Please explain in details:-
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.In life testing we are sometimes interested in establishing tolerance limits for the life of a component (see Section 15.7); in particular, we may be interested in a one-sided tolerance limit # for which we can assert with a (1 - o) 100% confidence that at least 100 - P percent of the components have a life exceeding t*. Using the exponential model, it can be shown that a good approximation is given by - 27, ( In P ) where Tr is as defined on page 494 and the value of y is to be obtained from Table 5 with 2 r degrees of freedom. (a) Using the data of Exercise 1, establish a lower tolerance limit for which one can assert with 95% confidence that it is exceeded by at least 80% of the lifetimes of the heaters. (b) Using the data of Exercise 2, establish a lower tolerance limit for which one can assert with 99% confidence that it is exceeded by at least 90% of the lifetimes of the given welds. Exercise 1 In a nonreplacement life test, 35 space heaters were put into continuous operation, and the first 5 failures occurred after 250, 380, 610, 980, and 1,250 hours. (a) Assuming the exponential model, construct a 99% confidence interval for the mean life of this kind of space heater. (b) To check the manufacturer's claim that the mean life of these heaters is greater than 5,000 hours, test the null hypothesis / = 5,000 against an appropriate alternative, so that the burden of proof is put on the manufacturer. Use a = 0.05. Exercise 2 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 system consists of 5 identical components connected in parallel. What must be the reliability of each component if the overall reliability of the system is to be 0.96?Using the fact that 2 Tr /p is a value of a random variable having the chi square distribution with 2 r degrees of freedom, derive the confidence interval for / given on page 494.Fifteen assemblies are put on accelerated life test without replacement, and the test is truncated after 4 failures. If the first 4 failures occurred at 16.5, the mean 19.2 20.8, and 37.3 hours, assuming an exponential model, (a) find a 90% confidence interval for the failure rate of such assemblies under these accelerated conditions; (b) test the null hypothesis that the failure rate is 0.004 failure per hour against the alternative that it is less than 0.004, using the 0.01 level of significance.A system consists of 6 components connected as in Figure. Find the overall reliability of the system, given that the reliabilities of A, B, C, D, E, and Fare, respectively, 0.95, 0.80, 0.90, 0.99, 0.90, and 0.85. Figure B A C E DOne hundred devices are put on life test and the times to failure (in hours) of the first 10 that fail are 7.0 14.1 18.9 31.6 52.8 80.0 164.5 355.4 451.0 795.1 Assuming a Weibull failure-time distribution, estimate the parameters o and 6 as well as the failure rate at 1,000 hours. How does this value of the failure rate compare with the value we would obtain if we assumed the exponential model?A sample of 200 switches was placed on life test consisting of repeated on-off cycles. The test was terminated after the third failure. The first three failure times were 2,076, 3,667, and 9,102. Find a 95% lower confidence limit for the mean life, in number of cycles, of the switches. Use the exponential model.Suppose that the flight of an aircraft is regarded as a system having the three main components A (aircraft), B (pilot), and C (airport). Suppose, furthermore, that component B can be regarded as a parallel subsystem consisting of 1(captain), 82 (first officer), and 83 (flight engineer); and C is a parallel subsystem consisting of C1 (scheduled airport) and C2 (alternate airport). Under given flight conditions, the reliabilities of components A, 81, 82, 83, C1, and C2 (defined as the probabilities that they can contribute to the successful completion of the scheduled flight) are, respectively, 0.9999, 0.9995, 0.999, 0.20, 0.95, and 0.85. (a) What is the reliability of the system? (b) What is the effect on system reliability of having a flight engineer who is also a trained pilot, so that the reliability of 83 is increased from 0.20 to 0.99? (c) If the flight crew did not have a first officer, what then would be the effect of increasing the reliability of 83 from 0.20 to 0.99? (d) What is the effect of adding a second alternate landing point, C3, with reliability 0.80?\fIn some reliability problems we are concerned only with initial failures, treating a component as if (for all practical purposes) it never fails, once it has survived past a certain time t = o. In a problem like this, it may be reasonable to use the failure rate for 0
