,598
Source: Data from E. B. Haugen, Probabilistic Mechanical Design Wiley, New York, 1980
(c) Determine the mean, median, and mode from the ungrouped data.
(d) Determine the range and standard deviation from the ungrouped data
(e) Plot the cumulative frequency distribution on normal-probability paper, and determine the mean and standard deviation.
(f), for the data given in Table .what are the 95 percent confidence limits on the mean of the population?
PROBLEM # 3 (1.0 point) (Use EXCEL and attach all spreadsheet analysis with solution)
Three sets of identical twenty five fatigue specimens were tested at the three different level of stresses.. The number of cycles to failure. The results are expressed as log, were as follows.
TABLE 1: FATIGUE LIFE DATA
NUMBER OF CYCLES TO FAILURES
No.
S1
S2
S3
380 MPa
340 MPa
300 MPa
1
34200
125500
954000
2
37700
156900
959400
3
42000
173600
1194600
4
42300
176900
1240500
5
48200
179400
1250400
6
52500
188500
1285500
7
55900
195100
1410500
8
58300
208100
1495100
9
61700
211900
1518700
10
64700
224100
1544700
11
65000
226000
1551400
12
65500
253000
1585900
13
70400
255500
1639100
14
71000
259000
1683700
15
72400
274000
1926100
16
75200
292000
2011300
17
77400
300400
2171800
18
77800
302300
2391500
19
87800
308300
2569400
20
93400
406300
2674900
21
94000
420700
2921700
22
97200
428500
3046500
23
99600
664800
3105500
24
116700
776100
3523200
25
122500
793900
4311700
Assume that the data at each stress level (S1, S2 and S3) is lognormally distributed.
Using directly the data in table determine
· What is the mean fatigue life ( μ) and its standard deviation (σ )?
· What is the mean Ln of fatigue life ( μlnN) and its standard deviation Ln of fatigue life (σlnN)?
· What are the Parameters of Lognormal distributions at S1,S2and S3.
· Fit lognormal distribution to the data using linear regression model using Excel or Use Statgraphics to fit the lognormal models to data for each stress level and comment on how good the fit is.
PROBLEM # 4 (1.5 point)
Q4. (a) The lifetime of a mechanical switch produced by a company has been determined to have a population mean of μ = 2000 h and σ = 200 h. The temper of a phosphor bronze leaf spring in the switch is changed slightly by the supplier. To determine whether this has changed the product, a sample of 100 switches is tested to give the sample values and . Has there been a change in the product? (0.75 point)
Q4. (b) A vendor of steel wire advertises a mean breaking load of 10,000 lb. A sample of eight tests shows a mean breaking load of 9250 lb and a standard deviation of 110 lb. Do our tests support the vendor’s claim? (0.75 point)
PROBLEM # 5 Control Charts (1 point)
Problem 5-Refer your Text Book above ( See Ebook provided as text book)- -Solve with appropriate calculations, tables and charts
5.1-Problem 8.1 Parts (a) and (b) –P454
5.2 -Problem 8.28 Parts (a) and (b) –P470
PROBLEM # 6
Regression Analysis (MUST USE STATGRAPHICS_ ATTACH COMPLETE REPORT PLUS ONE PAGE SUMMARY OF EACH FITTED MODEL) 2 points
Developing Cutting Forces Empirical Models of a Counter Boring Process in Aluminum.
Counter boring is an operation to enlarge the hole made using drilling. Counter boring or finish boring is a deep hole drilling process that requires a work piece with a pre-existing bore. Counter boring is used to enlarge the drilled hole to the proper depth and machine a square shoulder on the bottom to secure maximum clamping action from the faster. The drilling used to produce a circular hole by removing solid metal. The counter bore tool has a guide, called a pilot, which keep it positioned correctly in the hole. Counter boring tools are often used on low power machines were a small diameter solid boring tool is used for the pre-bore and then a counter boring tool is used to finish the job. Counter boring is also used when there is a heat treat process required after the initial hole is drilled or if a stepped hole is required. 31 d31
Pilot of diameter d , which is the predrilled hole size in the workpiece diameter of already drilled hole..
D Dia of the enlarged hole
Visit the link and download STATGRAPHICS FREE FOR 30 DAYS AND USE MULTIPLE REGRESSION MODULE (SEE EXAMPLES ON WEBSITE) TO DEVELOP FOLLOWING MODELS>
http://www.statgraphics.com/centurion-xvii
Regression Analysis http://www.statgraphics.com/regression-analysis
And Quality Control Module (PROICESS CAPABILITY BANALYSIS) from the link http://www.statgraphics.com/process-capability-analysis
Following are the results of Cutting Forces measurements experiments performed at KFUPM Workshop by Professor Anwar K Sheikh. The results are being shared for regression analysis learning objectives.
Cutting Force Fz as a Function of V,D,d and f
Fz
Newton
Speed , V
mm/minutes
Feed , f
Mm/revolution
d
mm
D
mm
Ln (FZ)
Ln(Speed)
Ln(feed)
Ln(D)
52
2463.007
0.03
3.5
6.5
3.951244
7.809138
-3.50656
1.252763
78
2463.007
0.05
3.5
6.5
4.356709
7.809138
-2.99573
1.252763
104
2463.007
0.08
3.5
6.5
4.644391
7.809138
-2.52573
1.252763
130
2463.007
0.12
3.5
6.5
4.867534
7.809138
-2.12026
1.252763
65
3903.426
0.03
3.5
6.5
4.174387
8.26961
-3.50656
1.252763
78
3903.426
0.05
3.5
6.5
4.356709
8.26961
-2.99573
1.252763
104
3903.426
0.08
3.5
6.5
4.644391
8.26961
-2.52573
1.252763
130
3903.426
0.12
3.5
6.5
4.867534
8.26961
-2.12026
1.252763
65
4948.004
0.03
3.5
6.5
4.174387
8.50674
-3.50656
1.252763
78
4948.004
0.05
3.5
6.5
4.356709
8.50674
-2.99573
1.252763
117
4948.004
0.08
3.5
6.5
4.762174
8.50674
-2.52573
1.252763
130
4948.004
0.12
3.5
6.5
4.867534
8.50674
-2.12026
1.252763
65
6157.516
0.03
3.5
6.5
4.174387
8.725429
-3.50656
1.252763
78
6157.516
0.05
3.5
6.5
4.356709
8.725429
-2.99573
1.252763
104
6157.516
0.08
3.5
6.5
4.644391
8.725429
-2.52573
1.252763
130
6157.516
0.12
3.5
6.5
4.867534
8.725429
-2.12026
1.252763
72
7806.851
0.03
3.5
6.5
4.276666
8.962757
-3.50656
1.252763
85
7806.851
0.05
3.5
6.5
4.442651
8.962757
-2.99573
1.252763
111
7806.851
0.08
3.5
6.5
4.70953
8.962757
-2.52573
1.252763
130
7806.851
0.12
3.5
6.5
4.867534
8.962757
-2.12026
1.252763
78
2463.007
0.03
5.5
10
4.356709
7.809138
-3.50656
1.704748
104
2463.007
0.05
5.5
10
4.644391
7.809138
-2.99573
1.704748
130
2463.007
0.08
5.5
10
4.867534
7.809138
-2.52573
1.704748
182
2463.007
0.12
5.5
10
5.204007
7.809138
-2.12026
1.704748
84
3903.426
0.03
5.5
10
4.430817
8.26961
-3.50656
1.704748
110
3903.426
0.05
5.5
10
4.70048
8.26961
-2.99573
1.704748
143
3903.426
0.08
5.5
10
4.962845
8.26961
-2.52573
1.704748
182
3903.426
0.12
5.5
10
5.204007
8.26961
-2.12026
1.704748
91
4948.004
0.03
5.5
10
4.51086
8.50674
-3.50656
1.704748
117
4948.004
0.05
5.5
10
4.762174
8.50674
-2.99573
1.704748
130
4948.004
0.08
5.5
10
4.867534
8.50674
-2.52573
1.704748
182
4948.004
0.12
5.5
10
5.204007
8.50674
-2.12026
1.704748
78
6157.516
0.03
5.5
10
4.356709
8.725429
-3.50656
1.704748
104
6157.516
0.05
5.5
10
4.644391
8.725429
-2.99573
1.704748
143
6157.516
0.08
5.5
10
4.962845
8.725429
-2.52573
1.704748
195
6157.516
0.12
5.5
10
5.273
8.725429
-2.12026
1.704748
91
7806.851
0.03
5.5
10
4.51086
8.962757
-3.50656
1.704748
117
7806.851
0.05
5.5
10
4.762174
8.962757
-2.99573
1.704748
143
7806.851
0.08
5.5
10
4.962845
8.962757
-2.52573
1.704748
195
7806.851
0.12
5.5
10
5.273
8.962757
-2.12026
1.704748
117
2463.007
0.03
7.5
15
4.762174
7.809138
-3.50656
2.014903
143
2463.007
0.05
7.5
15
4.962845
7.809138
-2.99573
2.014903
195
2463.007
0.08
7.5
15
5.273
7.809138
-2.52573
2.014903
234
2463.007
0.12
7.5
15
5.455321
7.809138
-2.12026
2.014903
123
3903.426
0.03
7.5
15
4.812184
8.26961
-3.50656
2.014903
156
3903.426
0.05
7.5
15
5.049856
8.26961
-2.99573
2.014903
208
3903.426
0.08
7.5
15
5.337538
8.26961
-2.52573
2.014903
234
3903.426
0.12
7.5
15
5.455321
8.26961
-2.12026
2.014903
123
4948.004
0.03
7.5
15
4.812184
8.50674
-3.50656
2.014903
169
4948.004
0.05
7.5
15
5.129899
8.50674
-2.99573
2.014903
208
4948.004
0.08
7.5
15
5.337538
8.50674
-2.52573
2.014903
247
4948.004
0.12
7.5
15
5.509388
8.50674
-2.12026
2.014903
130
6157.516
0.03
7.5
15
4.867534
8.725429
-3.50656
2.014903
169
6157.516
0.05
7.5
15
5.129899
8.725429
-2.99573
2.014903
208
6157.516
0.08
7.5
15
5.337538
8.725429
-2.52573
2.014903
247
6157.516
0.12
7.5
15
5.509388
8.725429
-2.12026
2.014903
143
7806.851
0.03
7.5
15
4.962845
8.962757
-3.50656
2.014903
182
7806.851
0.05
7.5
15
5.204007
8.962757
-2.99573
2.014903
208
7806.851
0.08
7.5
15
5.337538
8.962757
-2.52573
2.014903
247
7806.851
0.12
7.5
15
5.509388
8.962757
-2.12026
2.014903
156
2463.007
0.03
9.5
18
5.049856
7.809138
-3.50656
2.251292
208
2463.007
0.05
9.5
18
5.337538
7.809138
-2.99573
2.251292
260
2463.007
0.08
9.5
18
5.560682
7.809138
-2.52573
2.251292
338
2463.007
0.12
9.5
18
5.823046
7.809138
-2.12026
2.251292
208
3903.426
0.03
9.5
18
5.337538
8.26961
-3.50656
2.251292
234
3903.426
0.05
9.5
18
5.455321
8.26961
-2.99573
2.251292
260
3903.426
0.08
9.5
18
5.560682
8.26961
-2.52573
2.251292
364
3903.426
0.12
9.5
18
5.897154
8.26961
-2.12026
2.251292
208
4948.004
0.03
9.5
18
5.337538
8.50674
-3.50656
2.251292
234
4948.004
0.05
9.5
18
5.455321
8.50674
-2.99573
2.251292
260
4948.004
0.08
9.5
18
5.560682
8.50674
-2.52573
2.251292
364
4948.004
0.12
9.5
18
5.897154
8.50674
-2.12026
2.251292
208
6157.516
0.03
9.5
18
5.337538
8.725429
-3.50656
2.251292
260
6157.516
0.05
9.5
18
5.560682
8.725429
-2.99573
2.251292
286
6157.516
0.08
9.5
18
5.655992
8.725429
-2.52573
2.251292
338
6157.516
0.12
9.5
18
5.823046
8.725429
-2.12026
2.251292
234
7806.851
0.03
9.5
18
5.455321
8.962757
-3.50656
2.251292
286
7806.851
0.05
9.5
18
5.655992
8.962757
-2.99573
2.251292
312
7806.851
0.08
9.5
18
5.743003
8.962757
-2.52573
2.251292
Moment Mz as a Function of V,D,d and f
Mz Newton-meter
Speed , V
mm/minutes
Feed , f
Mm/ revolution
d
mm
D
mm
Ln (Mz)
Ln(Speed)
Ln(feed)
Ln(D)
39
2463.007
0.03
3.5
6.5
3.663562
7.809138
-3.50656
1.252763
59
2463.007
0.05
3.5
6.5
4.077537
7.809138
-2.99573
1.252763
72
2463.007
0.08
3.5
6.5
4.276666
7.809138
-2.52573
1.252763
104
2463.007
0.12
3.5
6.5
4.644391
7.809138
-2.12026
1.252763
39
3903.426
0.03
3.5
6.5
3.663562
8.26961
-3.50656
1.252763
52
3903.426
0.05
3.5
6.5
3.951244
8.26961
-2.99573
1.252763
78
3903.426
0.08
3.5
6.5
4.356709
8.26961
-2.52573
1.252763
117
3903.426
0.12
3.5
6.5
4.762174
8.26961
-2.12026
1.252763
39
4948.004
0.03
3.5
6.5
3.663562
8.50674
-3.50656
1.252763
59
4948.004
0.05
3.5
6.5
4.077537
8.50674
-2.99573
1.252763
78
4948.004
0.08
3.5
6.5
4.356709
8.50674
-2.52573
1.252763
117
4948.004
0.12
3.5
6.5
4.762174
8.50674
-2.12026
1.252763
39
6157.516
0.03
3.5
6.5
3.663562
8.725429
-3.50656
1.252763
52
6157.516
0.05
3.5
6.5
3.951244
8.725429
-2.99573
1.252763
78
6157.516
0.08
3.5
6.5
4.356709
8.725429
-2.52573
1.252763
124
6157.516
0.12
3.5
6.5
4.820282
8.725429
-2.12026
1.252763
39
7806.851
0.03
3.5
6.5
3.663562
8.962757
-3.50656
1.252763
59
7806.851
0.05
3.5
6.5
4.077537
8.962757
-2.99573
1.252763
72
7806.851
0.08
3.5
6.5
4.276666
8.962757
-2.52573
1.252763
117
7806.851
0.12
3.5
6.5
4.762174
8.962757
-2.12026
1.252763
52
2463.007
0.03
5.5
10
3.951244
7.809138
-3.50656
1.704748
72
2463.007
0.05
5.5
10
4.276666
7.809138
-2.99573
1.704748
98
2463.007
0.08
5.5
10
4.584967
7.809138
-2.52573
1.704748
163
2463.007
0.12
5.5
10
5.09375
7.809138
-2.12026
1.704748
65
3903.426
0.03
5.5
10
4.174387
8.26961
-3.50656
1.704748
84
3903.426
0.05
5.5
10
4.430817
8.26961
-2.99573
1.704748
110
3903.426
0.08
5.5
10
4.70048
8.26961
-2.52573
1.704748
169
3903.426
0.12
5.5
10
5.129899
8.26961
-2.12026
1.704748
65
4948.004
0.03
5.5
10
4.174387
8.50674
-3.50656
1.704748
98
4948.004
0.05
5.5
10
4.584967
8.50674
-2.99573
1.704748
110
4948.004
0.08
5.5
10
4.70048
8.50674
-2.52573
1.704748
169
4948.004
0.12
5.5
10
5.129899
8.50674
-2.12026
1.704748
65
6157.516
0.03
5.5
10
4.174387
8.725429
-3.50656
1.704748
98
6157.516
0.05
5.5
10
4.584967
8.725429
-2.99573
1.704748
117
6157.516
0.08
5.5
10
4.762174
8.725429
-2.52573
1.704748
169
6157.516
0.12
5.5
10
5.129899
8.725429
-2.12026
1.704748
65
7806.851
0.03
5.5
10
4.174387
8.962757
-3.50656
1.704748
98
7806.851
0.05
5.5
10
4.584967
8.962757
-2.99573
1.704748
130
7806.851
0.08
5.5
10
4.867534
8.962757
-2.52573
1.704748
163
7806.851
0.12
5.5
10
5.09375
8.962757
-2.12026
1.704748
81
2463.007
0.03
7.5
15
4.394449
7.809138
-3.50656
2.014903
130
2463.007
0.05
7.5
15
4.867534
7.809138
-2.99573
2.014903
195
2463.007
0.08
7.5
15
5.273
7.809138
-2.52573
2.014903
260
2463.007
0.12
7.5
15
5.560682
7.809138
-2.12026
2.014903
104
3903.426
0.03
7.5
15
4.644391
8.26961
-3.50656
2.014903
143
3903.426
0.05
7.5
15
4.962845
8.26961
-2.99573
2.014903
195
3903.426
0.08
7.5
15
5.273
8.26961
-2.52573
2.014903
260
3903.426
0.12
7.5
15
5.560682
8.26961
-2.12026
2.014903
117
4948.004
0.03
7.5
15
4.762174
8.50674
-3.50656
2.014903
156
4948.004
0.05
7.5
15
5.049856
8.50674
-2.99573
2.014903
221
4948.004
0.08
7.5
15
5.398163
8.50674
-2.52573
2.014903
260
4948.004
0.12
7.5
15
5.560682
8.50674
-2.12026
2.014903
130
6157.516
0.03
7.5
15
4.867534
8.725429
-3.50656
2.014903
156
6157.516
0.05
7.5
15
5.049856
8.725429
-2.99573
2.014903
195
6157.516
0.08
7.5
15
5.273
8.725429
-2.52573
2.014903
286
6157.516
0.12
7.5
15
5.655992
8.725429
-2.12026
2.014903
130
7806.851
0.03
7.5
15
4.867534
8.962757
-3.50656
2.014903
169
7806.851
0.05
7.5
15
5.129899
8.962757
-2.99573
2.014903
221
7806.851
0.08
7.5
15
5.398163
8.962757
-2.52573
2.014903
299
7806.851
0.12
7.5
15
5.700444
8.962757
-2.12026
2.014903
221
2463.007
0.03
9.5
18
5.398163
7.809138
-3.50656
2.251292
260
2463.007
0.05
9.5
18
5.560682
7.809138
-2.99573
2.251292
357
2463.007
0.08
9.5
18
5.877736
7.809138
-2.52573
2.251292
487
2463.007
0.12
9.5
18
6.188264
7.809138
-2.12026
2.251292
227
3903.426
0.03
9.5
18
5.42495
8.26961
-3.50656
2.251292
325
3903.426
0.05
9.5
18
5.783825
8.26961
-2.99573
2.251292
357
3903.426
0.08
9.5
18
5.877736
8.26961
-2.52573
2.251292
487
3903.426
0.12
9.5
18
6.188264
8.26961
-2.12026
2.251292
195
4948.004
0.03
9.5
18
5.273
8.50674
-3.50656
2.251292
292
4948.004
0.05
9.5
18
5.676754
8.50674
-2.99573
2.251292
357
4948.004
0.08
9.5
18
5.877736
8.50674
-2.52573
2.251292
520
4948.004
0.12
9.5
18
6.253829
8.50674
-2.12026
2.251292
292
6157.516
0.03
9.5
18
5.676754
8.725429
-3.50656
2.251292
445
6157.516
0.05
9.5
18
6.098074
8.725429
-2.99573
2.251292
650
6157.516
0.08
9.5
18
6.476972
8.725429
-2.52573
2.251292
747
6157.516
0.12
9.5
18
6.616065
8.725429
-2.12026
2.251292
357
7806.851
0.03
9.5
18
5.877736
8.962757
-3.50656
2.251292
487
7806.851
0.05
9.5
18
6.188264
8.962757
-2.99573
2.251292
650
7806.851
0.08
9.5
18
6.476972
8.962757
-2.52573
2.251292
Using STATGRAPHICS -MULTIPLE LINEAR REGRESSION MODULE develop the empirical model for cutting force Fz and Torque (Moment) Mz can be writing as following
Proposed Model 1 Force (Model 1 F)
Proposed Model 1 Moments (Model 1 M)
Proposed Model 2 Force (Model 1 F) .In spread sheet create a new column of val;ues.
Proposed Model 2 Moments (Model 1 M). In spread sheet create a new column of values.
Find A,f ,b and c d e etc in Fz model using first data table ,and Find B,d,e,f d using second Data table for each of the odel.. Write one page summary based upon completer regression report generated by STATGRAPHICS for each fitted model .Its goodness of fit as measured by R2 values and other important coefficients tabulated and plotted in the report. (Attach PDF copy of each full report of STATGRAPHICS output and your EXCEL File used as input data.
c
b
a
V
D
f
A
Fz
´
´
´
=
f
e
d
V
D
f
B
Mz
´
´
´
=
c
b
a
V
d
D
f
A
Fz
´
–
´
´
=
)
(
2
2
)
(
2
2
d
D
–
f
e
d
V
d
D
f
B
Mz
´
–
´
´
=
)
(
2
2
)
(
2
2
d
D
–
