STAT460 - Experimental Design
Daily Homework #9 - Power II

Question 1

Wagering on football games is often done "against the line." The "line" or "pointspread" is the predicted point difference between the two teams. If the Stetson football team is playing FSU, the line might be quoted as "Stetson +846." This would mean that 846 points would be added to Stetson’s score to determine the winner of the bet. (If FSU wins the game 847-0, then people who bet on FSU would win the bet. But if FSU only wins 845-0, then people who bet on Stetson would win the bet.) The idea behind this "line" is to make all bets 50/50 propositions. In practice, approximately half the gamblers will bet on each team. (The bookie can then use the loser’s bets to pay off the winners, and pocket the commission on the bets as profit.)

Balph Snerdwell believes he has a system that will enable him to pick the winners of college football game bets, against the line. He would like to validate the system empirically by testing it out on some games, before he invests his lunch money into actual bets using the system.

a) Balph decides to try his system out on 200 college football games. He will use the traditional α=.05. What is his rejection region?

b) Bookies typically charge a commission of $1 on every $10 wagered. (It costs $11 to place a $10 bet.) This means that, in order to break even, a bettor must win not half of the time, but 11/21 of the time. If Bolivar’s system is this good, what is the probability that his experiment will detect his abilities? That is, what is the power of the test against this particular alternative?

c) How large a sample would Bolivar need, to have power of 80% in this situation?

Question 2

Clorinda Cragdingle is testing the ping-pong ball machine used to select the winning number for the "Pick 3" lottery. The machine generates a random three-digit number (000 through 999) by making three random selections of the digits 0 through 9. We’ll simplify the task for now: Clorinda is simply testing to see whether "0" turns up with appropriate frequency as the first digit of the number. Hence, she will be testing H₀: π=.1 vs. H_A: π>.1 .

a) If she tests the machine 1000 times (and is traditional, using α=.05), what is the rejection region for her test?

b) What then would be the power of the test, if there is actually an 11% chance of getting a "0"?

c) She would like this power to be 98%. How large a sample size does she need?