12. A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under speci ed conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design.

a. De ne the parameter of interest and state the relevant hypotheses.

b. Suppose braking distance for the new system is normally distributedwith s10. Let X denote 1X  102/ 1s/1n2 z  1x  102/ 1s/1n2 x  10.1032 or x 9.8968 the sample average braking distance for a random sample of 36 observations. Which of the following rejection regions is appropriate: R1 

{ :  124.80}, R2  { :

115.20}, R3 

{ : either  125.13 or 114.87}?

c. What is the signi cance level for the appropriate region of part (b)? How would you change the region to obtain a test with a  .001?

d. What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the appropriate region from part

(b) is used?

e. Let . What is the signicance level for the rejection region {z: z

2.33}? For the region {z: z

2.88}?