Let's turn our attention back to the random variable D we defined earlier. As we know, D follows a normal distribution with mean E(D) and variance Var(D), both of which we found above.

2. Check Your Head , Part I Over the last few years , there has been growing concern about the long-term effect of callsion* induced concussions in professional sports , particularly football . Specifically , collisions that result in a player's head undergoing large rotational accelerations are thought to be the most dangerous - collisions where a player is struck from the side near the chin , for example . In response to this concern , sports equipment companies are attempting to design helmets that reduce rotational acceleration by directing the force of impact in directions least likely to rotate the head . Mi Ke De , a sports equipment company is developing such a helmet and currently has two prototypes , Prototype* and Prototype { . The team of engineers who designed the helmets plan to conduct side impact tests with 25 Prototype* helmets and 25 Prototype {' helmets . using accelerometers to measure the rotational acceleration caused by each impact ( the units of rotational acceleration are in kiloradians per second squared , or* ^] . Suppose that the designers know the standard deviation of the rotational acceleration far side impacts on Pro- tatype* helmets is or = 1.5brake and the standard deviation of the rotational acceleration for side impacts on Prototype { helmets is ar - 1.5^`^ . Let ur he the true average rota - tional acceleration for side impacts on Prototype* and let us he the true average rotational acceleration for side impacts on Prototype {. If the designers find evidence that ux _ MY Spring 2010 Stat $0.5 Homework !) Thursday , April 25 then they will use Prototype {. However , if the designers instead find evidence that MY _ MIX then they will use Prototype*` Since the researchers have not vet collected any data , we can consider the side- impact test. results to he independent random variables , using $ 1 , $7 . .... {} for the side- impact test results using Prototype* and * 1 . 12 ..... In for the side- impact test results using Prototype* Y . We can also define * = = = $ 1 + 75 }2 + ... + =_$2{ and }" = = 1 1 + 75 /2 + ... + = $25. ( a ) What is FIX! ? ( h; What is Vari* ] ?" ( C ) What is the distribution of* ? How do we know that ?" ( d ) What is EIVY ? ( e ) What is Vari> > ?! IF ) What is the distribution of !" ? How do we know that ?"\f\f\f