608 Chapter 14*:Life Testing
20.What is the Bayes estimate ofλ=1/θin Problem 18 if the prior distribution on
λis exponential with mean 1/30?
21.The following data represent failure times, in minutes, for two types of electrical
insulation subject to a certain voltage stress.Type I 212, 88.5, 122.3, 116.4, 125, 132, 66
Type II 34.6, 54, 162, 49, 78, 121, 128Test the hypothesis that the two sets of data come from the same exponential
distribution.
22.Suppose that the life distributions of two types of transistors are both exponential.
To test the equality of means of these two distributions,n 1 type 1 transistors are
simultaneously put on a life test that is scheduled to end when there have been
a total ofr 1 failures. Similarly,n 2 type 2 transistors are simultaneously put on a life
test that is to end when there have beenr 2 failures.
(a) Using results from Section 14.3.1, show how the hypothesis that the means
are equal can be tested by using a test statistic that, when the means are equal,
has anF-distribution with 2r 1 and 2r 2 degrees of freedom.
(b) Supposen 1 =20,r 1 =10 andn 2 =10,r 2 =7 with the following data
resulting.
Type 1 failures at times:
10.4, 23.2, 31.4, 45, 61.1, 69.6, 81.3, 95.2, 112, 129.4
Type 2 failures at times:
6.1, 13.8, 21.2, 31.6, 46.4, 66.7, 92.4
What is the smallest significance levelαfor which the hypothesis of equal
means would be rejected? (That is, what is thep-value of the test data?)
23.IfXis a Weibull random variable with parameters (α,β), show thatE[X]=α−1/β (1+1/β)where (y) is the gamma function defined by(y)=∫∞0e−xxy−^1 dxHint:WriteE[X]=∫∞0tαβtβ−^1 exp{−αtβ}dt