Språkstatistik HT97:12

Exercise class 4

Deadline

These exercises are supposed to be handed in on Monday October 06. They deal with the chapters 16-18 and 26 (without 26.6) of the course book.

Exercises

1. In an icehockey competition teams have 40% chance of winning, 20% chance on a draw and 40% chance of losing. Winning a game earns two points, a draw earns one point and losing a game earns no points. Make a box model for the number of points that a team can obtain in a game.

2. (continuation of previous question) Team A plays 4 games. How many points do you expect the team to have after those games? What is the standard error of that expectation?

3. (continuation of previous question) We observe that team A looses all four games and obtains zero points. Can the difference between this observation and the result of exercise 2 be explained by a chance error? Or is the box model of exercise 1 a bad model for this team? Motivate your answer.

4. We have two dice which instead of the standard number sequence 1, 2, 3, 4, 5, 6 contain the numbers 1, 2, 3, 5, 7 and 12. Draw the probability histogram for the sum of the numbers of a throw with this pair of dice.

5. We take one of the dice of question 4 and throw it 100 times. Estimate the probability of the sum of the results being larger than 426.

6. Estimate the probability of getting less than 14 (exclusive) sevens in the 100 dice throws of the previous question.

7. We will bet on the outcome of a series of coin flips. If the number of heads in the series is 10 or more higher than the expected value then we will win a dollar. What is better: betting on a series of 123 coin flips or betting on 321 coin flips? Motivate your answer.

8. A box only contains cards with number 1 and cards with number 0. We do not know how many cards there are in the box. We only know that there are twice as many 1 cards as 0 cards. Compute the average and the standard deviation of the box.

9. A safari park is assumed to contain equal numbers of lions, elephants and crocodiles and no other animals. However a tourist driving around in the park reports that of the first 100 animals she saw 45 were crocodiles. We assume that the animal spottings are independent of each other. Can the difference between the assumption and the observation be explained by a chance error? Motivate your answer.

10. A gambler reasons in the following way:

    In a roulette game the probability that the ball lands on a square is 1/38 for all squares. However they only pay 36 times the money I bet on a square. I will start by looking what happens in the first three games without participating. In the fourth game I will put a dollar on each of the 35 squares that weren't selected in the first three games. Now the ball will land on one of these and I will win 36-35=1$. Then in the fifth game I will put a dollar on each of the 34 squares that weren't selected in the first four games. Now I will win 36-34=2$. I will continue playing like this and win a lot of money.

    Will this playing system work? If you think it is correct then compute the gain of the player in 38 games (including the first three). If you think it is wrong then state why.

Each exercise is worth 1 point.

Related Exercises

If you want to make extra exercises you can try the following optional exercises from the second edition of the book:

Law of averages: 16B4, 16C5.
Expected value and SE: 17A1, 17A3, 17A6, 17B2, 17B4, 17C1, 17D1, 17D4, 17E3, 17E6.
Normal approximation: 18A2, 18B4, 18C1.
Tests of significance: 26A4, 26A5, 26B3, 26C2, 26C4, 26D2, 26D5, 26E2, 26E4.

A code like 16B4 points to the fourth exercise of section B from chapter 16. R stands for review exercise. The answers to these exercises can be found in the final part of the book.