Cycles to failure Position in ascending order 0.5 f(x)) (x;) Problem 44 Marsha, a renowned cake scientist, is trying to determine how long different cakes can survive intense fork attacks before collapsing into crumbs. To simulate real-world cake consumption, she designs a test where cakes are subjected to repeated fork stabs and bites, mimicking the brutal reality of birthday parties. After rigorous testing, Marsha records 10 observations of how many stabs each cake endured before structural failure. Construct P-P plots for (a.) a normal distribution, (b.) a lognormal distribution, and (c.) a Weibull distribution (using the information included in the table below). Which distribution seems to be the best model for the cycles to failure for this material? Explain your answer in detail. Observation Empirical cumulative Probability distribution Cumulative distribution Inverse of cumulative distribution F-1 (-0.5) F(x)) (S) n 4 3 1 0.05 9 5 2 0.15 7 7 3 0.25 1 10 4 0.35 3 12 5 0.45 Normal 0.01818 0.02315 0.02826 0.0352 0.03865 10 15 6 0.55 0.04106 5 18 7 0.65 0.03965 0.02617 0.03346 2 22 8 0.75 0.03262 0.01783 8 27 9 0.85 0.02013 6 35 10 0.95 0.00535 0.01116 0.00551 Lognormal Weibull Normal Lognormal Weibull 0.02947 0.02926 0.10074 0.02836 0.05504 0.05224 0.03769 0.14201 0.11314 0.12269 0.05848 0.04246 0.19344 0.22605 0.20337 7.40962 0.05242 0.04454 0.28902 0.39499 0.33517 11.65941 9.16308 10.33315 0.04525 0.04341 0.36308 0.49277 0.4234 14.18011 11.0885 12.61728 0.03476 0.03926 0.48357 0.61244 0.548 16.61989 13.33659 15.05101 0.60558 0.70329 0.65734 19.14059 16.13899 17.78218 0.02506 0.75171 0.79019 0.77433 21.94776 19.95822 21.06514 0.01577 0.88394 0.86124 0.87548 25.46141 26.03647 25.50287 0.00616 0.97826 0.92495 0.9588 31.36779 40.70696 33.6899 Normal Lognormal Weibull -0.56779 5.33859 8.85224 3.63286 2.82503 5.67983 5.70478 8.07935
Cycles to failure Position in ascending order 0.5 f(x)) (x;) Problem 44 Marsha, a renowned cake scientist, is trying to determine how long different cakes can survive intense fork attacks before collapsing into crumbs. To simulate real-world cake consumption, she designs a test where cakes are subjected to repeated fork stabs and bites, mimicking the brutal reality of birthday parties. After rigorous testing, Marsha records 10 observations of how many stabs each cake endured before structural failure. Construct P-P plots for (a.) a normal distribution, (b.) a lognormal distribution, and (c.) a Weibull distribution (using the information included in the table below). Which distribution seems to be the best model for the cycles to failure for this material? Explain your answer in detail. Observation Empirical cumulative Probability distribution Cumulative distribution Inverse of cumulative distribution F-1 (-0.5) F(x)) (S) n 4 3 1 0.05 9 5 2 0.15 7 7 3 0.25 1 10 4 0.35 3 12 5 0.45 Normal 0.01818 0.02315 0.02826 0.0352 0.03865 10 15 6 0.55 0.04106 5 18 7 0.65 0.03965 0.02617 0.03346 2 22 8 0.75 0.03262 0.01783 8 27 9 0.85 0.02013 6 35 10 0.95 0.00535 0.01116 0.00551 Lognormal Weibull Normal Lognormal Weibull 0.02947 0.02926 0.10074 0.02836 0.05504 0.05224 0.03769 0.14201 0.11314 0.12269 0.05848 0.04246 0.19344 0.22605 0.20337 7.40962 0.05242 0.04454 0.28902 0.39499 0.33517 11.65941 9.16308 10.33315 0.04525 0.04341 0.36308 0.49277 0.4234 14.18011 11.0885 12.61728 0.03476 0.03926 0.48357 0.61244 0.548 16.61989 13.33659 15.05101 0.60558 0.70329 0.65734 19.14059 16.13899 17.78218 0.02506 0.75171 0.79019 0.77433 21.94776 19.95822 21.06514 0.01577 0.88394 0.86124 0.87548 25.46141 26.03647 25.50287 0.00616 0.97826 0.92495 0.9588 31.36779 40.70696 33.6899 Normal Lognormal Weibull -0.56779 5.33859 8.85224 3.63286 2.82503 5.67983 5.70478 8.07935
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 15EQ
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Question
Transcribed Image Text:Cycles to
failure
Position in
ascending
order
0.5
f(x))
(x;)
Problem 44
Marsha, a renowned cake scientist, is trying to determine how long different cakes can survive intense fork attacks before collapsing into crumbs.
To simulate real-world cake consumption, she designs a test where cakes are subjected to repeated fork stabs and bites, mimicking the brutal
reality of birthday parties. After rigorous testing, Marsha records 10 observations of how many stabs each cake endured before structural failure.
Construct P-P plots for (a.) a normal distribution, (b.) a lognormal distribution, and (c.) a Weibull distribution (using the information included in the
table below). Which distribution seems to be the best model for the cycles to failure for this material? Explain your answer in detail.
Observation
Empirical
cumulative
Probability distribution
Cumulative distribution
Inverse of cumulative
distribution F-1 (-0.5)
F(x))
(S)
n
4
3
1
0.05
9
5
2
0.15
7
7
3
0.25
1
10
4
0.35
3
12
5
0.45
Normal
0.01818
0.02315
0.02826
0.0352
0.03865
10
15
6
0.55
0.04106
5
18
7
0.65
0.03965
0.02617
0.03346
2
22
8
0.75
0.03262
0.01783
8
27
9
0.85
0.02013
6
35
10
0.95
0.00535
0.01116
0.00551
Lognormal Weibull Normal Lognormal Weibull
0.02947 0.02926 0.10074 0.02836 0.05504
0.05224 0.03769 0.14201 0.11314 0.12269
0.05848 0.04246 0.19344 0.22605 0.20337
7.40962
0.05242 0.04454 0.28902 0.39499 0.33517 11.65941 9.16308 10.33315
0.04525 0.04341 0.36308 0.49277 0.4234 14.18011 11.0885 12.61728
0.03476 0.03926 0.48357 0.61244 0.548
16.61989 13.33659 15.05101
0.60558 0.70329 0.65734 19.14059 16.13899 17.78218
0.02506 0.75171 0.79019 0.77433 21.94776 19.95822 21.06514
0.01577 0.88394 0.86124 0.87548 25.46141 26.03647 25.50287
0.00616 0.97826 0.92495 0.9588 31.36779 40.70696 33.6899
Normal
Lognormal
Weibull
-0.56779
5.33859
8.85224
3.63286
2.82503
5.67983
5.70478
8.07935
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