MATH 106 FINAL EXAMINATION This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Record your answers and work on the separate answer sheet provided. There are 25 problems. Problems #1 12 are Multiple Choice. Problems #13 15 are Short Answer. (Work not required to be shown) Problems #16 25 are Short Answer with work required to be shown. MULTIPLE CHOICE 1. Julie purchases a car for $24,000, makes a down payment of 20%, and finances the rest with a 3-year car loan at an annual interest rate of 5.7% compounded monthly. What is the amount of her monthly loan payment? 1. A. $780.87 B. $726.87 C. $624.53 D. $581.49 2. Find the result of performing the row operation (4)R 1 + R 2 R 2 2. 2 1 1 0 3 2 A. 2 1 9 0 3 2 B. 2 1 9 4 3 10 C. 2 1 8 4 3 12 D. 2 1 12 4 3 4 Page 1 of 10
3. Find the values of x and y that maximize the objective function 5x + 2y for the feasible region shown below. 3. A. (x, y) = (0, 20) B. (x, y) = (5, 15) C. (x, y) = (8, 10) D. (x, y) = (10, 0) 4. The purchases made by customers at a convenience store are normally distributed, with a mean of $7.00 and a standard deviation of $3.00. What is the probability that a randomly chosen customer makes a purchase between $4.00 and $10.00? 4. A. 0.5000 B. 0.6826 C. 0.7580 D. 0.9544 Page 2 of 10
5. Determine which shaded region corresponds to the solution region of the system of linear inequalities x + 3y 6 2x + y 4 x 0 y 0 5. GRAPH A. GRAPH B. GRAPH C. GRAPH D. Page 3 of 10
For #6 and #7: A merchant makes two raisin nut mixtures. Each box of mixture A contains 15 ounces of peanuts and 3 ounces of raisins, and sells for $5.10. Each box of mixture B contains 8 ounces of peanuts and 4 ounces of raisins, and sells for $3.60. The company has available 6,200 ounces of peanuts and 2,000 ounces of raisins. The merchant will try to sell the amount of each mixture that maximizes income. Let x be the number of boxes of mixture A and let y be the number of boxes of mixture B. 6. State the objective function. 6. A. 15x + 3y B. 15x + 8y C. 23x + 7y D. 5.1x + 3.6y 7. Since the merchant has 6,200 ounces of peanuts available, one inequality that must be satisfied is: 7. A. 15x + 8y 6,200 B. 15x + 3y 6,200 C. 15x + 8y 6,200 D. 5.1x + 3.6y 6,200 8. A jar contains 16 red jelly beans, 8 yellow jelly beans, and 20 orange jelly beans. Suppose that each jelly bean has an equal chance of being picked from the jar. If a jelly bean is selected at random from the jar, what is the probability that it is not red? 8. A. 16 1 B. 11 4 C. 7 4 7 D. 11 Page 4 of 10
9. When solving a system of linear equations with the unknowns x 1 and x 2 the following reduced augmented matrix was obtained. 1 3 0 0 7 1 What can be concluded about the solution of the system? 9. A. There are infinitely many solutions. The solution is x 1 = 3t 7 and x 2 = t, for any real number t. B. There are infinitely many solutions. The solution is x 1 = 3t 7 and x 2 = t, for any real number t. C. The unique solution to the system is x 1 = 3 and x 2 = 7. D. There is no solution. 10. Which of the following statements is NOT true? 10. A. The variance must be a nonnegative number. B. The variance is a measure of the dispersion or spread of a distribution about its mean. C. The variance is the square root of the standard deviation. D. If all of the data values in a data set are identical, then the standard deviation is 0. 11. In a certain manufacturing process, the probability of a type I defect is 0.08, the probability of a type II defect is 0.06, and the probability of having both types of defects is 0.04. Find the probability that neither defect occurs. 11. A. 0.96 B. 0.90 C. 0.86 D. 0.82 12. Which of the following is NOT true? 12. A. If an event cannot possibly occur, then the probability of the event is a negative number. B. A probability must be less than or equal to 1. C. If only two outcomes are possible for an experiment, then the sum of the probabilities of the outcomes is equal to 1. D. If events E and F are mutually exclusive events, then P(E F) = 0. Page 5 of 10
SHORT ANSWER: 13. Let the universal set U = {1, 2, 3, 4, 5, 6, 7}. Let A = {1, 3, 5, 7} and B = {1, 2, 3, 6}. Determine the set A' B. (Be sure to notice the complement symbol applied to A.) 14. Consider the following graph of a line. (a) State the x-intercept. (b) State the y-intercept. (c) Determine the slope. (d) Find the slope-intercept form of the equation of the line. (e) Write the equation of the line in the form Ax + By = C where A, B, and C are integers. Page 6 of 10
15. 400 employees at a particular company were asked their status (full-time or part-time) and their primary means of transportation to and from work. The following table was obtained. Primary Full-time Part-time Total Transportation Car 188 60 248 Bus 32 44 76 Subway 30 27 57 On Foot 10 9 19 Total 260 140 400 (Report your answers as fractions or as decimal values rounded to the nearest hundredth.) Find the probability that a randomly selected employee: (a) travels by car and is part-time. (b) travels by car or is part-time. (c) travels by car, given that the employee is part-time. SHORT ANSWER, with work required to be shown, as indicated. 16. For a four year period, Ned deposited $400 each quarter into an account paying 3.6% annual interest compounded quarterly. (Round your answers to the nearest cent.) (a) How much money was in the account at the end of 4 years? Show work. (b) How much interest was earned during the 4 year period? Show work. Ned then made no more deposits or withdrawals, and the money in the account continued to earn 3.6% annual interest compounded quarterly, for 6 more years. (c) How much money was in the account after the 6 year period? Show work. (d) How much interest was earned during the 6 year period? Show work. 17. A contest has 12 finalists. One finalist is awarded first prize, another finalist is awarded second prize, and another is awarded third prize. How many different ways could the prizes be awarded? Show work. Page 7 of 10
18. There is a collection of 16 different coins from a variety of civilizations. 10 of the coins are silver and 6 of the coins are gold. (a) In how many ways can 7 of the 16 coins be chosen for a museum display? Show work. (b) In how many ways can the 7 coins be chosen from the collection of 16 coins, if 4 of the coins must be silver and 3 of the coins must be gold? Show work. (c) If the 7 coins are selected at random from the collection of 16 coins, what is the probability that the coins consists of 4 silver coins and 3 gold coins? Show work. 19. In the year 2000, the U.S. Consumer Price Index (CPI) was 172. In 2010, the CPI was 218. Let y be the U.S. Consumer Price Index in year x, where x = 0 represents the year 2000. (a) Which of the following linear equations could be used to predict the U.S. Consumer Price Index y in a given year x, where x = 0 represents the year 2000? Explain/show work. A. y = 4.6x 9028 B. y = 4.6x + 172 C. y = 46x 90,280 D. y = 46x + 1720 (b) Use the equation from part (a) to predict the CPI in the year 2015. Show work. (c) Fill in the blank: The average rate of change of the CPI with respect to time is per year. 20. Solve the system of equations using elimination by addition or by augmented matrix methods (your choice). Show work. x + 4y = 2 3x + 13y = 4 Page 8 of 10
21. The feasible region shown below is bounded by lines 2x y = 4, x + 3y = 4, and y = 0. Find the coordinates of corner point A. Show work. 22. A survey of 70 adult residents of Metropolis found the following: 30 of those surveyed read the city newspaper in printed form. 38 read newspaper in electronic form. 44 read the city newspaper in printed form or electronic form (or both). (a) How many of the surveyed respondents read both the printed form and the electronic form of the newspaper? Show work. (b) Let circle P = {survey respondents reading the printed form of the newspaper} and circle E = {survey respondents reading the electronic form of the newspaper } Determine the number of adults belonging to each of the regions I, II, III, IV. U P I II III E IV Page 9 of 10
23. Consider the sample data 34, 50, 80, 34, 65, 73, 91. (a) State the mode. (b) Find the median. Show work/explanation. (c) State the mean. (d) The sample standard deviation is 22.4. What percentage of the data fall within one standard deviation of the mean? Show work/explanation. (d) A. 57% B. 68% C. 71% D. 75% 24. If the probability distribution for the random variable X is given in the table, what is the expected value of X? Show work. x i 30 10 15 30 p i 0.20 0.10 0.40 0.30 25. According to a recent report, 0.56 is the probability that an American adult owns a smartphone. Six American adults are randomly selected. Find the probability that exactly 2 of the 6 American adults owns a smartphone. Show work. Page 10 of 10