# Probability, Statistics, & Data Analysis (PSD) Numbers: Concepts & Properties (NCP)

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1 Name ACT Prep PSD/NCP Probability, Statistics, & Data Analysis (PSD) Numbers: Concepts & Properties (NCP) Table of Contents: PSD 40: Calculate the missing data value, given the average and all data values but one. Page 3 PSD 404: Exhibit knowledge of simple counting techniques.. Page 4 PSD 50: Calculate the average, given the frequency counts of all the data values.. Page 5 PSD 503: Compute straightforward probabilities for common situations. Page 6 PSD 504: Use Venn diagrams in counting...page 7 PSD 603: Apply counting techniques.. Page 8 PSD 604: Compute a probability when the event and/or sample space are not given...page 9 NCP 40: Absolute Value..Page 0 NCP 508: Determine when an expression is undefined. Page 0 NCP 504: Work with Scientific Notation Page NCP 505/507: Work with squares and square roots/cubes and cube roots of numbers.. Page 2 NCP 509: Exhibit some knowledge of the complex numbers Page 3 NCP 604: Apply rules of exponents Page 4 BOA 40/50: Solve arithmetic problems that involve planning, percents, converting and rate. Page 5 Answer Key Page 6

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3 PSD 40: Calculate the missing data value, given the average and all data values but one EX: A data set has 5 data points. For this data set, the mean, the median, and the mode are each equal to 0. Four of the 5 points are 8, 0, 0, and 5. Which of the following is the 5 th data point? 7 B. 8 C. 9 D. 0 E. 7. There are 4 tests, each worth 00 points, in Marlene s Earth Science class. Her scores on the first 3 tests are 86, 79 and 78. What is the minimum score she can earn on the fourth test in order to guarantee a test average of at least 80 points? 6 B. 77 C. 80 D. 8 E Marlon is bowling in a tournament and has the highest average after 5 games, with scores of 20, 225, 254, 23, and 280. In order to maintain this exact average, what must be Marlon s score for his 6 th game? 200 B. 20 C. 23 D. 240 E So far, a student has earned the following scores on four 00- point tests this grading period: 65, 73, 8 and 82. What score must the student earn on the fifth and last 00- point test of the grading period to earn an average test grade of 80 for 5 tests? 75 B. 76 C. 78 D. 99 E. The student cannot earn an average of Vince earned scores of 75, 70, 92, 95 and 97 points (a total of 429 points) on the first 5 tests in Foods II. Solving which of the following equations for s gives the score he needs to earn on the 6 th test to average exactly 85 points for all 6 tests? s = 85 B s = 85 C. s D. s E. s = 85 = 85 =

4 PSD 404: Exhibit knowledge of simple counting techniques EX: Rudi has 5 pairs of slacks, 6 blouses, and 2 sweaters in her closet. How many different outfits, composed of a pair of slacks, a blouse, and a sweater, can she choose from this closet? 60 B. 32 C. 3 D. 6 E. 2. In the school cafeteria, students choose their lunch from 3 sandwiches, 3 soups, 4 salads, and 2 drinks. How many different lunches are possible for a student who chooses exactly sandwich, soup, salad, and drink? 2 B. 4 C. 2 D. 36 E Andrea has 3 sweaters, 2 scarves, and 2 skirts that go together well in any combination. How many different outfits can she put together consisting of sweater, scarf, and skirt? 3 B. 6 C. 7 D. 0 E How many distinct orders can 5 students stand in line to buy yearbooks? 5 B. 5 C. 25 D. 20 E. 3,25 4. Three friends will run a race. If there are no ties, in how many distinct orders can these 3 friends finish the race? 2 B. 3 C. 4 D. 5 E Boaters use 5 small flags of different colors to send messages to people in other boats. All 5 flags flown on pole, and the messages are conveyed by the order of the flags from the top of the pole downward. In how many different orders can the 5 flags be flown? 5 B. 5 C. 24 D. 25 E. 20 4

5 PSD 50: Calculate the average, given the frequency counts of all the data values Example: In a town of 500 people, the 300 males have an average age of 45 and the 200 females have an average age of 35. To the nearest year, what is the average age of the town s entire population? 40 B. 4 C. 42 D. 43 E. 44. High school students are surveyed as to how many hours they slept the night before. The results are summarized in the table below. What was the average number of hours slept among those surveyed rounded to the nearest tenth? 2. Four coins are flipped and the number of heads is counted. This experiment is repeated several times and the results are reported in the table below. The average number of head per experiment is 2.2. How many experiments resulted in 4 heads? 5.2 B. 7.0 C. 7.3 D. 8.0 E. 8.2 Hours Slept Number of Respondents B. C. 2 D. 3 E. 4 Number of Heads Frequency x 3. At ACT High School there are year olds, year olds, year olds, year olds, year olds, 3 9- year olds, and 2 20 year- olds. What is the average age of the students at EGHS rounded to the nearest tenth? 5.8 B. 9.9 C. 7.0 D E

6 PSD 503: Compute straightforward probabilities for common situations EX: A die is rolled 3 times. What is the probability of getting three 2 s in a row? 26 B. 72 C. 8 D. 6 E. 2. Event A is flipping a coin and getting a Head. Event B is rolling a die and getting an even number. Which of these events has the higher probability? Event A B. Event B C. Event A and B occurring simultaneously D. Same E. Cannot be determined. 3. Each of 6 historical events occurred in a different year. You are asked to arrange the 6 event in ascending order by the years they occurred. You know the earliest and the latest. You randomly order the other events. What is the probability that you order the 6 events correctly? 720 B. 20 C. 24 D. 6 E A regular octahedron, with faces numbered through 8, is given 2 fair rolls. What is the probability that both numbers rolled are even? 64 B. 56 C. 6 D. 2 E A jar contains exactly 2 gumballs, each of which is a solid color. There are 5 blue and 7 white gumballs. Chris will draw one gumball at random and then, without replacement, will draw another gumball from the jar at random. Which of the following expressions gives the probability that Chris will draw 2 blue gumballs? B. 5 2 i 5 2 C. 5 2 i 4 2 D. 5 2 i 5 E. 5 2 i 4 5. The 6- member drama club needs to choose a student government representative. They decide that the representative, who will be chosen at random, CANNOT be any of the 3 officers of the club. What is the probability that Adrian, who is a member of the club but NOT an officer, will be chosen? 0 B. 6 C. 3 D. 3 6 E. 3 6

7 PSD 504: Use Venn diagrams in counting EX: A health club surveyed 75 members about which types of equipment they had used in the past month. Of the 75 members, 7 had used treadmills, 89 had used stationary bikes, and 53 had used both types of equipment. Some members had used neither type of equipment. Of the 75 members, how many had used treadmills, stationary bikes, or both? 53 B. 8 C. 22 D. 34 E. 53. The intramural program at Ellington School has 300 students participating in one or more sports. The Venn diagram below indicates the percentages of these students who participate in soccer and/or volleyball. How many of the 300 students participate in neither soccer nor volleyball? 75 B. 20 C. 35 D. 50 E After polling a class of 20 music students by a show of hands, you find that 8 students play the guitar and 9 students play the piano. Given that information, what is the minimum number of students in this music class who play both the guitar and the piano? 0 B. C. 8 D. 9 E In a large high school, some teachers teach only subject, and some teachers teach more than subject. Using the information given in the table below about the math, science, and gym teachers in the school, how many teachers teach math only? B. 2 C. 5 D. 6 E. 24 7

8 PSD 603: Apply counting techniques EX: Each of 6 historical events occurred in a different year. You are asked to arrange the 6 events in ascending order by the years they occurred. You know the earliest and the latest. You randomly order the other events. What is the probability that you order the 6 events correctly? 720 B. 20 C. 24 D. 6 E. 4. Happy Soup Company stamps a 6- character product code on each can of soup it produces. Each product code consists of 5 letters (from the 26- letter alphabet) followed by a single digit (from the digits 0 to 9). The letters may repeat. How many such product codes are possible? 5(26)(0) B. 5(4)(3)(2) C. 5 (0) D. 26(25)(24)(23)(22)(0) E (0) 2. If you guessed randomly on this question, the probability of getting the correct answer would be 5. What would be the probability of getting all 4 answers correct if you guessed randomly and independently on 4 such questions? 3,25 B. 625 C. 20 D. 5 E

9 PSD 604: Compute a probability when the event and/or sample space are not given or obvious EX: A bag contains 6 red marbles, 5 yellow marbles, and 7 green marbles. How many additional red marbles must be added to the 8 marbles already in the bag so that the probability of randomly drawing a red marble is 3 5? 2 B. 6 C. 8 D. 24 E. 36. Jay starts with 0 white and 20 red marbles in a bag. He places 6 additional white marbles in the bag. How many red marbles, if it is possible, must also be placed in the bag so that the ratio of white to red marbles is the same as when Jay started? 3 B. 6 C. 9 D. 2 E. The ratio cannot be the same, because he would need a fractional number of marbles. 2. Of the 40 marbles in Dakarai s bag, all are solid in color and 6 are blue. The probability that he will randomly choose a yellow marble from the bag is 8. How many yellow marbles are in Dakarai s bag? 2 B. 3 C. 5 D. 6 E Snake- eyes occur when you roll two s on a pair of regular, 6- sided dice numbered from to 6. On any roll, what is the probability of rolling snake- eyes? 36 B. 25 C. 8 D. 6 E. 3 9

10 NCP 40: Exhibit knowledge of elementary number concepts such as absolute value EX: 5( 4)+ 3( 6) =? - 2 B. 2 C. 0 D. 9 E =? - 9 B. - 3 C. D. 3 E =? - 42 B. - 6 C. - D. 6 E. 42 NCP 508: Determine when an expression is undefined. EX: For what values of x is the expression 3 only B. All real numbers except 3 and - 2 C. All real numbers except - 2 and 2 D. All real numbers except - 3 and 2 E. All real numbers x 2 4 x 2 + x 6 defined.. What is the domain of the function f ( x) = All real numbers B. x x is a real numberand x 3 x + 3 x 2 2x 3 { } { } { } { } C. x x is a real numberand x 3 and x D. x x is a real numberand x 3,x,and x 3 E. x x 0 2. What is the sum of the values of x that makes the 3 x( x 2) undefined. 4 x B C. 0 D. 4 E. 34 0

11 NCP 504: Work with Scientific Notation ( b 0 ) 4 c 0.00 EX: For all nonzero b and c, b 0,000 0 B. C. 0 D. 0 7 E. b c ( ) ( )( c 0 ) =? 3. Water is considered contaminated when the level of zinc in the water reaches 5 parts of zinc per million parts of water. What is this level of zinc contamination written in scientific notation? B C D E Traveling at approximately 86,000 miles per second about how many miles does a beam of light travel in 2 hours? B C D E If there are hydrogen molecules in a volume of cubic centimeters, what is the average number of hydrogen molecules per cubic centimeter? B C D E Let a and b be integers (whole numbers), what is ( 0 ) a b expressed in scientific notation? 0 ab B. 0 ab C. 0 a+b D. 0 a b a b E. 0

12 NCP 505/507: Work with squares and square roots/cubes and cube roots EX: What is the simplified form of ( 62) 3? B C. 9 2 D. 9 2 E x is a real number such that x 3 = 64, then x 2 + x =? 4 B. 0 C. 8 D. 20 E Express in simplest radical form: B C. 3 5 D E =? 4. What is the simplifed form of the radical expression ? 8 B. 4 C D E. 6 0 B. 3 C. 4 D E

13 NCP 509: Exhibit some knowledge of complex numbers EX: Find the value of i 25. i B. - i C. D. - E. No Solution. For i 2 =, (4 + i ) 2 =? 5 B. 7 C i D i E i 2. What does the quotient 3+ i 2+ 3i i B. 6 3i i C i D i E. 3 equal? Find the value i 52. i B. i C. D. - E. No Solution Which of the following expression is the simplified form of 2i i i B. 0 2i C. 8 4i D. 8 0i E. 6 2i ( ) + 8? 5. The imaginary number, i, is defined such that i 2 =. What does i + i 2 + i i 49 equal? i B. i C. - D. 0 E. 3

14 NCP 604: Apply rules of exponents ( 3a 4 b ) 3 2 EX: Which of the following expression is equivalent to ab 5 9a 7 b B. 9a 9 b C. 9a 7 b D. 9a 9 b E. 9a7 b ( ) 3 ( 3a 2 b) 2 is equivalent to:. 2ab 2 6a 5 b 7 B. 6a 2 b If x 36 3 = x p for all x 0, then p 2 =? B. 25 C. 2 C. 36a 7 b 7 D. 72a 7 b 8 E. 6 5 a 2 b 2 D E For all a >, the expression 3a4 3a 6 equals: B. a 2 C. a 2 D. a 2 4. For any nonzero value of y, ( y ) 5 3 =? y 5 B. y 2 C. y 8 D. y 5 E. y 25 E. a 2 5. Whenever x and z are nonzero numbers, x 4 z 2 B. x 4 z C. x 4 z x 2 z ( ) simplifies to: x2 z 2 6. If xa x b = x3 for all x 0, which of the following must be true? a b = 3 B. a + b = 3 C. a b = 3 D. a b = 3 E. ab = 3 D. z 2 E. z 4

15 BOA 40/50: Solve arithmetic problems that involve planning, percents, converting and rate EX: How many minutes would it take a train to travel 90,000 meters at a constant speed of 20 kilometers per hour? 30 B. 40 C. 45 D. 80 E. 90. On September, a dress was priced at \$90. On October, the price was reduced by 20%. On November, the price was further reduced by 25% of the October price and marked FINAL. What percent of the original price was the FINAL price? 40% B. 45% C. 55% D. 60% E. 77.5% 3. To park a car at a short- term parking lot costs \$.75 for the st hour or any part thereof, \$.50 for the 2 nd hour or any part thereof, and \$0.75 for each additional hour or any part thereof after the 2 nd hour. Your ticket shows that you parked your car in this lot from 0:47 a.m. to 4:35 p.m. on the same day. What is the cost of parking your car, according to this ticket? (Note: Prices include all applicable sales tax.) \$4.75 B. \$5.50 C. \$5.86 D. \$6.0 E. \$ Vehicle A averages 7 miles per gallon of gasoline, and Vehicle B average 45 miles per gallon of gasoline. At these rates, how many more gallons of gasoline does Vehicle A need than Vehicle B to make a,530- mile trip? 3 B. 34 C. 56 D. 62 E Ken baked, frosted, and decorated a cake for the last Math Club meeting. The Math Club will pay Ken \$5.00 for preparing a cake and will also pay him for the cost of the cake mix at \$.73, the frosting mix at \$2.67, and the sales tax of 5% on these 2 items. What is the total amount the Math Club will pay Ken? \$4.67 B. \$9.40 C. \$9.45 D. \$9.62 E. \$ A bus company always keeps 3 tires in stock for every bus it owns, plus an additional 30 tires in stock for emergencies. According to this policy, the bus company needs to have a total of 20 tires in stock. How many buses does the company own? 30 B. 35 C. 40 D. 45 E Issa bought 2 cases of 2- ounces cans of soda. Each case contained 24 cans. Issa could have bought the same amount of soda by buying how many 6- ounce bottles of soda? 8 B. 8 C. 32 D. 36 E. 48 5

16 Answer Key: PSD 40. B 2. D 3. D 4. D PSD 404. E 2. E 3. D 4. E 5. E PSD 50. C 2. D 3. A PSD 503. D 2. E 3. C 4. E 5. C PSD 504. B 2. A 3. C PSD 603. E 2. B PSD 604. D 2. C 3. A NCP 40. B 2. B NCP 508. C 2. B NCP 504. D 2. E 3. C 4. A NCP 505/507. C 2. B 3. B 4. A NCP 509. D 2. C 3. C 4. B 5. A NCP 604. D 2. A 3. E 4. A 5. E 6. A BOA 40/50. D 2. D 3. E 4. A 5. C 6. D 6

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