The times to failure of certain electronic components in

Question

The times to failure of certain electronic components in accelerate environment tests are 15, 28, 3, 12, 42, 19, 20, 2, 25, 30, 62, 12, 18, 16, 44, 65, 33, 51, 4, and 28 minutes. Looking upon these data as a random sample from an exponential population, use the results of Exercise 21 and Theorem 2 to test the null hypothesis θ = 15 minutes against the alternative hypothesis θ  15 minutes at the 0.05 level of significance. (Use ln 1.763 = 0.570.)

Exercise 21

A random sample of size n is to be used to test the null hypothesis that the parameter θ of an exponential population equals θ₀ against the alternative that it does not equal θ₀.

(a) Find an expression for the likelihood ratio statistic.

(b) Use the result of part (a) to show that the critical region of the likelihood ratio test can be written as

Theorem 2

† For large n, the distribution of -2 · ln approaches, under very general conditions, the chi-square distribution with 1 degree of freedom.