Chapter 10.5, Problem 69E

Optical fibers are used in telecommunications to transmit light. Suppose current technology allows production of fibers that transmit light about 50 km. Researchers are trying to develop a new type of glass fiber that will increase this distance. In evaluating a new fiber, it is of interest to test H₀: μ = 50 versus H_a: μ > 50, with μ denoting the mean transmission distance for the new optical fiber.

1. a. Assuming σ = 10 and n = 10, use Appendix Table 5 to approximate β, the probability of a Type II error, for each of the given alternative values of μ for a test with significance level 0.05:
   1. i. 52
   2. ii. 55
   3. iii. 60
   4. iv. 70
2. b. What happens to β in each of the cases in Part (a) if σ is actually larger than 10? Explain your reasoning.