Problems 607
(b) Test at the .05 level of significance the hypothesis thatθ =20 versus the
alternative thatθ=20.
(c) Determine a 95 percent confidence interval forθ.14.Using the fact that a Poisson process results when the times between successive
events are independent and identically distributed exponential random variables,
show thatP{X≥n}=Fχ (^22) n(x)
whenX is a Poisson random variable with meanx/2 andFχ (^22) nis the chi-square
distribution function with 2n degrees of freedom. (Hint: Use the results of
Section 14.3.2.)
15.From a sample of items having an exponential life distribution with unknown
meanθ, items are tested in sequence. The testing continues until either therth
failure occurs or after a timeTelapses.
(a) Determine the likelihood function.
(b) Verify that the maximum likelihood estimator ofθis equal to the total time
on test of all items divided by the number of observed failures.
16.Verify that the maximum likelihood estimate corresponding to Equation 14.3.9
is given by Equation 14.3.10.
17.A testing laboratory has facilities to simultaneously life test 5 components. The
lab tested a sample of 10 components from a common exponential distribution
by initially putting 5 on test and then replacing any failed component by one still
waiting to be tested. The test was designed to end either at 200 hours or when all
10 components had failed. If there were a total of 9 failures occurring at times 15,
28.2, 46, 62.2, 76, 86, 128, 153, 197, what is the maximum likelihood estimate
of the mean life of a component?
18.Suppose that the remission time, in weeks, of leukemia patients that have under-
gone a certain type of chemotherapy treatment is an exponential random variable
having an unknown meanθ. A group of 20 such patients is being monitored and,
at present, their remission times are (in weeks) 1.2, 1.8∗, 2.2, 4.1, 5.6, 8.4, 11.8∗,
13.4∗, 16.2, 21.7, 29∗, 41, 42∗, 42.4∗, 49.3, 60.5, 61∗, 94, 98, 99.2∗where
an asterisk next to the data means that the patient’s remission is still continuing,
whereas a data point without an asterisk means that the remission ended at that
time. What is the maximum likelihood estimate ofθ?
19.In Problem 17, suppose that prior to the testing phase and based on past experi-
ence one felt that the value ofλ=1/θcould be thought of as the outcome of a
gamma random variable with parameters 1, 100. What is the Bayes estimate ofλ?