The authors of the article “A Probabilistic Insulation Life Model for Combined Thermal-Electrical Stresses” (IEEE Trans. on Elect. Insulation, 1985:519–522) state that “the Weibull distribution is widely used in statistical problems relating to aging of solid insulating materials subjected to aging and stress.”They propose the use of the distribution as a model for time (in hours) to failure of solid insulating specimens subjected to AC voltage. The values of the parameters depend on the voltage and temperature; suppose α = 2.5 and β = 200 (values suggested by data in the article).
a. What is the
b. What is the probability that a specimen’s lifetime is between 100 and 250?
c. What value is such that exactly 50% of all specimens have lifetimes exceeding that value?
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Chapter 4 Solutions
Probability and Statistics for Engineering and the Sciences
- 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 = 16 sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? E(X) = GPa GPa x = (b) Answer the questions posed in part (a) for a sample size of n = 64 sheets. E(X) GPa GPa ox = (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 decreased variability of X that…arrow_forwardYoung'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) = GPa GPa ox= (b) Answer the questions posed in part (a) for a sample size of n = 256 sheets. E(X) = ox GPa GPa (c) For which of the two random samples, the one of part (a) or the one of part (b), is 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 (a). This is due to the decreased variability of X that…arrow_forwardYoung'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 = 16 sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? E(X) = GPa ? X = GPa (b) Answer the questions posed in part (a) for a sample size of n = 64 sheets. E(X) = GPa ? X = GPaarrow_forward
- 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 = 16 sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? E(X) = GPa ? X = GPa (b) Answer the questions posed in part (a) for a sample size of n = 256 sheets. E(X) = GPa ? X = GPa (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. X is more likely to be within 1 GPa of the mean in part (a). This is due…arrow_forwardAn experiment was conducted to investigate leaking current in a SOS MOSFETS device. The purpose of the experiment was to investigate how leakage current varies as the channel length changes. Four channel lengths were selected. For each channel length, five different widths were also used, and width is to be considered a nuisance factor. The data are as follows: (a) Test the hypothesis that mean leakage voltage does not depend on the channel length using a= 0.05. (b) Analyze the residuals from this experiment. Comment on the residual plots. (c) The observed leakage voltage for channel length 4 and width 5 was erroneously recorded. The correct observation is 4.0. Analyze the corrected data from this experiment. Is there evidence to conclude that mean leakage voltage increases with channel length?arrow_forwardUrinary fluoride concentration (in parts per million) was measured for both a sample of livestock that had been grazing in an area previously exposed to fluoride pollution and a similar sample of livestock that had grazed in an unpolluted region. Do the accompanying data indicate strongly that the mean fluoride concentration for livestock grazing in the polluted region is larger than that for livestock grazing in the unpolluted region? Assume that the distributions of urinary fluoride concentration for both grazing areas have the same shape and spread. Use a significance level of 0.05. Group 1 (Polluted): 21.3 || 18.7 || 23.0 || 17.1 || 16.8 || 20.9 || 19.7 Group 2 (Unpolluted): 14.2 || 18.3 || 17.2 || 18.4 || 20.0 What is/are the Mann-Whitney Rank Sum critical value(s) for this problem? Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a 43 b 56 n 58,33 d 60arrow_forward
- Lemons and Car Crashes Listed below are annual data for various years. The data are weights (metric tons) of lemons imported from Mexico and U.S. car crash fatality rates per 100,000 population [based on data from “The Trouble with QSAR (or How I Learned to Stop Worrying and Embrace Fallacy),” by Stephen Johnson, Journal of Chemical Information and Modeling, Vol. 48, No. 1]. Is there sufficient evidence to conclude that there is a linear correlation between weights of lemon imports from Mexico and U.S. car fatality rates? Do the results suggest that imported lemons cause car fatalities?arrow_forward.. .. 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 = 16 sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? E(X) GPa %3D GPa (b) Answer the questions posed in part (a) for a sample size of n = 64 sheets. E(X) = GPa GPa (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. X is more likely to be within 1 GPa of the mean in part (b). This is due to the decreased variability of X that…arrow_forwardThe accompanying data set consists of observationson shower-flow rate (L/min) for a sample of n 5 129houses in Perth, Australia (“An Application of BayesMethodology to the Analysis of Diary Records in aWater Use Study,” J. Amer. Stat. Assoc., 1987:705–711):4.6 12.3 7.1 7.0 4.0 9.2 6.7 6.9 11.5 5.111.2 10.5 14.3 8.0 8.8 6.4 5.1 5.6 9.6 7.57.5 6.2 5.8 2.3 3.4 10.4 9.8 6.6 3.7 6.48.3 6.5 7.6 9.3 9.2 7.3 5.0 6.3 13.8 6.25.4 4.8 7.5 6.0 6.9 10.8 7.5 6.6 5.0 3.37.6 3.9 11.9 2.2 15.0 7.2 6.1 15.3 18.9 7.25.4 5.5 4.3 9.0 12.7 11.3 7.4 5.0 3.5 8.28.4 7.3 10.3 11.9 6.0 5.6 9.5 9.3 10.4 9.75.1 6.7 10.2 6.2 8.4 7.0 4.8 5.6 10.5 14.610.8 15.5 7.5 6.4 3.4 5.5 6.6 5.9 15.0 9.67.8 7.0 6.9 4.1 3.6 11.9 3.7 5.7 6.8 11.39.3 9.6 10.4 9.3 6.9 9.8 9.1 10.6 4.5 6.28.3 3.2 4.9 5.0 6.0 8.2 6.3 3.8 6.0 b. What is a typical, or representative, flow rate?arrow_forward
- Q.1 An Electronic equipment used for the analysis and storage of the data in an industry has three identical input devices/ports. Different components of the electronic equipment are connected to each other as shown in Figure Q.1. Assume the following estimated reliability values of the components as shown in Table Q.1. Calculate the reliability of the electronic equipment operating successfully with at least one input device/port for 1000 hours? Also, estimate Mean Time to Failure (MTTF) for the electronic equipment for 1000 hrs. Table Q.1 Component Failure rate (2) x 10°, per hr Input device/Port Storage drive Central Processing Unit (CPU) Manual Key Pad Monitor/screen to display results 45 35 7 5 10 Input device 1 Monitor/ Storage device Manual Input device 2 CPU Кeурad screen Input Figure Q.1 device 3arrow_forwardA study was undertaken to compare the average high-density lipoprotein (HDL) levels of normal and obese adults in a certain community. HDL levels of a random sample of adults from this community were measured (in milligrams per deciliter or mg/dL) and presented below. Assume that the HDL levels of both the normal and obese adults follow the normal distribution with unknown but equal population variances. Is there a significant difference between the average HDL levels of normal and obese adults? Use a 5% level of significance.arrow_forward
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