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…

Problem 3 In their lab, engineer Daniel and Paulina are desperately trying to perfect time travel. But the problem is that their machine still struggles with power inconsistencies-sometimes generating too little energy, other times too much, causing unstable time jumps. To prevent catastrophic misjumps into the Jurassic era or the far future, they must calibrate the machine's power output. After extensive testing, they found that the time machine's power output follows a normal distribution, with an average energy level of 8.7 gigawatts and a standard deviation of 1.2 gigawatts. The Time Travel Safety Board has set strict guidelines: For a successful time jump, the machine's power must be between 8.5 and 9.5 gigawatts. What is the probability that a randomly selected time jump meets this precision requirement? Daniel suggests that adjusting the mean power output could improve time-travel accuracy. Can adjusting the mean reduce the number of dangerous misjumps? If yes, what should the…

Problem 5 ( Marybeth is also interested in the experiment from Problem 2 (associated with the enhancements for Captain America's shield), so she decides to start a detailed literature review on the subject. Among others, she found a paper where they used a 2"(4-1) fractional factorial design in the factors: (A) shield material, (B) throwing mechanism, (C) edge modification, and (D) handle adjustment. The experimental design used in the paper is shown in the table below. a. Run A B с D 1 (1) -1 -1 -1 1 2 a 1 -1 -1 1 3 bd -1 1 -1 1 4 abd 1 1 -1 1 5 cd -1 -1 1 -1 6 acd 1 -1 1 -1 7 bc -1 1 1 -1 abc 1 1 1 -1 paper? s) What was the generator used in the 2"(4-1) fractional factorial design described in the b. Based on the resolution of this design, what do you think about the generator used in the paper? Do you think it was a good choice, or would you have selected a different one? Explain your answer in detail.