MATH 3200 PROBABILITY AND STATISTICS M3200SP081.1

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1 MATH 3200 PROBABILITY AND STATISTICS M3200SP081.1 This examination has twenty problems, of which most are straightforward modifications of the recommended homework problems. The remaining 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 binompdf( 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 that you don t need to wrap the text the way it appears on the small calculator screen). 1. A team of 3 people is randomly formed from a group of 2 managers, 10 analysts, and 10 salespersons to study the marketing needs of a small company. Find the probability that the team is composed only of analysts. 2. Find the exponent of x 3 y 5 in (2x+3y) 8.

2 MATH 3200 PROBABILITY AND STATISTICS M3200SP A team of 3 people is randomly formed from a group of 4 managers, 5 analysts, and 6 computer technicians in order to assess the computer needs of a small company. Find the probability that the team is composed of one person of each type: That is, one manager, one analyst, and one computer technician. 4. The accuracy of a medical diagnostic test, in which a positive result indicates the presence of a disease, is often stated in terms of its sensitivity, the proportion of diseased people that test positive or P(Pos Disease), and its specificity, the proportion of people who test negative or P(Neg No Disease). Suppose that 3% of the population has the disease (called the prevalence rate). A diagnostic test for the disease has 95% sensitivity (P(Pos Disease)) and 90% specificity (P(Neg No Disease)). Given that a person s test result is positive, what is the probability that the person actually has the disease?

3 MATH 3200 PROBABILITY AND STATISTICS M3200SP Consider the following function: f(x) = P(X = x) = c(1/3) x for x = 1, 2, 3, and f(x) = 0 for all other values of x. Find the constant c so that f(x) is a p.m.f. (or a discrete p.d.f.). 6. A random variable X has p.d.f. f(x) = 0.3 for -1 x 0, f(x)=cx 2 for 0 x 3, and f(x) = 0 for all other values of x. Find the constant c so that f(x) is a p.d.f.

4 MATH 3200 PROBABILITY AND STATISTICS M3200SP A random variable X has p.d.f. f(x)=cx(1-x) for 0 x 1 and f(x) = 0 for all other values of x. Find the constant c so that f(x) is the p.d.f. of X and use it to find E(X) and E(X 2 ). 8. A random variable X has p.d.f. f(x) = x 2 for x 1 and f(x) = 0 for all other values of x. Find the c.d.f. of X and use it to find the median of X.

5 MATH 3200 PROBABILITY AND STATISTICS M3200SP Let X 1 and X 2 be two uncorrelated random variables, each with variance σ 2. Show that Y 1 = X 1 + X 2 and Y 2 = X 1 X 2 are uncorrelated. 10. A husband and wife invest their $4000 IRAs in two different portfolios. After one year the husband s portfolio has 0.4 probability of losing $500 and 0.6 probability of gaining $500. The wife s portfolio has 0.2 probability of losing $1000 and 0.8 probability of gaining $1000. 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.

6 MATH 3200 PROBABILITY AND STATISTICS M3200SP The number of customers of an insurance company with policy type x=1,2 and credit rating y=1,2,3 is given by the table y=1 y=2 y= x= x= Let X,Y be random variables whose joint distribution is given by the proportions of customers in the table. Find the conditional means E(Y X=1) and E(Y X=2). 12. A grocery store has 12 checkout lanes. During a busy hour the probability that any given lane is occupied (has at least one customer) is 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?

7 MATH 3200 PROBABILITY AND STATISTICS M3200SP A public speaking course has 12 one-hour meetings spread out over 12 weeks. A topic for each meeting is announced one week in advance, but the speaker is chosen at random from the students just before each meeting begins. Suppose that there are 7 students attending the course. If you are one of the 7 students, find the probability that you will be asked to speak 3 times over the 12 weeks. 14. Suppose that X has a hypergeometric distribution with N = 60, M = 20, and n = 10. Find P(X = 7).

8 MATH 3200 PROBABILITY AND STATISTICS M3200SP A typist makes a typographical error at the rate of 1 every 20 pages. Let X be the number of errors in a manuscript of 75 pages. Assuming that X has a Poisson distribution, calculate the probability that the manuscript has at most 5 errors. 16. Let X = the time to failure of a light bulb. Assume that X is exponentially distributed with a mean time to failure of 500 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 that the failures are independent, what is the variance of T?

9 MATH 3200 PROBABILITY AND STATISTICS M3200SP The weight of coffee in a can is normally distributed with a mean of 16.2 oz. and standard deviation of 0.8 oz. What is the lower 10 th percentile of the distribution of coffee weight? (Ten percent of the cans contain coffee less than this weight.) 18. In an extremely delicate 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 the shaft will fit into the bearing?

10 MATH 3200 PROBABILITY AND STATISTICS M3200SP Lilith received a score of 96.1 on a chemistry examination for which the mean was 61 and the standard deviation was 22. Assuming that the scores for the entire class are normally distributed, what fraction of the class scored higher than she did? (Assume that, as a result of partial credit, the scores are continuously distributed.) 20. Most people think that the normal adult body temperature is 98.6 F. A recent study showed that a more accurate figure for human temperature is 98.2 F with a standard deviation appeared of 0.7 F. Assume that a normal model is appropriate. How many degrees less than 98.6 are the coolest 25% of people?