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599x0711 1/8/07 ECO251 QBA1 FIRST EXAM ECO500 Business Statistics Name: _____________________ Student Number : _____________________ Remember – Neatness, or at least legibility, counts. In all questions an answer needs a calculation or explanation to count. Show your work! Part I. (10 points) The following numbers are a sample of 20. 22, 11, 22, 23, 17, 25, 17, 20, 26, 14, 23, 15, 34, 12, 15, 11, 19, 16, 34, 24 Compute the following: Show your work! a) The Median (1) b) The Standard Deviation (3) c) The Interquartile Range (3) d) The Coefficient of variation (1) e) A measure of skewness (2) Part II. (At least 40 points. Parentheses give points on individual questions. Brackets give cumulative point total.) Exam is normed on 50 points. 1. For the sample above, find the two numbers that would be at the limits of an interval of 2.5 standard deviations from the mean x 2.5s . If these were population parameters instead of sample statistics, what would be the minimum fraction of the data between these two numbers according to Tchebyschev's rule? The Empirical rule says that about 98.8% of the data should be between these two points. How does the fraction of the data actually between these two points compare? (5) 2. (Most of the following problems are due to Mansfield and Schmidt) As you leave a restaurant the attendant gives you a randomly selected coat from the coatroom. If 10% of the coats are worth $200, 40% are worth $100 and 50% are worth $50, show that you have a valid distribution and compute the expected value and standard deviation of the value of the coat you receive. (5) 3. If the gross profit on each calculator sold is $5 and the fixed cost is $100000 per year, so that if the firm sells 50000 calculators, annual profit is 550000 100000 150000 , what is the expected value and standard deviation of the firm’s annual profit? ( I should have said that average sales are $75000 units with a standard deviation of $30000.) (4) [14] 4. If a population has a mean weight of 140 pounds and a standard deviation of 30 pounds, what is the mean and standard deviation of the total weight of 3 randomly selected people? (3) [17] 5. Two cards are drawn at random from a deck of cards. Identify the following events: A1 an ace on the first draw and A2 an ace on the second draw. Find the following: P A1 , P A1 A2 , P A2 A1 , P A2 A1 , P A1 A2 , P A2 . Can you explain why P A1 A2 P A1 P A2 ? Repeat the exercise for the following events H 1 , a head on the first flip and H 2 , a head on the second flip. By contrasting P A1 A2 and PH 1 H 2 , show that only one of these pairs of events are independent. (8) [25] 6. Your significant other is supposed to pick you up at 5pm. The probability that said other remembered to buy gas is 60%. If said other remembered there is a 90% probability that you will be picked up on time. If buying gas was forgotten the probability that you will be picked up on time is 60%. No one shows at 5pm. What is the probability that your significant other forgot to buy gas? To keep us all together on this, let G 1 599x0711 1/8/07 be the event that gas is remembered and T be the event that you are picked up on time. There is partial credit her for finding the probability of events like T G or T G . (5) [30] 7. The formula for the probability of x successes in n tries is Px C xn p x q n x , where p is the probability of a success on 1 try and q is the probability of failure. For example, if your probability of success on 1 try is .20 and the probability of failure is .80, the probability of 3 successes on 7 tries is 7! .20 3 .80 4 . This may help you in the following: You roll one die and then toss a coin the C 37 p 3 q 4 3!4! number of times indicated by the die; for example if you roll a two you toss the coin twice, Let y be the number of heads you get. Find the possible values of y and their probabilities. Show that y has a valid distribution and find the probabilities of each value of y . Find E y and y . (6) [36] 8. As everyone knows, a Veeblefetzer has two components, a Phillinx and a Flubberall. Depending on the design of the Veeblefetzer, the Veeblefetzer will work a) as long as both components work or b) as long as either component works. Let us assume that rule b) is in effect and that the life of a phillinx, x1 , is a random variable with a continuous uniform distribution between 0 and 4, so that the probability of it failing in the first year is P0 x1 1 . The life of a Flubberall, x 2 , is a random variable with a continuous uniform distribution between 0.5 and 2.7. What is the probability that the Veeblefetzer fails in Year 1? Year 2? After Year 2? If you will lose $2million if it fails in the first year, break even if it fails in the second year and make $5 million of it lasts beyond the second year, what is the standard deviation of your earnings? (5) [41] A1 A2 A3 B1 .05 .05 .20 9. The following joint probability represents returns for two different stocks. B 2 .05 .30 .05 B3 .20 .05 .05 Find the following: P A1 B2 , P A1 B2 , P A1 B2 The returns on stock 1 are A1 x1 10 % , A2 x1 20% and A3 x1 30% The returns on stock 1 are B1 x 2 5% , B 2 x 2 10% and B 3 x 2 15% Find E x1 , E x 2 , x21 , x22 , x x , x x 1 2 1 2 If your return is x1 x 2 , find the mean and standard deviation of the return. If your return is P1 x1 P2 x 2 and P1 P2 1 , find the proportions P1 and P2 that will lead to a minimum risk. (9) [50] 2