Stat 88 midterm review questions - Fall 2017 Stoyanov PDF

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STATISTICS 88 Fall 2017

Shobhana M. Stoyanov Practice problems

• Please make sure to go over the problems of each chapter as well.

1. The public radio show ”A Prairie Home Companion,” features news from the fictional town of Lake Wobegon, MN, home to many Norwegian bachelor farmers, and where ”all the women are strong, all the men are good looking, and all the children are above average.” Suppose average means average for the town. True or false, and explain: Such a town could not possibly exist. 2. Evans Hall is home to the University of California, Berkeley Statistics Department. In 150 trips on the Evans Hall elevator, the average waiting time for the car to arrive was 6 minutes. What is the largest number of trips (among those 150) in which it took 18 minutes or longer for the car to arrive? 3. A certain population has an average IQ of 108 with an SD of 10. What can you say about the fraction of people in that population with IQs between 83 and 133? 4. If the SD of a list of numbers is zero, what can we say about the list (circle all that apply): (a) All the numbers in the list are 0 (b) All the numbers in the list are equal (c) The mean of the list is 0 5. Which of the above conditions will ensure that the SD is 0? 6. You can find a list of the CEOs with the highest compensations on the internet at various sites. Interestingly, the list seems to be site-dependent. In any case, the list from the AFL-CIO website has the salaries for the 100 highest-paid CEOs. If we consider the salaries in millions of dollars, the average salary is about 29.5 million dollars, the median salary about 25 million dollars, and the SD of the salaries about 13.5 million dollars. What would be an upper bound on the percentage of CEOs (from this list of the top 100 earners) that make at least 50 million dollars? Give the best upper bound that you can. 7. How many five-letter code words are possible using the letters in HOUSE if: (a) The letters may be repeated? (b) The letters may not be repeated? 8. A pair of dice are thrown. Find the probability that both dice show the same number of spots. 9. Show that if A and B are independent events, then Ac and B c must also be independent. 10. A, B and C are mutually independent events that occur with probabilities P (A) = 0.3, P (B) = .2, P (C) = 0.5. (a) Find the probability that at least one of the events occurs. (b) Find the probability that exactly 2 of the events occur. 11. In a game of poker, 5 cards are dealt from a well-shuffled standard deck. (A standard deck has 52 cards: 4 suits, with 13 cards in each suit.) (a) How many 5-card hands can be dealt? (b) What is the probability that a 5-card hand will contain a full house (3 cards of one value, and 2 of another value)? 12. A (biased) coin is flipped until a head appears for the first time. Let X be the number of tails that occur, and let 1 P (H ) = . 3 (a) Write down the probability that X = k, where k = 0, 1, 2, . . . (b) Find P (X = 3|X > 2)

(c) Now, suppose the coin is flipped until we see three heads, so we stop after the third head. Let Y be the number of tails in this situation. Write down the probability that Y = k, where k = 0, 1, 2, . . . 13. 14. In a large statistics course, the scores for the final followed the normal curve closely. The average was 70 points, and three-fourths of the class scored between 60 and 80 points. The SD of the scores was: (a) larger than 10 points (b) smaller than 10 points (c) impossible to say with the information given. 15. A and B are independent events. The chance of A is 0.3, and the chance of B is 0.5. Fill in the blanks, choosing from the options given below. Explain your answer. (a) The chance of both A and B happening is (i)0.15 (ii) 0.65 (iii)0.8 (iv) 0.3

. (v) 0.5

(vi) 0.35

(b) The chance of A happening, given that B has happened is (i)0.15 (ii) 0.65 (iii)0.8 (iv) 0.3 (v) 0.5

(vi) 0.35

(vii) Need more information.

(vi) 0.35

(vii) Need more information.

(vi) 0.35

(vii) Need more information.

(c) The chance of either A or B happening is (i)0.15 (ii) 0.65 (iii)0.8 (iv) 0.3

(v) 0.5

(d) The chance of neither A nor B happening is (i)0.15 (ii) 0.65 (iii)0.8 (iv) 0.3

(v) 0.5

(vii) Need more information. .

. .

16. Consider a standard deck of 52 cards (4 suits: hearts, spades, clubs, diamonds, 13 cards of each suit: 2-10, ace, king, queen, jack). Two cards are dealt from a well-shuffled deck. (a) What is the probability that the first card is a heart? (b) What is the probability that the second card is a heart? (c) What is the probability that both cards are hearts? (d) What is the probability that exactly one of the two cards dealt is a heart? (e) What is the probability that neither of the two cards dealt is a heart? 17. Suppose that A, B and C are three events with probabilities 0.7, 0.5 and 0.2 respectively. Explain your reasoning. (a) What is the largest that P (A ∩ B) can be?

(1 point)

(b) What is the smallest that P (A ∩ B) can be?

(1 point)

(c) What is the smallest that P (A ∩ B ∩ C) can be?

(1 point)

(d) Suppose now that B and C are independent. Can P (B ∩ C) = 0? If yes, explain. If no, write what this probability should be. (2 points) 18. You and a friend are rolling a fair die. If you roll it 10 times, what is the chance that you will see no sixes? 19. Now you begin betting money on the rolls. You keep rolling the die over and over. If you roll a six, your friend gives you a dollar. If you don’t, you give your friend a dollar. You stop if your net gain is two dollars. (a) What is the probability that you will stop after exactly two rolls? (b) What is the probability that you will stop after exactly three rolls? (c) What is the probability that you will stop after exactly four rolls?...