Your company supplies an electronic component that is critical to the navigational systems of large jet aircraft. The operating life of the component has an exponential distribution with some mean value \(v\), where operating life is measured in 100,000 hour units. You are seeking a contract to supply these components to a major aircraft manufacturer on the West Coast. The contract calls for a minimum mean operating life of 750,000 hours for the component, and you must provide evidence on the reliability of your product. You have a random sample of observations on tests of 300 of your components that provide measurements on the components' operating lives. According to the tests performed, the mean operating life of the components was 783,824 hours.

(a) In designing a UMP level \(\alpha\) test in this situation, should the null hypothesis be defined as \(\theta \geq 7.5\) or \(\theta \leq 7.5\) ? Base your discussion on the characteristics of the power function of each of the tests.

(b) Design a UMP size \(\alpha\) test of whichever null hypothesis you feel is appropriate based on you discussion in (a). Choose whatever size test you feel is appropriate, and discuss your choice of size.

(c) Conduct the hypothesis test. Should your company get the contract? Why or why not?