An article in International Journal of Electrical Power & Energy Systems "Stochastic Optimal Load Flow Using a Combined Quasi-Newton and Conjugate Gradient Technique" (1989, Vol. 11(2), pp. 85-93) considered the problem of optimal power flow in electric power systems and included the effects of uncertain variables in the problem formulation. The method treats the system power demand as a normal random variable with 0 mean and unit variance. What is the probability that the power demand is positive? Round your answer to two decimal places (e.g., 98.8)

1) Below are data of foam indexes for three different espresso makers. Test the hypothesis that foam indexes differ between the three. (Note: these data are highly kurtotic, so normality cannot be assumed.) Set a at 0.05. Which test should you use and why? State the hypothesis, provide the critical value and test statistic and state your conclusions. Hyper-Espresso Method (HIP) I-Espresso System (IT) 56.19 Bar Machine (BM) 36.64 70.84 39.65 46.68 36.67 35.35 40.11 37.74 73.19 35.96 57.78 38.52 48.61 33.52 37.12 37.33 72.77 21.02 24.81 65.04 32.68 48.33 34.18 62.53 23.08 54.26

3. A certain actuarial model of mortality rate is fitted to some given data. Table 2 below reports the Z-score statistic of the model graduation. (a) List the key steps involved in developing an actuarial model. (b) Explain why we need graduation. What do we actually test statistically? (c) What are the drawbacks of a x²- test? Explain how the sign test may overcome these. (d) Write a statistical hypothesis to test the adherence of the model to the data. (e) The life table data below provides an estimate of mortality rates obtained from a three-parameter Poisson model. Give the quality of goodness-of-fit of the Poisson model by performing statistical tests based on:, and draw the conclusions. i. x²-test. ii. Standardized deviations test. [Consider the following subintervals for the data range: A₁ = (-∞, −3], A₂ = (−3,−2], A3 = (−2, −1], A4 = (–1, 0], A5 = (0, 1], A6 = (1, 2], A7 = (2, 3], and Ag = (3, ∞).] iii. Sign test iv. Change-of-sign test v. Grouping-of-sign test vi. Lag-2 serial…

Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article "Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis" (Intl. J. of Advanced Manuf. Tech., 2010: 117-134)). (a) If X is the sample mean Young's modulus for a random sample of n = 64 sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? E(X) ох = GPa GPa (b) Answer the questions posed in part (a) for a sample size of n = 256 sheets. E(X) GPa GPa ох = (c) For which of the two random samples, the one of part (a) or the one of part (b), is X more likely to be within 1 GPa of 70 GPa? Explain your reasoning. O X is more likely to be within 1 GPa of the mean in part (b). This is due to the increased variability of X that…