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Math Analysis CP WS- Matrix Multiplication All work must be done and neatly shown on a separate sheet of paper. 1. A furniture manufacturer produces upholstered chairs and wood tables. For each chair, it takes 4 hours for a woodworker to build the frame, 2 hours for a finisher to sand, stain and varnish the legs, and 12 hours for an upholsterer to install the padding and fabric. For each table, it takes 18 hours for a woodworker to cut and assemble the pieces and 15 hours for a finisher to sand, stain, and varnish the table. No upholstery work is required for the table. a. Organize this information in a 2 x 3 matrix called A. b. The factory receives an order for 10 chairs and 3 tables. Represent this information in a 1 x 2 matrix, B. c. Multiply the two matrices to obtain the number of hours required for each type of worker. d. If woodworkers are paid $15 per hour, finishers are paid $9 per hour and upholsterers earn $12 per hour, perform matrix multiplication to find the total labor costs for the order. 2. In the problem above, suppose that in addition to the order for 10 chairs and 3 tables, the company receives three other orders: 8 chairs and 2 tables, 0 chairs and 4 tables, and 2 chairs and 1 table. a. Write a 4 x 2 matrix to represent the four orders. b. Perform matrix multiplication to find the labor cost for each order. 3. Every electrical appliance has a kilowatt rating. If you multiply the number of kilowatts by the number of hours the appliance is in use, you obtain a unit of energy, the kilowatt-hour (kwh). If a clothes iron has a kilowatt rating of 1.2 and it is used for 2 hours, then it has used 2.4 kwh of energy. a. An electric room heater is rated at 1.6 kilowatts, an electric dryer is rated at 6 kilowatts, and an air conditioning unit is rated at 5 kilowatts. Represent this data in a 1 x 3 matrix called A. b. In the non-summer months, the heater is used 2000 hours, the dryer is used 80 hours, and the air conditioner is not used. In the summer months, the heater is not used, the dryer is used 40 hours, and the air conditioner is used 400 hours. Represent this information in a 3 x 2 matrix called B. c. In the non-summer months, the electric company charges $0.09 for each kwh used. In the summer months, it charges $0.12 for each kwh. Represent this data in a 2 x 1 matrix called C. d. Find AB. What does it represent? e. Find ABC. What does it represent? 4. A roofer needs plywood, building paper, shingles, nails, and flashing material. He has three distributors available to him. Distributor #1 charges $8 per unit of plywood, $10 per unit of building paper, $30 per unit of shingles, $2 per unit of nails, and $6 per unit of flashing. Distributor #2 charges $6 per unit of plywood, $8 per unit of building paper, $35 per unit of shingles, $3 per unit of nails, and $5 per unit of flashing. Distributor #3 charges $10, $6, $32, $2, and $7 per unit, respectively. a. Represent this data in a 3 x 5 matrix called A. b. The roofer has two jobs for which he must buy materials. The first job requires 6 units of plywood, 7 units of building paper, 20 units of shingles, 8 units of nails, and 3 units of flashing. The second job requires 100, 10, 30, 12 and 8 units of these materials. Represent this data in a 5 x 2 matrix called B. c. Find the matrix AB. What does it represent? d. The roofer wishes to purchase all of the materials for each job from one distributor because then the distributor will deliver the goods to the job site at no extra cost. Which distributor should he choose for the first job? Which distributor should he choose for the second job? 5. A candy company packages caramels, chocolates, and hard candy in three different assortments: traditional, deluxe and superb. The table below gives the number of each type of candy in each assortment, the number of calories for each type of candy, and the cost to make each piece. Caramels Chocolates Hard Candy Traditional Deluxe Superb Calories per piece 10 12 10 16 8 16 15 25 8 60 70 55 Cost per piece (in cents) 10 12 6 The company receives an order for 300 traditional, 180 deluxe and 100 superb assortments. Use matrix multiplication to find: a. The number of each type of candy needed to fill the order. b. The number of calories in each type of assortment. c. The cost of producing each type of assortment. d. The cost to fill the order. Math Analysis CP WS- Eschelon Word Problems For problems #1-6, define the variables and write a system of equations. Then write the augmented matrix, the RREF, and give the solutions. All work must be done and neatly shown on a separate sheet of paper. 1. Find the quadratic equation that fits the given points: (-10, -88) (-2, -8) (2, -40) 2. Find the cubic equation that fits the given points: (0, 1) (1, 2) (3, 21) (-1, 0) 3. A mile offshore on Lake Michigan, a fisherman searches for salmon. With an electric thermometer, he finds the following readings (depth in feet, temperature in °F): (5, 71) (20, 65) (30, 60) (45, 50) a. Find a cubic equation to model the data. b. If salmon swim in water at 55 °F, at what depth should he fish? 4. The pollution count for Big City on a particular day is 600. Assume that this pollution is produced by three industries: coal, steel, and plastics. The coal industry contributes twice as much to the pollution count as the steel industry. It is known that the total pollution count would be 500 if the pollution count from the coal industry were reduced by 50%. Find the pollution count for each industry. 5. In an experiment involving mice, a zoologist needs a food mix that contains, among other things, 23 g of protein, 6.2 g of fat, and 16 g of moisture. She has on hand mixes of the following compositions: Mix A contains 20% protein, 2% fat and 15% moisture. Mix B contains 10% protein, 6% fat and 10% moisture. Mix C contains 15% protein, 5% fat and 5% moisture. How many grams of each mix should be combined to produce the desired food mix? 6. A garment industry manufactures three different shirt styles. Each shirt requires the services of three departments: cutting, sewing, and packaging. The maximum hours available for each department are 1160, 1560, and 480 labor-hours per week, respectively. Style A requires 12 minutes for cutting, 18 min. for sewing and 6 min. for packaging. Style B requires 24 minutes for cutting, 30 min. for sewing and 12 min. for packaging. Style C requires 18 minutes for cutting, 24 min. for sewing and 6 min. for packaging. How many of each shirt style must be produced each week to operate at full speed? For problems #7 & 8, define the variables and write a system of equations. If there is one solution, give it. If not, give the parametric formulas for the solutions, state the domain, and give one specific solution. 7. A person receives $306 per year in simple interest from three investments totaling $3200. Part of the money is invested at 8%, part at 9%, and part at 10%. There is $1800 more invested at 10% than at 8%. Find the amount invested at each interest rate. 8. A chemistry laboratory has available three kinds of hydrochloric acid (HCl) solution containing 10%, 30% and 50% HCl. How many liters of each solution should be mixed to obtain 100 liters of 25% HCl? -------------------------------------------------------------------------------------------------------------Math Analysis CP WS- RREF Systems of Equations For each system, write both the original (augmented) matrix and the solution (RREF) matrix. Give the unique solution; or write “no solution” or “infinitely many solutions.” If there are infinitely many solutions, write the possible solutions in parametric form and give one specific solution. 2 x 4 y 10z 2 1. 3 x 9 y 21z 0 x 5 y 12z 1 4. 2x 3y z 1 x 2y 2z 2 3 x 8 y z 18 2. 2 x y 5z 8 2 x 4 y 2 z 4 2 x1 2 x2 2 5. x1 2 x2 3 3 x2 6 3 x1 4 x2 x3 1 7. 2 x1 3 x2 x3 1 x1 2 x2 3 x3 2 8. 2 x y 4z 0 x y 3z 1 5 x1 3 x2 2 x3 12 11. 2 x1 4 x2 3 x3 9 Selected Answers: 1. 2, 3, 1 7. No solution 4 x1 2 x2 5 x3 13 3. 2t 3, t 2, t 9. 2t2 t1 1, t2, t1 2 x1 x2 3 x3 8 x1 2 x2 7 2x y 0 6. 3 x 2y 7 x y 2 3 x 2 y z 7 3 x 3 y 3z 15 10. 3 x 2y 5z 19 5 x 4 y 2z 2 3. 9. 2 x1 4 x2 2 x3 2 3 x1 6 x2 3 x3 3 w 2x 4y z 7 12. 2w 5 x 9 y 4z 16 w 5 x 7 y 7z 13 5. No solution 11. 0.854, 1.77, 1.208 Math Analysis CP WS- Matrix Review Solve each system of equations. Give the unique solution; or write “no solution” or “infinitely many solutions.” If there are infinitely many solutions, write the possible solutions in parametric form and give one specific solution. 2x y z 6 x 2y z 7 x 2 y 2z 8 2. 2 x y 3z 2 3. 3 x y z 6 1. 3 x y z 2 2 x 3 y 2z 7 5y z 6 4 x 2y 2z 12 4. Four landscapers want to purchase three types of trees. The trees are available at three different nurseries. a. George wishes to purchase 18 Spruce, 35 Pine and 75 Birch. Wally wishes to purchase 75 Spruce, 50 Pine and 15 Birch. Chuck wants 25 Spruce, 15 Pine and 14 Birch, and Marty wishes to purchase 45 Spruce, 50 Pine and 32 Birch. Display this information in a 4 x 3 matrix. b. Nursery A sells Spruce saplings for $1.25, Pine for $1.50, and Birch for $1.55. Nursery B sells Spruce saplings for $1.40, Pine for $1.45, and Birch for $1.50. Nursery C sells Spruce saplings for $1.35, Pine for $1.30, and Birch for $1.60. Display this information in a 3 x 3 matrix. c. Use matrix multiplication to determine which nursery offers the best overall deal for each of the four landscapers. For each problem, define the variables and write a system of equations. If there is one solution, give it. If not, give the parametric formulas for the solutions, state the domain, and give one specific solution. 5. One group of customers bought 8 deluxe hamburgers, 6 orders of large fries, and 6 large colas for $26.10. A second group of customers ordered 10 deluxe hamburgers, 6 orders of large fries, and 8 large colas for $31.60. Determine the price of each food item. 6. To the information given in #4, add a third group that purchased 3 deluxe hamburgers, 2 orders of large fries, and 4 large colas for $10.95. Determine the price of each item. Answers: 1. (2, -1, 3) 2. No solution 3. 0, t 6, t 4. Nursery B for George, Nursery C for Wally, Nursery A for Chuck and Nursery C for Marty. 1 41 5. t 2.75, t , t 3 60 6. (1.95, 0.95, 0.80) 5. If for all real numbers x , a function f x is Cumulative Review- Chapter 02 Matrices Analysis CP All work is to be neatly shown on a separate piece of paper. Please write and box your answer, including the multiple choice answer letter. This is due the school day after the chapter test. It will be graded on completeness and correctness. 1. If r s r s , then which of the following must be true? (A) r s (B) s 0 (C) r 0 (D) r s (E) s 0 2. If f x x 10 , for which of the following 2, x 13 defined by f x , then 4, x 13 f 15 f 14 (A) -2 (B) 0 (C) 1 (D) 2 (E) 4 Written Response Question 6. If the ratio of sec x to csc x is 1:4, then the ratio of tan x to cot x is (A) 1:16 (B) 1:4 (C) 1:1 (D) 4:1 (E) 16:1 values of x does f x f x ? 7. If sin (A) -10 only (B) -10 and 10 only (C) All real x (D) All real x except 10 (E) All real x except -10 and 10 (A) (A) x 1 (C) x 2 1 (D) x 1 2 (E) x 2 2 9. If f x x 2 5 x 6 , for what value of x will f x have its minimum value? (E) k 2 1 (A) -3 4. If x mod y is the remainder when x is divided by y , then 61 mod 7 5 mod 5 (B) 3 2 (B) x 2 2 (D) j 2 1 (A) 2 7 9 1 and , then cos 2 4 4 3 2 2 7 (D) (E) 1 (B) (C) 3 3 9 8. If f x x 1 , for all x 0 , then f 1 x j jk k 3. If jk 0 , then j k j 2 (A) k k j2 (B) j 2 2 k (C) jk 1 (C) 4 (D) 5 (E) 6 10. The polar equation r sin 1 defines the graph of (A) a line (B) a circle (C) an ellipse (D) a parabola (E) a hyperbola (B) 5 2 (C) -2 (D) 0 (E) 5 2 A complete response requires the following: Express your thinking in words Label any figures you draw Identify any formulas used Make clear the source of numbers used Full credit will not be earned if your work cannot clearly be followed. The final answer is important, but meaningless if you cannot show somebody how to get it. ---------------------------------------------------------------Old McDonald has a garden enclosed by a triangular fence of 40 feet on each side. The garden is surrounded by a field of grass. a. His goat, Billy, is tethered outside the garden on a 30 foot chain attached to a corner of the fence. Draw a figure that represents the grazing region and label appropriate angles and lengths. Find the area of the grazing region. Show work that leads to your answer. b. Suppose instead that the chain is attached to the midpoint of a side of the fence. Draw a figure that represents the grazing region and label appropriate angles and lengths. Find the area of the grazing region. Show the work that leads to your answer. Math Analysis CP WS 3A.1- Limits (Graphically) Math Analysis CP WS 3A.2- Piecewise Continuity f(x) 4 2 Find the following: 1. a. 2. a. 3. a. 4. a. 5. a. 6. a. lim f x b. lim f x b. lim f x b. lim f x b. lim f x b. lim f x b. x 0 x 1 x 1 x 2 x 2 x 0.5 f x 7. a. xlim 0 lim f x c. lim f x c. lim f x c. lim f x lim f x c. lim f x c. lim f x c. x 0 x 1 x 1 x 2 x 2 x 0.5 lim f x x 0 -2 lim f x x 1 x 2 1. For function f x above: lim f x a. Locate all points of discontinuity and identify each as a hole or a jump. For each hole, identify the point limit at that location. b. Write the piecewise function that created the graph. x 2 lim f x x 0.5 b. f 1 ? c. Is f continuous at x 1 ? f x 9. a. xlim 1 b. f 1 ? c. Is f continuous at x 1 ? f x 10. a. xlim 2 b. f 2 ? c. Is f continuous at x 2 ? lim f x b. f 2 ? c. Is f continuous at x 2 ? lim f x b. f 0.5 ? c. Is f continuous at x 0 ? 11. a. 12. a. lim f x x 2 x 0.5 13. Function f is discontinuous at which of the following values of x? -3 -2 -1 0 1 x -2 lim f x c. Is f continuous at x 0 ? x 1 4 x 1 b. f 0 ? 8. a. 2 2 3 For problems 2-7, graph each piecewise function and locate all points of discontinuity. 1 x 2. f x 5 x if x 1 if x 1 x 2 3. f x 2 x 1 x 4. f x 5 x if x 2 x 2 if x 2 5. f x 2 x if x 2 if x 0 1 if x 0 7. f x 0 1 x if x 0 x 6. f x 1 x if x 2 if x 0 if x 0 if x 0 if x 1 if x 1 Math Analysis CP WS 3A.3- Continuity and End Behavior Math Analysis CP WS 3A.1- Limits (Algebraic) Determine whether each function is continuous at the given x-value. Justify your answer using the continuity test. Evaluate each limit. 1. y 2 3x 2 ; x x4 2. y ; 2 2 x 1 3. y x 3 2 x 2; x 1 4. y x 2 ; x4 x 1 x 3 x 2 36 x 6 x 6 3. lim x 4 5. lim Describe the end behavior of each function. 5. y 2 x 5 4 x 1. lim x 2 3 x 8 x 2 x 2 x2 4 2. lim 2 x 7 x 2 x 2 5x 6 x 3 x 2 4. lim 6. lim x 3 x2 9 x 3 27 6. y 2 x 6 4 x 4 2 x 1 7. lim 7. y x 4 2 x 3 x 8. y 4 x 3 5 9. Given the graph of the function, determine the interval(s) for which the function is increasing and the interval(s) for which the function is decreasing. 3 x 2 9 x 0 x 2x 3 6 x 2 x 3 x 3 x 3 9. lim 8. lim x 4 10. lim x 2 2x 1 x2 x x 3 2 x 2 5x 7 f(x) 12 11. lim n 8 3n 5 8n 12. lim n n3 4n 2 4 -4 -3 -2 -1 1 -4 -8 2 3 4 x 13. lim n 2n 2 4n 5n 2 12 14. lim n 5 4n 2 6n 2 1 Cumulative Algebra Review- Chapter 03A Analysis CP All work is to be neatly shown on a separate piece of paper. Please write and box your answer, including the multiple choice answer letter. This is due the school day after the chapter test. 1. For which of the following functions f is f 1 a function? I. f x x2 II. f x x3 III. f x x 6. If x i 1 , then x 2 2 x 2 (A) 2i 4 (B) 2i 4 (C) 0 (D) i (E) -2 7. Two coins are removed from a purse containing three nickels and eight dimes. What is the probability that both coins will be dimes? 14 49 28 32 64 (B) (C) (D) (E) (A) 55 110 55 55 121 8. Vectors v and w have components 3, 4 and 12, 5 (A) I only (B) II only (C) I and III only (D) II and III only (E) I, II, and III x x ? x 1 (A) -2 (B) -1 (C) 1 (E) The limit does not exist respectively. If z v w , then z has components (A) 9, 9 (B) 5, 13 a. Write an expression in terms of an integer a for the sum of any three consecutive integers and show that this sum is divisible by 3. (C) 5, 13 (D) 9, 9 3 2. What is lim x 1 9 9 (E) , 2 2 (D) 2 3. A cube is inscribed in a sphere of radius 6. What is the volume of the cube? (A) 36 3 (B) 36 (C) 216 (D) 192 3 (E) 216 3 4. If the graph of the equation y 2 x 6 x c is tangent to the x-axis, then the value of c is (A) 3 (B) 3.5 (C) 4 (D) 4.5 (E) 5 2 5. The system of equations given by 2x 3y 7 10 x cy 3 has solutions for all values of c EXCEPT (A) -15 (B) -3 (C) 3 (D) 10 (E) 15 9. What is the value of 6 3i ? (A) -3 (B) 3 2 (C) 3 5 (D) 9 (E) 15 b. Write an expression in terms of an integer a for the sum of any four consecutive integers and show that this sum is not divisible by 4. c. 10. In the figure, rectangle J contains all points x, y . What is the area of a rectangle that contains all points 2 x, y 1 ? 0, 3 6, 3 0, 0 6, 0 (A) 12 A complete response requires the following: Express your thinking in words Label any figures you draw Identify any formulas used Make clear the source of numbers used Full credit will not be earned if your work cannot clearly be followed. The final answer is important, but meaningless if you cannot show somebody how to get it. ---------------------------------------------------------------If p and q are integers, we say q is divisible by p if q r p where r is an integer. For example, 6 is divisible by 3, since 6 2 3 . But 7 is not divisible 7 7 by 3 since 7 3 , and is not an integer. 3 3 Consecutive integers beginning with a can be represented by a, a 1, a 2 ,…, etc. (B) 18 (C) 24 (D) 36 (E) 48 Written Response Question Pick an integer n greater than 4. Is the sum of every n consecutive integers divisible by n? Choose a different integer m greater than 4. Is the sum of every m consecutive integers divisible by m? Explain your reasoning. d. Based on your results in Parts A, B, and C, make a conjecture about the divisibility by an integer k of an arbitrary sum of k integers. Be sure to use clear and coherent sentences. Math Analysis CP WS 4.X- Section 4.1-4.4 Review Math Analysis CP WS 4.X- Section 4.6-4.8 Review Complete each question without the use of a graphing calculator. 1. Solve each equation or inequality: 1. Compare the a 6 d. 2 x 3 2 2 a. meaning a2 a2 of the 2 6 e. 3 2m 1 3 words: 5 b. w w 1 roots, f. a 1 5 a 6 x2 9 zeros c. 0 2 and 2x 5 x 3 factors. 3 2 2. Determine whether -3 is a root of x 3 x x 1 0 . Show your work. 3. Write a polynomial equation of least degree with roots 3, -1, 2i, and -2i. How many times does the graph of the related function intersect the x-axis? 4. What is the polynomial of least degree that has roots i , 0, 1 ? 5. Give an equation for a third degree polynomial that has roots at 3 and -2. 6. What do you know about the roots of a quadratic equation if its discriminant is 0? 7. Solve 3 x 2 6 x 2 0 by completing the square. 8. Find the discriminant of 15 x 2 4 x 1 and describe the nature of the roots of the equation. Then solve the equation using the quadratic formula. 9. How many times is -2 a root of x 4 8 x 2 16 0 ? 10. Find the remainder of x 3 6 x 9 x 3 and state whether the binomial is a factor of the polynomial. 2. Use the graph of f(x) to solve f(x) < 0. f(x) 10 5 0 x -5 -10 -10 -5 0 5 10 3. Consider the data below: 11. Find the remainder of x 4 6 x 2 8 x 2 and state whether the binomial is a factor of the polynomial. 12. Find the value of k so that the remainder of x 3 5 x 2 kx 2 x 2 is 0. 13. Find k such that x 1 is a factor of f ( x ) x x 2kx 3 x 5 . 4 3 2 14. One of the factors of 2 x 3 7 x 2 4 x 3 is x 3 . What is another factor? 15. List all the possible rational roots of 2x 4 x 3 6 x 3 0 . 16. Find all the zeros of f x 3 x 3 x 2 6 x 2 . 17. Find all the roots and factors of a 4 a2 2 0 . 18. Find all roots of the equation 4 x 4 3 x 2 1 0 . 19. In the 6 x 8 rectangular garden, the paths (shaded), have equal widths. The garden’s planting regions are shown as unshaded rectangles. If the total area of the shaded and unshaded regions is equal, how wide is each garden path? x y -1 48.6 a. b. c. d. -0.5 0.1 0 -9.3 .5 -0.3 1 11.6 1.5 18.1 2 14.6 2.5 6.4 3 0.4 3.5 12.2 4 63.8 Sketch a scatter plot of the data. What type of equation would best model the data? Find an equation f(x) to model the data. Find the approximate values of x for which f(x) = 2. 4. A Broadway theater sells 250 tickets for every performance. Each ticket costs $80. The company wants to increase the ticket price. They estimate that for each $3 increase in ticket price, 5 customers will be lost. Determine the ticket price that will allow the theater to increase its revenue by $1000. Cumulative Algebra Review- Chapter 04 Analysis CP Runner A travels how much further than Runner B, in feet? (A) a 60b All work is to be neatly shown on a separate piece of paper. Please write and box your answer, including the multiple choice answer letter. This is due the school day after the chapter test. (B) a 2 60b 2 (C) 360a b (D) 60 a b 1. What is the equation of a line with a y-intercept of 3 and an x-intercept of -5? (A) y 0.6 x 3 (B) y 1.7 x 3 (C) y 3 x 5 (D) y 3 x 5 (E) y 5 x 3 (E) 60 a 60b 7. If f x g f x (A) 2 (B) x 2 (C) 2 x 2 x2 (D) x 1 2x 1 (E) x 1 2. If the second term in an arithmetic sequence is 4, and the tenth term is 15, what is the first term in the sequence? (A) 1.18 (B) 1.27 (C) 1.38 (D) 2.63 (E) 2.75 1 3. If f x x 2 6 x 11 , then what is the 2 minimum value of f x ? (A) -8.0 (D) 6.0 (B) -7.0 (E) 11.0 4. What value does (C) 3.2 x2 x 6 approach as x 3x 6 approaches -2? (A) -1.67 (B) -0.60 (D) 1.00 (E) 2.33 (C) 0 5. If the greatest possible distance between two points in a rectangular solid is 12, then which of the following could be the dimensions of this solid? (A) 3 3 3 (B) 3 6 7 (C) 3 8 12 (D) 4 7 9 (E) 4 8 8 6. Runner A travels a feet every minute. Runner B travels b feet every second. In one hour, 1 1 and g x 1 , then x 1 x 8. x! x 2! (A) 0.5 (B) 2.0 (C) x (D) x 2 x (E) x 2 2 x 1 9. In order to disprove the hypothesis "No number divisible by 5 is less than 5," it would be necessary to (A) prove the statement false for all numbers divisible by 5 (B) demonstrate that numbers greater than 5 are often divisible by 5 (C) indicate that infinitely many numbers greater than 5 are divisible by 5 (D) supply one case in which a number divisible by 5 is less than 5 (E) show that a statement true of numbers greater than 5 is also true of numbers less than 5 10. The expression x2 3x 4 2 x 2 10 x 8 what value(s) of x? (A) x 1, 4 is undefined for (B) x 1 (C) x 0 (D) x 1, 4 (E) x 0, 1, 4 ---------------------------------------------------------------Written Response Question A complete response requires the following: Express your thinking in words Label any figures you draw Identify any formulas used Make clear the source of numbers used Full credit will not be earned if your work cannot clearly be followed. The final answer is important, but meaningless if you cannot show somebody how to get it. ---------------------------------------------------------------Matt says that if a and b are any positive numbers, then a 2 b2 must be smaller than a b . a. Choose three specific pairs of numbers to test Matt's statement. b. Suppose a2 b2 a b . Use algebra to determine what this implies about the value of a or b. Hint: Start by squaring both sides. c. Based on what you discovered in part b, what can you determine about Matt's initial assumption about a and b? d. For any triangle, the sum of the smaller two sides must be greater than the length of the third side. Using the given triangle, determine if Matt's statement is true or false. c a b Math Analysis CP WS 3B.X- Analyzing Polynomials Worksheet Provide all required information and sketch an accurate graph on graph paper without using the graphing calculator. f (x ) x3 8x 3 Type of function Total number of roots/ Number of real roots y-intercept x-intercepts (zeros) Multiplicity of zeros? Local (relative) Min/Max Absolute Min/Max Inflection Point(s) Interval where graph is concave up Interval where graph is concave down End behavior Location and Type of Discontinuity Domain Range f ( x ) x 4 8 x 2 16 g( x ) 6x3 x 2 5x 2 y 4 x 3 4 x 2 5 x 3 Math Analysis CP WS 3B.X- Analyzing Rational Functions Analyzing Rational Functions Worksheet Provide all required information and sketch an accurate graph on graph paper without using the graphing calculator. f (x) Type of function y-intercept x-intercept(s) Vertical Asymptote(s) Horizontal Asymptote Slant Asymptote Hole(s) End behavior Locations and Type of Discontinuity Domain Range Interval where graph is increasing Interval where graph is decreasing Graph – on graph paper 2x 5 x 3 g(x ) 2 2x 2 5 x 3 f (x) 3x 2 4x 5 x 3 y x 2 3x 4 x2 1 Cumulative Algebra Review- Chapter 03B Analysis CP All work is to be neatly shown on a separate piece of paper. Please write and box your answer, including the multiple choice answer letter. This is due the school day after the chapter test. 1. A parallelogram has vertices 0, 0 , 5, 0 , and 2, 3 . What are the coordinates of the fourth vertex? (A) 3, 2 (C) 7, 3 (B) 5, 3 (E) It cannot be determined from the information given. 2. In the function g x A sin Bx C D , constants are represented by A, B, C, and D. If g x is to be altered in such a way that both its period and amplitude are increased, which of the following must be increased? (A) A only (B) B only (C) C only (D) A and B only (E) A and C only 3. The height of a cylinder is equal to one-half of n, where n is equal to one-half of the cylinder's diameter. What is the surface area of this cylinder in terms of n? 3 n 2 (B) 2 n 2 (C) 3 n 2 (A) 2 n (E) 2 n 2 n (D) 2 n 2 2 f g 2.3 ? (A) 0.1 (D) 1.8 (B) 1.2 (E) 2.3 1 x 1, then 2 (C) 1.3 5. If logy 2 8 , then y (A) 0.25 (D) 2.83 (B) 1.04 (E) 3.00 and the line y mx b . If m b , then what is the volume of the cone generated by rotating this triangle around the x-axis? (C) 1.09 Written Response Question 3 (A) (B) 9 3 (C) (D) 3 (E) 9 7. If sin x m and 0 x 90 , then tan x (A) (D) (D) 10, 5 4. If f x 3 x and g x 6. A triangle is formed by the x-axis, the y-axis, 1 m2 m 1 m2 (B) (E) m 1 m2 (C) 1 m2 m m2 1 m2 8. When 4 x 2 6 x L is divided by x 1 , the remainder is 2. Which of the following is the value of L? (A) 4 (B) 6 (C) 10 (D) 12 (E) 15 9. The menu of a certain restaurant lists 10 items in column A and 20 items in column B. A family plans to share 5 items from column A and 5 items from column B. If none of the items are found in both columns, how many different combinations of items could the family choose? (A) 25 (B) 200 (C) 3,425 (D) 3,907,008 (E) 5.63 1010 10. x y y x (A) 0 (B) x y (C) y x (D) 2 x y (E) 2 x y A complete response requires the following: Express your thinking in words Label any figures you draw Identify any formulas used Make clear the source of numbers used Full credit will not be earned if your work cannot clearly be followed. The final answer is important, but meaningless if you cannot show somebody how to get it. ---------------------------------------------------------------Prove that there exists exactly one right triangle whose side lengths are consecutive integers. (Hint: Remember to generalize. A random integer is x, so three consecutive integers would be x , x 1 , and x 2 .) Math Analysis CP WS- Exponent/Log Review Day 1 Math Analysis CP WS- Exponent/Log Review Day 2 Do all work neatly on a separate sheet of paper. This should be done WITHOUT using your calculator. Do all work neatly on a separate sheet of paper. This should be done WITHOUT using your calculator. Solve each equation. Solve each equation. 1. 34 3 x1 2. 2 3 3. 3 27n 3 23. log3 2 x log3 3 x 5 24. log10 5 y log10 13 26. log2 3 x 1 log2 x 7 25. log9 x 2 x log9 6 Simplify. 4. 7 y 7 4 y 9 n 2 2 3 5. 5 6 2 6. 8 3 16 2 Use log10 3 0.4771 and log10 5 0.6990 to evaluate each expression. 27. log10 15 28. log10 Change each equation to logarithmic form. 7. 52 25 9. 32 8. 1 9 25 32 10. 53 5 3 29. log10 75 Solve each equation. 1 125 30. log3 x log3 4 3 31. log6 x log6 x 1 log6 4 32. log5 x 2 log5 x 3 log5 5 33. log7 x 1 log7 x 1 log7 8 Change each equation to exponential form. 11. log3 81 4 13. log4 8 3 2 12. log9 81 2 14. log10 0.01 2 Use the change of base formula to write in terms of common logs only. 35. log1 3 9 36. log8 0.25 34. log2 15 Evaluate each expression. Solve each equation. 15. log2 x 3 16. log2 x 0.5 1 2 18. logx 27 3 17. logx 8 Evaluate each expression. 20. log2 0.5 21. 8log8 9 22. logx x 3 G2. y 3 x 2 x 39. eln10 41. 4e x 2 14 42. 3ln 5 x 2 12 Solve each equation. 40. ln 3 x 1 ln 2 x 8 G5. y log3 x Graph the following on graph paper. 1 G3. y 3 38. loge e2 Graph the following on graph paper. Identify each domain and range. 19. log64 8 G1. y 3 x 37. lne3 1 G4. y 2 3 x G6. y 2 log3 x G7. y log3 x 2 Math Analysis CP WS- Radioactive Decay and Logistics Day 1 Math Analysis CP WS- Radioactive Decay and Logistics Day 2 Do all work neatly on a separate sheet of paper. Do all work neatly on a separate sheet of paper. 1. The half-life of the radioactive krypton-19 is 10 seconds. If 16 grams of the element are initially present, how many grams are present 29 seconds? 2. The half-life of radioactive plutonium-239 is 25,000 years. If 16 grams of the element are initially present, how many grams are present after 80,000 years? 3. Prehistoric cave paintings were discovered in a cave in France. The paint contained 15% of the original carbon-14. Estimate the age of the paintings is the half-life for carbon-14 is 5700 years. 4. Skeletons were found at a construction site in San Francisco in 1989. The skeletons contained 88% of the expected amount of carbon-14 found in a living person. How old were the skeletons in 1989 if the half-life for carbon14 is 5700 years? 5. A bird species in danger of extinction has a population of 1400 five years ago and today only 1000 of the birds are alive. Once the population drops below 100, the situation will be irreversible and extinction occurs. When will this happen? 6. The population of a single-celled organism in a pond doubles every 5 days. If the initial count of organisms is 5000 and the final count is 25,000, how many days have passed? 7. A certain strain of bacteria increases from an initial count of 1000 to a final count of 35,000 in six hours. How long does it take for this strain to triple? 8. A culture of bacteria grew from 2500 at 2:00pm to 4000 at 5:00pm. What will the population be at 10:00pm? 9. During the first week of flu season, 100 people became ill. After the 4th week, 1080 had the flu, and after the 7th week, 5100 were ill. a) Determine the logistics model for this data b) When will the flu outbreaks “level” off? 10. The percentage of 1-year olds with some heart disease is 0.5%, for 20year olds with some heart disease is 4% while the percentage of 80-year olds is 89%. a) Determine the logistics model for this data b) What percentage of 40-year olds have some coronary heart disease. 11. Shown is world population, in billions, for seven selected years. Find the logistic regression to model this data. According to the model, what is the limiting size of the population that the Earth will eventually sustain? What does this mean in terms of the statement made by the U.S. National Academy of Sciences that 10 billion is the maximum that the world can support with some degree of comfort and individual choice? x, Number of Years After 1949 1 (1950) 11 (1960) 21 (1970) 31 (1980) 41 (1990) 51 (2000) 54 (2003) y, World Population (Billions) 2.6 3.0 3.7 4.5 5.3 6.1 6.3 Determine the regression, exponential or logistics, you would choose to model each data set and why. Miscarriages in Women Women's Percent of Age Miscarriages 22 9 27 10 32 13 37 20 42 38 47 52 Number of Countries Connected to the Internet Year # of Countries 1985 11 1991 91 1994 146 1997 195 2002 220 Number of Illegal Immigrants Living in the US # of Illegal Year Immigrants (Millions) 1992 3.4 1996 5.0 2000 7.0 2004 8.0 # of US Households with Pets # with Pets Year (Millions) 1998 54.0 1999 58.2 2000 61.1 2001 63.0 2002 64.2 Alcohol Use by US High School Seniors Year % Usage 30 days Prior to Survey 1980 72.0 1985 65.9 1990 57.1 1995 51.3 2000 50.0 2002 48.6 2003 47.5 Math Analysis CP WS- Compound Interest Review Math Analysis CP WS- Future Value Do all work neatly on a separate sheet of paper. Do all work neatly on a separate sheet of paper. 1. The value of a new car purchased for $18,000 decreases by 12% each year. Estimate the value of the car after two years. 2. You deposit $1000 in an account that pays 5% annual interest. Find the balance at the end of three years if your interest is compounded: a. Yearly? b. Quarterly? c. Monthly? d. Continuously? 1. A chemist deposits $300 in a savings account that pays 4% interest compounded annually and adds $300 at the end of each year for 4 years. How much money does she have at the end of 5 years? 2. Grandparents of a 4th grader decided to start a college fund so that in 8 years their grandchild will have $40,000 saved toward college tuition. What monthly payments must they make if they find a bank paying 8% interest? 3. A 55-yr old man would like to have $100,000 in his account when he retires in 10 yrs. What monthly payments should he make to an account that pays 6% monthly? 4. A math teacher deposits $1000 in a savings account at the end of each quarter for 10 yrs. How much money does she have at the end of 10 yrs if the account pays 8.25% compounded quarterly? 5. In 5 yrs, a company wants to buy a new computer system costing $100,000. They establish a sinking fund that pays 6% compounded semiannually. To accumulate $100,000 in 5 yrs, what is the payment every 6 months? 6. A manufacturer deposits $1000 each month into an account that pays 5.5% compounded monthly. He plans to do this for 7 years, how much money will be in the account at the end of that time period? 3. A newly married couple has $15,000 toward a new home. How long will the money have to be invested at 10% compounded quarterly to grow to the estimated $20000 needed for a down payment? 4. Sue has saved $7000 toward the purchase of a $9000 used car. How long will Sue's money need to be invested at 9% compounded monthly until she can buy the car? 5. What is the average rate of inflation if a desk that cost $250 in 1990 cost $275 in 1997? (Note: Inflation is compounded continuously) 6. The world population is approximately 6.9 Billion people and growing at 1.1% compounded continuously. In what year will the population reach 9 Billion people? -------------------------------------------------------------------------------------------------------------WS- Effective Annual Yield Do all work neatly on a separate sheet of paper. Find the effective annual yield for each investment. Then find the value of a $1000 investment after 5 years for each situation. 1. 10% compounded quarterly 2. 8.5% compounded monthly 3. 9.25% compounded continuously 4. 7.75% compounded continuously 5. 6.5% compounded daily (365 day year) 6. Which investment yields more interest – 9% compounded continuously or 9.2% compounded quarterly? Also complete Page 751: 31-51 odd Math Analysis CP WS- Present Value Math Analysis CP WS- Review- Day 1 Do all work neatly on a separate sheet of paper. Do all work neatly on a separate sheet of paper. 1. 2. 3. What amount of money must be invested today at 6% compounded monthly so that payments of $100 per month can be made from this fund for 5 years? A television is purchased for $100 down and $30 a month for 12 months. If the finance charge is 15% compounded monthly, find the original price of the set. What are the monthly payments to finance a $12,000 car at 13% interest for 5 years? How much interest was paid? What is the total price of the car? 1. 2. 3. 4. 5. 6. 4. What is the highest priced price a person can afford if he is willing to pay monthly car payments of $350 for the next 5 years with the interest rate at 12%? 5. You decide to purchase a house for $450,000. Your parents give you the 20% down payment. Find the monthly house payments if you are able to get a 30-year loan at 5 ½% compounded monthly. How much interest was paid? What was the total cost of the house? 6. In order for you not to work while in college, a fund is set up to pay you $500 a month for 4 years. How much should be deposited in the fund which is paying 6% compounded monthly to achieve this goal? How many months were you able to “live” off your interest? 7. Radon has a half-life of 3.8 days. How long would it take 2 grams of decay to 0.4 grams? How much should you deposit initially in an account paying 10% compounded semi-annually in order to have $25,000 in 10 years? Which is the better investment and why: 9% compounded quarterly or 9.25% compounded annually? A company establishes a sinking fund to replace equipment in 6 years at an estimated cost of $50,000. They plan to make monthly payments that pay 6% compounded monthly. How much should each payment be? If an account has $20,000, after an initial investment of $5000, how long did it take to accumulate this amount at 12% with simple interest? You can afford monthly deposits of $200 into an account that pays 7.98% compounded monthly. How long will it be until you have $2500 to purchase a used car? At noon, a culture of bacteria has 2.5 106 members and at 3:00pm the culture had grown to 4.5 106 . When will the population be 8.0 106 ? 8. You decide to buy a house for $250,000 and are able to make a 10% down payment. What is your monthly payment if the loan is 7% compounded monthly for 30 years? How much did you pay in total? How much interest? -------------------------------------------------------------------------------------------------------------Math Analysis CP WS- Review- Day 2 Do all work neatly on a separate sheet of paper. 9. 10. 11. 12. 13. 14. 15. What is the interest rate if $3500 grew to $10,000 over 3 years, compounded quarterly? A city's population grew from 50,000 to 90,000 from 1970-1980. How many people will there be in 2015? In what year will the population be 2,000,000? The half-life of a substance is 80 days. How much remains after 25 days? An account has $20,000 after an initial investment of $3000. How long was the account open if the money was compounded continuously at 5%? What is the inflation rate if a car cost $15,500 in 1986 and now costs $23,000? At what interest rate would you need to have your money double in 5 years compounded daily? A scholarship committee wishes to establish a scholarship that will pay $1500 per quarter to a student for two years. How much should be deposited now, at 8% compounded quarterly, to establish this scholarship? How much will the student receive in two years? 16. Find an exponential function of the form y ab x whose graph passes through the points 2, 4 and 6, 8 Math Analysis CP WS- Sequence and Series Classwork Arithmetic Sequence Arithmetic Series Geometric Sequence Geometric Series Infinite Geometric Series Example 1, 5, 9, 13, 17… Formula Variables Sample Problem And Solution! Word Problem And Solution! Sn n a1 an 2 a1- first term an – nth term r – common ratio n – number of terms 1 Evaluate n 1 2 5 n 1 On the first swing, a pendulum travels 8 in. On the next swing, it travels 6 in, then 4.5 in and so on. How far does it travel before it stops swinging? Math Analysis CP WS- Sets and Venn Diagrams Math Analysis CP WS- Extra Problems 35 Complete each problem on a separate piece of paper. 5 20 1. An auto club’s emergency service has determined that when club members call to report that their cars will not start, the probability that the engine is flooded is 0.5 and the probability that the battery is dead is 0.4. a) Assuming that the events are independent, what is the probability that both the engine is flooded and the battery is dead? b) Are the two events mutually exclusive (disjoint)? c) What is the probability that the next person to report that a car will not start has a neither a flooded engine nor a dead battery? 2. In order to test a new drug for adverse reactions, the drug was administered to 1000 test subjects with the following results: 60 subjects reported that their only adverse reaction was a loss of appetite, 90 subjects reported that their only adverse reaction was a loss of sleep, and 800 subjects reported no adverse reactions at all. a) What is the probability that a person suffered from loss of appetite and sleep? b) If a randomly selected test subject suffered a loss of appetite, what is the probability they also suffered a loss of sleep? c) If a randomly selected test subject suffered a loss of sleep, what is the probability they also suffered a loss of appetite? 3. Let A be the event that a randomly selected student completes his/her homework. Let B be the event that the student does well on an exam. a) Use the probability tree to find each probability: 40 1. Use the Venn Diagram at right to determine the number of elements in each set: a. A b. U c. B d. A’ e. B’ f. U’ g. A B h. A B j. A B ' i. A ' B k. A B ' l. A B ' 2. A marketing survey of 1,000 car commuters found that 600 listen to the news, 500 listen to music, and 300 listen to both. Let N = Set of commuters in the sample who listen to news M = Set of commuters in the sample who listen to music a. Draw a Venn Diagram to represent the information. b. How many elements are in each of the following sets? i. N M ii. N M vi. N M ' v. N ' M iii. N M ' iv. N M ' c. 3. How many commuters are in each of the following sets? Commuters who listen to either news or music Commuters who listen to both news and music Commuters who do not listen to either news or music Commuters who do not listen to both news and music Commuters who listen to music but not news Commuters who listen to news but not music 150 Analysis students were asked if they were going to take Statistics or Discrete Math during their senior year. 80 students said Statistics, 60 students said Discrete, and 10 students said they will be taking both math courses. Let S = number of students who will take Statistics D = number of students who will take Discrete Describe in words the sets defined by each of the following AND find the number of students in each set. 4. a. S e. S D b. D f. S D i. S D ' j. S D ' c. S’ g. S ' D Find the number of employees in each set AND describe the set in words: a. H b. S d. Y c. S P e. H Y f. H Y g. S N h. S N i. S P j. Y N ii. P(A’) v. P A B vii. P(B) viii. P(B’) iii. P(B|A) vi. P A ' B .7 .3 b) c) .8 B .2 B’ .5 B A A’ State in words what each probability above describes. B’ Are the events “a student does homework” and “a student does well on an exam” independent? Explain how you know. 4. A new test has been developed to screen for a particular type of cancer. Approximately 2% of all adults are known to have this type of cancer. A random sample of 1000 adults is given the new test, and it is found that the test indicates cancer in 96% of those who actually have it. The test also indicates cancer in 1% of patients who do not have the disease. Based on these results, what is the probability that a randomly chosen person who tests positive with the new test does not actually have cancer? 5. Shaquille O’Neil has a long-term free throw percentage of 55%. (Meaning that the probability he makes a given free throw is 0.55) Find the probability that Shaq makes: a) Exactly 7 of his next 10 free throws b) 7 or more of his next 10 free throws 6. A five-question multiple-choice test has 4 answer choices (A, B, C, D) for each question. A student who did no studying for the test is randomly guessing on each question. d. D’ h. S D ' In a study to determine employee views about an upcoming strike, 1000 employees were selected at random and classified by their method of payment ( H= Hourly, S = Salary, P = Percentage of profits) and whether they were in favor of the strike (Y = Yes, N= No). The responses are summarized below : Hourly Salary Percentage of Profits Yes to Strike 400 180 20 No to Strike 150 120 130 i. P(A) iv. P(B|A’) a) b) c) d) e) What is the probability of answering a single question correctly by random guessing? What is the probability of answering all 5 questions correctly? What is the probability of answering exactly 3 questions correctly? What is the probability of failing the quiz if a grade below 60% is a fail? What score is the student most likely to earn on the quiz? Math Analysis CP WS- Probability Review 1. A survey of residents in a certain town indicates 170 own a humidifier, 130 own a snow blower, and 80 own both a dehumidifier and snow blower. How many own either a dehumidifier or snow blower? 2. From a group of 10 people, in how many ways can we choose a chairperson, vice chair, treasurer and secretary, assuming one person cannot hold more than one position? 3. A teacher has 14 math books on her shelf. In how many ways can the books be arranged if there are 3 (identical) Algebra books, 4 (identical) Analysis books, 5 (identical) Statistics books, and two different Calculus books? 4. Find the number of ways eight people can be seated at an octagonal table (relative to each other). 5. A software company employs 9 sales reps and 8 technical reps. In how many ways can the company select 5 of these employees to send to a computer convention if at least 4 technical reps must attend? 6. An experiment consists of drawing a ball from a box that contains nine balls numbered 1 to 9. What is the probability that the ball number on the ball is even or divisible by 3? 7. Eight cards are drawn from a standard deck of cards. What is the probability that there are 5 face cards and 3 non-face cards? 8. In a law firm consisting of 20 lawyers, 9 are criminal lawyers, 6 are divorce lawyers and 4 can practice either criminal or divorce law. If a lawyer from the firm is chosen at random, what is the probability that he or she isn’t a criminal or divorce lawyer? 9. A pair of dice is tossed. Find the odds that the sum of the dice facing up is 5. 10. Two marbles are drawn in succession out of a box that contains 3 green and 8 yellow marbles without replacement. Find the probability that exactly one green marble was drawn. 11. A little girl has 15 socks in her drawer, 7 are pink and 8 are purple. If she selects two socks at random, what is the probability that at least one of the socks is pink? 12. Each person in a group of students was identified by his/her hair color and then asked whether he or she preferred taking math classes in the morning or afternoon. The results are shown in the table below. Class Time Preference Blonde Brunette Redhead Morning 45 25 5 Afternoon 40 20 20 a. Find the probability that a randomly selected student from this group prefers afternoon math classes? b. Find the probability that a randomly selected redhead from the group prefers afternoon math classes? c. Are the events “prefers afternoon math classes” and “being a redhead” independent? Explain how you know. 13. A soccer team is to play two games in a tournament. The probability of winning the first game is 0.60. If the first game is won, the probability of winning the second game is 0.25. If the first game is lost, the probability of winning the second game is 0.15. What is the probability the team won its first game if we know the second game was lost? 14. Each question on an 8-question multiple choice test has 5 possible answer choices, only one of which is correct. If a student randomly guesses on all 8 questions, find the probability the student gets more than 5 questions correct. 15. Four coins are tossed. Find the probability that three are heads or two are tails. Math Analysis CP WS- Statistics Day 1: Probability Distributions Math Analysis CP WS- Statistics Day 2: Describing Distributions Do all work neatly on a separate piece of paper. Do all work neatly on a separate piece of paper. 1. 1. Describe the shape of each of the distributions you constructed for Day 1. 2. The following data show the salaries (in thousands of dollars) of 28 employees of a small company: (Note: Data is the same as Day 1, # 4) 30, 33, 35, 41, 42, 45, 45, 50, 52, 53, 55, 57, 60, 60, 61, 62, 63, 65, 67, 70, 71, 72, 73, 98, 105, 125, 150, 175 2. 3. A student randomly guesses the answer for each question of a 6-question True/False exam. Let the random variable X be the number of questions answered correctly. a. List the possible numerical outcomes (sample space) of X. b. Find the probability of each outcome (Hint: it’s binomial!) c. Construct a histogram showing the probability distribution. Be sure to label your horizontal and vertical axis. d. What is the probability that the student will get 4 or more questions correct? e. What is the expected value of the number of questions the student will answer correctly? c. d. e. Now you will construct an experimental probability distribution for rolling a die 50 times. Use your calculator MATH PRB randInt (1, 6) to simulate rolling a die one time. Repeat 50 rolls of the die and record the frequency of each outcome in a probability distribution table. 3. a. 4. b. 4. a. b. You are going to roll a die 50 times. a. What is the sample space? b. What is the probability of each outcome? c. Construct a histogram showing the probability distribution. Label your horizontal and vertical axes. d. What is the expected value of a single dice roll? Will you ever actually roll that value? e. Does your probability distribution represent theoretical or experimental probabilities? Use your data to construct a frequency table and a probability distribution. Outcome 1 2 3 4 Observed Frequency Probability 5 6 In the Pick 3 Lottery you choose a 3 digit number. If your number matches the number selected by the lottery board, you win $500. Otherwise, you win nothing. a. What are the possible numerical outcomes for your winnings? b. What is the probability of each outcome? c. Construct a histogram showing the probability distribution. d. Find the expected value of your winnings. e. A game is considered “fair” if the cost to play is equal to the expected value of the winnings. What would be a fair price to charge for a lottery ticket? n 1 to calculate the standard deviation of 0, 2, 5, 8, 10. What The heights of a random sample of 19 men are recorded below: a. b. c. d. e. f. 30, 33, 35, 41, 42, 45, 45, 50, 52, 53, 55, 57, 60, 60, 61, 62, 63, 65, 67, 70, 71, 72, 73, 98, 105, 125, 150, 175 5. s 2 69.9, 71.8, 72.1, 73.1, 73.8, 70.6, 69.4, 69.6, 76.2, 71.8, 74.6, 66.9, 69.1, 66.7, 70.4, 71.8, 69.3, 72.3, 71.5 Construct a probability histogram for your data. Compare your experimental distribution to the theoretical distribution you drew for #2. Construct a frequency table for the data. What class intervals will you use? Draw a frequency histogram for the data. Include a title and label your axes. Suppose that an adult is randomly selected from this company. What is the probability that s/he earns more than $75,000? Use the formula x x does this value tell you about the data? The following data show the salaries (in thousands of dollars) of 28 employees of a small company: a. b. c. What are the mean and median salaries? Which one is a better measure of center? Why? Find the range and IQR of the data. Which one is a better measure of the spread of the data? Why? Find the standard deviation. Use the 1.5 IQR rule to determine if the data has any outliers. Draw a box plot of the data showing outliers (if present). 5. Use your calculator to construct a histogram of the data. Use class interval widths of 2 inches. Describe the shape of the distribution. Find the mean, median, and mode (class interval with the highest frequency). What do you notice about the three measures of center? Construct a box plot for the data. Are there any outliers? What interval of heights contains the middle 50% of the men? What height interval contains the shortest 25% of the men? The tallest 25%? A student randomly guesses the answer for each question of a 6-question True/False exam. (Note: You constructed the probability distribution on Day 1, Q #1) a. b. c. d. e. f. Enter the numerical values of the sample space into L1 of your calculator. Enter the probability of each outcome into L2. To have your calculator construct the probability histogram, press 2nd STAT PLOT 1: PLOT 1…On. Choose the histogram. Set Xlist: L1 and Freq: L2. (This allows the probabilities in L2 to be used as frequencies). Use the same window as you used for the graph yesterday. Is the calculator’s graph the same as yours? Follow these instructions carefully: Press STAT CALC 1-Var Stats ENTER L1, L2 ENTER . (The calculator will know to use the values in L2 as frequencies for L1). What is the mean of the distribution? Is it the same as the expected value you found yesterday? Does the calculator report a sample standard deviation? For a probability distribution, the calculator provides the population standard deviation . What is this value? Report Q1, Median, and Q3 for the distribution. Use your calculator to construct a box plot. Math Analysis CP WS- Statistics Day 3: Normal Distributions Math Analysis CP WS- Statistics Day 4: Review Do all work neatly on a separate piece of paper. Do all work neatly on a separate piece of paper. Table entries show the percent (P) of observations in a Normal distribution that are less than the z-score. 1. A basket ball player who consistently makes 80% of his free throws attempts 5 freethrows during a game. Let the random variable X be the number of free throws made. a. List the possible numerical outcomes for X. b. Find the probability of each outcome. c. Construct a histogram to display the probability distribution. d. Describe the shape of the distribution. e. What is the expected value of X. Explain what this number tells you. 2. The weight in pounds of children in a 4th grade class are given below: z P -3.0 0.13 1. The lengths of babies born at City Hospital last year were approximately Normally distributed with a mean of 20 inches and a standard deviation of 1 inch. a. Draw a Normal curve showing the mean and ± 1, ± 2, ± 3 std. dev. b. What percent of the babies born were longer than 20 inches? c. What percent of the babies born were between 19 and 22 inches? d. Would it be unusual for a baby to be more than 23 inches? Explain your answer. e. What interval centered about the mean would contain 95% of all babies born? 2. 3. 4. 5. -2.5 0.6 -2.0 2.3 -1.5 6.7 -1.0 15.9 -.50 30.9 0 50% .50 69.1 1.0 84.1 1.5 93.3 2.0 97.7 2.5 99.4 3.0 99.87 The amount of cola that a machine puts into soda cans is approximately Normally distributed with a mean of 355 mL and a standard deviation of 2 mL. Assume that the machine fills 2000 can a day. a. Draw a Normal curve showing the mean and ± 1, ± 2, ± 3 std. dev. a. About how many cans will contain more than 359 mL of cola? b. About how many cans will contain between 353 and 357 mL of cola? c. About how many cans will contain less than 352 mL of cola? d. If the size of the can is 360 mL, about how many cans will overflow? 64 84 a. b. c. d. e. 3. Scores on the Weschler Adult Intelligence Scale are approximately Normally distributed with a mean of 110 and a standard deviation of 25. a. Approximately what percent of adults will score below 98? b. Approximately what percent of adults will score above 135? c. MENSA is an elite organization that will only admit the top 2% of applicants. Approximately what score on the Weschler exam would an adult need to qualify for MENSA? 71 60 57 68 67 72 74 91 65 77 59 69 62 76 67 88 72 99 Use your calculator to construct a histogram of the data. Describe the shape, center and spread of the data. Use the 1.5 IQR rule to determine if there are any outliers. Construct a box plot of the data. What weight interval contains 50% of the students? A survey of 100 randomly selected families in a small community asked how many pets lived in their household. The responses are given below. Number of Pets Frequency a. b. c. 0 14 1 22 2 20 3 18 4 10 5 8 6 6 7 2 Identify the random variable and its possible numerical outcomes. What is the probability that a family selected at random has 5 or more pets? What is the expected value for the number of pets in a household in this community? A math teacher will be grading a test using a Normal curve. The mean score on the test was 80 and the standard deviation was 8. a. What percent of the students will get A’s if a score of 88 or above is an A? b. What percent of students will get B’s if a score of 80 to 88 is a B? c. What percent of students will get C’s if a score of 68 to 80 is a C? d. The teacher only wants about 2% of his students to fail. What score should be the lowest passing (D) grade? 5. When administered to 4th graders, a test of reading ability has a mean of 75 and standard deviation of 12. Sixth graders have a mean score of 85 with a standard deviation of 8 on a similar test. A young student scored 71 on the reading test as a 4th grader and 79 as a 6th grader. a. Compute the z-score of the student as a 4th grader and a 6th grader. b. Relative to his classmates, is his reading performance as measured by the test improving? c. Do you need to know if the scores are Normally distributed to answer part a & b? The commute times for employees at a large company follow an approximately Normal distribution with a mean of 33 minutes and a standard deviation of 7 minutes. a. Draw a Normal curve showing the mean and + 1, + 2, + 3 std. dev. b. What percent of the company’s employees commute less than 26 minutes? c. What is the commute time of the middle 95% of the company’s employees? d. What percent of employees commute between 26 and 47 minutes? e. Would you be surprised to hear that many employees have commute times over 50 minutes? 6. The midterm scores in Professor Normal’s statistic course had a mean of 84 and a standard deviation of 8. The final exam scores had a mean of 77 and a standard deviation of 11. Student A scored 72 on the midterm and 66 on the final. Has her test score relative to the other students in the class improved? 4. Use the formula s x x n 1 2 to calculate the standard deviation of 7, 8, 9, 10, 11, 12, 13. Explain what this value tells you about the data.
