MATH 3200 PROBABILITY AND STATISTICS M3200FL081.1

Transcription

1 MATH 3200 PROBABILITY AND STATISTICS M3200FL081.1 This examination has twenty problems, of which the first seventeen are modifications of the recommended homework problems. The remaining three problems are in the same spirit. Each problem will be graded on a three-point scale, with 3=fully correct, 2=mostly correct, 1=mostly incorrect, and 0=fully incorrect. Please draw a box or oval around each final answer so I can locate it easily. Of course, for me to assign partial credit, I will also need to see your steps. This is especially important when you use the statistical capabilities of your calculator. For example, if you use the normalcdf function to solve a problem, one step should be to write down the function and its arguments exactly as they appear on your calculator screen (except you don t need to wrap the text the way it appears on the small calculator screen). 1. In a state lottery game you must match 3 out of 6 numbers drawn at random from 1 to 60 without replacement in order to win the fourth prize. The order of the numbers is irrelevant. Find the probability of winning the fourth prize. 2. A team of three people to assess computer needs of a company is randomly formed from a group of 4 managers, 15 analysts, and 30 technicians. Find the probability that two team members are from the same job category and the third member is from a different category.

2 MATH 3200 PROBABILITY AND STATISTICS M3200FL Let A and B be two events. Suppose that P(A) = 0.3, P(B) = p, and P(A U B) = 0.9. For what value of p will A and B be independent? 4. Consider the function f(x) = P(X = x) = c(⅔) x for x = 1,2,3,4 and 0 otherwise. Find the c.d.f. of X and draw a graph of it. (Hint: first find that value of c that makes f(x) a p.m.f.)

3 MATH 3200 PROBABILITY AND STATISTICS M3200FL A random variable X has p.d.f. f(x) = cx 4 for x 1 and f(x) = 0 for all other values of x. Find the value of c and then find the variance of X. 6. Find the variance of a Bernoulli random variable from its moment generating function pe t + q.

4 MATH 3200 PROBABILITY AND STATISTICS M3200FL A husband and wife invest their $4000 IRAs in two different portfolios. After one year the husband s portfolio has 0.25 probability of losing $1000 and 0.75 probability of gaining $900. The wife s portfolio has 0.15 probability of losing $500 and 0.85 probability of gaining $600. Let X denote the husband s gain, and Y denote the wife s gain. Assume that X and Y are independent. Find the standard deviation of X + Y. 8. A grocery store has 20 checkout lanes. During a busy hour the probability that any given lane is occupied (has at least one customer) is 0.9. Assume that the lanes are occupied or not occupied independently of each other. What is the probability that a customer will find at least one lane unoccupied?

5 MATH 3200 PROBABILITY AND STATISTICS M3200FL An office supply warehouse receives an order for ten staplers. The warehouse has fifty staplers in stock, of which five are defective. The order is filled by randomly drawing from the staplers in stock. What is the probability that the customer receives exactly one defective stapler in the shipment of ten? 10. An experiment measures Higgs Boson emissions from the Large Hadron Collider. The number of emissions has a Poisson distribution with rate λ = 0.2 per week. What is the probability of at least two emissions in a given week?

6 MATH 3200 PROBABILITY AND STATISTICS M3200FL A car travels between two cities A and C which are sixty miles apart. If the car has a breakdown, the distance X from the breakdown to city A is distributed U[0,60]. The driver is a member of an automotive service that has contracts with garages in cities A, B, and C, where city B is between cities A and C, twenty miles from city A. If the car breaks down, it is towed to the closest garage. Find the probability that the car is towed more than 10 miles. 12. Let X = the time to failure of a light bulb. Assume that X is exponentially distributed with a mean time to failure of 2000 hours. Suppose that a light bulb is replaced immediately after it burns out by an identical one (i.e., one that has the same failure distribution). Let T denote the total time until failure of the third bulb. Assuming the failures are independent, what is the variance of T?

7 MATH 3200 PROBABILITY AND STATISTICS M3200FL The weight U of a man is normally distributed with mean 180 lb and standard deviation 25 lb. The weight V of a woman is normally distributed with mean 110 lb and standard deviation 20 lb. A man and woman are randomly selected. What is the probability that the man is at least 50 lb. heavier than the woman? 14. In a shaft and bearing assembly, the diameters of the bearings, X, are normally distributed with mean = inches and standard deviation = inches. The diameters of the shafts, Y, are normally distributed with mean = inches and standard deviation = inches. Assuming independence of X and Y, what is the probability that out of twenty randomly selected shaft-bearing pairs, all will fit properly (i.e., the shafts fit into the bearings)?

8 MATH 3200 PROBABILITY AND STATISTICS M3200FL Confounding is present in the following study. Explain the nature of the confounding and why the conclusion drawn may not be valid. A cross-country coach thinks that a particular technique will improve the times of her runners. As an experiment she offers an extra daily practice to work on this technique for runners who want to participate. At the end of the season the coach concludes that the technique is effective, because runners who participated in the extra practices have shorter average times than those who did not. 16. A Chicago radio station frequently broadcasts an opinion question during the afternoon rush hour and gives a telephone number to call for a Yes response and another for a No response. Poll results are announced at the end of afternoon rush hour. Why are such call-in polls likely to be biased?

9 MATH 3200 PROBABILITY AND STATISTICS M3200FL A national sample of college students is to be taken, but a comprehensive list of all college students doesn t exist. A list of the 4140 colleges and universities (public and private) does exist. For any particular college or university, a list of all students can be obtained. Tell which sampling design you would use and why you would use it. 18. Two plants A and B ship appliances to a warehouse. Plant A produces 70% of the warehouse s inventory, with a 4% defect rate. Plant B produces 30% of the warehouse s inventory, with a 6% defect rate. Suppose that a randomly selected appliance is defective. What is the probability that it came from Plant A?

10 MATH 3200 PROBABILITY AND STATISTICS M3200FL By traveling though a wormhole to the Delta Quadrant, the crew of Starship Voyager have discovered that its inhabitants like baseball just as much as we do. Their Interstellar Series has a nine game playoff, with the following probabilities of ending at 5 through 9 games: 5 games games games games games 0.39 What are the mean and standard deviation of this random variable? 20. Suppose a kindly teacher gives such easy exams that the scores are continuously distributed with the triangular density function shown below: / / / / / That is, it is not possible to score below 70 points, and the density function continues to increase linearly until it hits a maximum at 100 points. What are the mean and standard deviation of this random variable?