STA 103: Midterm I. Print clearly on this exam. Only correct solutions that can be read will be given credit.

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1 STA 103: Midterm I May 30, 2008 Name: } {{ } by writing my name i swear by the honor code Read all of the following information before starting the exam: Print clearly on this exam. Only correct solutions that can be read will be given credit. You may use a calculator. You may tear off the scap sheets, and ask for more blank papers if necessary. For problem 1, write your answers on the blank space after each question. Be concise, and do not write more than the questions ask. For problems 3, write your answers on the underlines. Circle your answer for the True/False questions and multiple choice questions. You NEED to show your calculations for problem 3, and answers without supporting supporting work won t be given full credit. However, for the True/False questions and multiple choice questions, you do NOT need to explain the reasons NOR show the steps you get the answers. Carry out all calculations to 2 decimal places of accuracy. You should not leave the answers in fractions. This test has 10 problems and is worth 100 points. In addition, there is a bonus question worthy of 10 extra points. It is your responsibility to make sure that you have all of the pages! Good luck!

2 1. (10 points) Marijuana use and short-term memory. Ten marijuana users, aged 14 to 16, were randomly drawn from patients enrolled in a drug abuse program and compared to nine drug free volunteers of the same age group. Neuropsychological tests for short-term memory were given, and the marijuana group average was found to be significantly lower than the control group average. The marijuana group was held drug-free for the next six weeks, at which time a similar test was given with essentially the same result. The researcher concluded that (1) marijuana use caused adolescents to have short-term memory deficits; (2) the deficits continue for at least six weeks after the last use of marijuana. a. (4 pts) Can genuine casual relationship be established from this study? Explain your reasons in at most 2 sentences. b. (4 pts) Can the results be generalized to all year-olds? Explain your reasons in at most 2 sentences. c. (2 pts) Can you think of one potential confounding variables? (do not write more than one thing, otherwise you won t get any point)

3 2. (15 points) True or False. Circle T or F on the blank line. a. (3 pts) T F 20 people take IQ tests, and half of them are male. The median IQ among the males is 90, and the median IQ among the females is 110. Therefore, the median IQ in these 20 people is 100. b. (3 pts) T F Suppose you come to me to bet on who would win the Democratic US president nominee. I would like to offer you two different payoff odds for Clinton and Obama. If I want to set up a fair gambling game, and the payoff odds I offered for Clinton to win is 3 to 2, then the payoff odds for Obama should be 2 to 3. This also shows that I believe that Obama has a better chance to win. c. (3 pts) T F Bob tosses a fair coin just one time. The event that he gets a head and the event that he gets a tail are independent. d. (3 pts) T F A national survey wants to investigate the relationship between unemployment and education. The probability of getting a job for those graduated from college is no less than the probability that a person gets a job and graduates from college, no matter what the survey says. e. (3 pts) T F If two random variables X and Y are independent, they must be un-correlated.

4 3. (15 points) Parking Lots. Near a convention center, there are six square parking lots, whose sides have length 20, 20, 20, 30, 30, 60 meters respectively. Please write the answer on the underlines and show your calculations on the blank space under the question. a. (5 pts) The 75 th percentile of the areas of the lots is square meters. b. (5 pts) The standard deviation of the areas of the lots is square meters. c. (5 pts) Suppose now we use feet instead of meters to measure the lots. The standard deviation of the areas of those lots is square feet. Note that 1 meter equals to 3.28 feet.

5 4. (5 points) Choose the best answer Here is a histogram of the salaries of 506 people. Each bin has width 10, in unit of thousand dollars. One can deduct that 75 th percentile is in the interval (a) [0, 10] (b) [10, 20] (c) [20, 30] (d) [30, 40] (e) [40, 50]

6 5. (5 points) Choose the best answer Here is a box plot of the age of 506 people. approximately One can estimate that interquantile range is (a) 20 (b) 30 (c) 40 (d) 50 (e) 60 Box plot of Age

7 6. (10 points) Calculate the standard deviation. a. (5 pts) If a fair six-face die is tossed six times, denote P is the proportion of times that the die lands on six. What is the standard deviation of P? (a) (b) (c) (d) (e) none of the above b. (5 pts) A continuous random variable takes value in the interval [0, 2], with density function f(x) = x/2 when 0 x 2 and 0 elsewhere, which is shown in the Figure below. Which of the following numbers is most close to the standard deviation of X? (a) 0.25 (b) 0.3 (c) 0.4 (d) 0.5 (e)

8 7. (10 points) Five Face Dice. There are 5-sided dice that are balanced in a way that makes all faces (numbered 1, 2, 3, 4, 5) come up with equal probability. a. (5 pts) I roll two dice together. What is the probability that the sum of the numbers showed up is greater than 5? (a) 0.2 (b) 0.4 (c) 0.5 (d) 0.6 (e) 0.8 b. (5 pts) Suppose I roll two dice together, and I didn t show the result to you. However, I tell you that the sum of the numbers showed up is greater than 5. What is the chance you should guess for at least one 5 showed up in the two dice I rolled? (a) 0.2 (b) 0.4 (c) 0.5 (d) 0.6 (e) 0.8

9 8. (10 points) Guess the Factory. Two factories manufactures light bulbs. 3% of the light bulbs manufactured in factory A are defective. while only 2% of the light bulb manufactured in factory B are defective. I received a shipment of 1000 light bulbs, but it does not know they are manufactured in which factory. Suppose there is a half chance for each factory before I examine the shipment. a. (5 pts) I randomly picked one light bulb from the shipment, and found that it is defective. What is the probability that the shipment is from factory B? (a) 0.3 (b) 0.4 (c) 0.5 (d) 0.6 (e) 0.7 b. (5 pts) After seeing the defective sample, I decided to take two more samples. The result is one good, and one defective. Now what is the probability that the shipment is from factory B? (a) 0.3 (b) 0.4 (c) 0.5 (d) 0.6 (e) 0.7

10 9. (10 points) Tire mileage. Suppose a factory manufactures a new model of tires called LA 50 tire. Life tests shows that the mileage for each LA 50 tire is normally distributed with mean 47,900 miles, and standard deviation 2,050 miles. a. (5 pts) The factory wants to set up the guaranteed mileage so that no more than 5% of the tires will have to be replaced before the guaranteed mileage. What guaranteed mileage should the factory announce? (a) 43, 882 (b) 44, 528 (c) 47, 900 (d) 51, 272 (e) 51, 918 b. (5 pts) Suppose I bought two tires, and their performances can be assumed to be independent. What is the probability that the total mileage of these two tires can deliver is greater than 100,000 miles? (a) less than (b) (c) (d) (e) 0.153

11 10. (10 points) Tennis Tournament. In many tennis tournaments, the winning player is the rst to win two out of three sets. Suppose that two players named Venus and Serena are playing in a tournament match. Assume Venus has a 40% chance of winning any set against Serena, and that all sets are independent, i.e. what happens in one set does not affect the outcome of other sets. a. (5 pts) Given the match lasts three sets, which of the following numbers is most close to the expected number of sets that Venus wins? (a) 1.0 (b) 1.2 (c) 1.4 (d) 1.6 (e) 1.8 b. (5 pts) Given the match lasts three sets, which of the following numbers is most close to the standard deviation of the number of sets that Venus wins? (a) 0.4 (b) 0.5 (c) 0.6 (d) 0.7 (e) 0.8

12 11. (10 points) Extra Credit. There are a lot of colored marble balls in a box. 2/7 of them are red, 1/7 are blue, 1/7 are white and the remaining 3/7 are green. Suppose I randomly draw 5 balls from the box, what is the expected number of different colors in those 5 balls? Show your calculations and round the answer with 2 decimal places of accuracy.

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