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A holiday wreath has 8 light bulbs connected in series. What would have to be the reliability of each bulb if there were a 95% chance of the string's lighting after a year's storage?To investigate the average time to failure of a certain weld subjected to continuous vibration, 7 welded pieces were subjected to specified frequencies and amplitudes of vibration and their times to failure were 211, 350, 384, 510, 539, 620, and 715 thousand cycles. (a) Assuming the exponential model, construct a 95% confidence interval for the mean life (in thousands of cycles) of such a weld under the given vibration conditions. (b) Assuming the exponential model, test the null hypothesis that the mean life of the weld under the given vibration conditions is 500,000 cycles against the two-sided alternative / # 500,000. Use the level of significance 0.10.A certain component has an exponential life distribution with a failure rate of a = 0.0045 failures per hour (a) What is the probability that the component will fail during the first 250 hours it is in operation? (b) What is the probability that two such components will both survive the first 100 hours of operation?A sample of 300 high-reliability capacitors was placed on life test until the first four failures occurred and the test was then terminated. The first 4 failures were 3,582, 8,482, 8,921, and 16,303 hours. Find a 95% lower confidence limit for the mean life of the capacitors assuming an exponential model.To investigate the performance of a logic circuit for a small electronic calculator, a laboratory puts 75 of the circuits on life test without replacement under specified environmental conditions, and the first 10 failures are observed after 28, 46, 50, 63, 81, 101, 116, 137, 159, and 175 hours. Using the Weibull model, estimate the mean life of such a circuit. How does this value compare with the mean life that would have been obtained under the exponential assumption?Using the estimates of the parameters of the Weibull model obtained in Exercise 1, estimate the probability that this kind of circuit will perform satisfactorily for at least 100 hours. Exercise 1 To investigate the performance of a logic circuit for a small electronic calculator, a laboratory puts 75 of the circuits on life test without replacement under specified environmental conditions, and the first 10 failures are observed after 28, 46, 50, 63, 81, 101, 116, 137, 159, and 175 hours. Using the Weibull model, estimate the mean life of such a circuit. How does this value compare with the mean life that would have been obtained under the exponential assumption?With reference to Exercise 1, make (a) a total time on test plot (b) a Weibull plot Exercise 1 To investigate the performance of a logic circuit for a small electronic calculator, a laboratory puts 75 of the circuits on life test without replacement under specified environmental conditions, and the first 10 failures are observed after 28, 46, 50, 63, 81, 101, 116, 137, 159, and 175 hours. Using the Weibull model, estimate the mean life of such a circuit. How does this value compare with the mean life that would have been obtained under the exponential assumption?(Stress-strength models for reliability) An alternative model used in reliability treats the environmental stress as a random variable X, with probability density ((x), and the strength of the component to withstand this stress as an independent random variable Y having probability density g()). Then, the reliability is defined as R = P(Y > X] = fx) gly)dxdy -00 = where F(x) is the distribution function of X. Evaluate this reliability when (a) X has an exponential distribution with o = 0.01 and Y has an exponential distribution with failure rate 0.005; (b) X has an exponential distribution with o = 0.005 and Y has an exponential distribution with failure rate 0.005; (c) In X has a normal distribution with / = 60 and o = 5 and In Y has a normal distribution with p = 80 and o = 5.In a laboratory experiment, 18 determinations of the coefficient of friction between leather and metal yielded the following results: 0.59, 0.56, 0.49, 0.55, 0.65, 0.55, 0.51, 0.60, 0.56, 0.47, 0.58, 0.61, 0.54, 0.68, 0.56, 0.50, 0.57, and 0.53. Use the sign test at the 0.05 level of significance to test the null hypothesis a = 0.55 against the alternative hypothesisThe quality-control department of a large manufacturer obtained the following sample data (in pounds) on the breaking strength of a certain kind of 2-inch cotton ribbon: 153, 159, 144, 160, 158, 153, 171, 162, 159, 137, 159, 159, 148, 162, 154, 159, 160, 157, 140, 168, 163, 148, 151, 153, 157, 155, 148, 168, 152, and 149. Use the sign test at the 0.01 level of significance to test the null hypothesis x = 150 against the alternative hypothesis 7 > 150
