Slide 1 / 203 Slide 2 / 203 New Jersey Center for Teaching and Learning Algebra II Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of Statistics and Probability students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning 2013-04-16 community, and/or provide access to course materials to parents, students and others. www.njctl.org Click to go to website: www.njctl.org Slide 3 / 203 Slide 4 / 203 Table of Contents click on the topic to go to that section Sets Sets Independence and Conditional Probability Measures of Central Tendency Standard Deviation and Normal Distribution Two-Way Frequency Tables Sampling and Experiments Return to Table of Contents Slide 5 / 203 Slide 6 / 203 Sets Sets Why do we need this? Goals and Objectives Being able to categorize and describe situations allows us to analyze problems that we are presented with in their most basic forms. Many different fields need to categorize elements they use or study. Businesses need to look at what they are offering, Students will be able to use characteristics of problems, including Biologists need to organize material they are studying and even unions, intersections and complement, to describe events with you will need to categorize different options for your living appropriate set notation and Venn Diagrams. situation, such as insurance, in the future.
Slide 7 / 203 Slide 8 / 203 Sets Sets Create a Venn Diagram to match the information. Vocabulary and Set Notation Teacher U 4 Sample Space - Set of all possible outcomes. 7 A 2 B 9 0 Universe ( U ) - Set of all elements that need to be considered in the problem. 10 8 1 Empty Set ( # ) - The set that has no elements. 6 5 3 Subset - a set that is a part of a larger set. A = {0, 2, 3, 7, 9} Sets are usually denoted with uppercase letters and listed B = {1, 3, 7, 10} with brackets. For example: A = {-5, -2, 0, 1, 5} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Slide 9 / 203 Slide 10 / 203 Sets Sets Venn Diagrams are one example of a sample space that helps Use a sample space that helps organize the data effectively. Teacher us organize information.You can also use charts, tables, graphs and tree diagrams just to name a few more. For example, would you be able to effectively display a coin toss in a Venn Diagram or on a chart? Decide Tree Diagram for tossing a coin: Chart for rolling dice (sums): how to display the following information. H 6 1. Survey results about what subject students like in school. 7 8 9 10 11 12 H T 2. The different ways you can deal two cards from a deck of cards. 5 6 7 8 9 10 11 H H 3. Results that compare the number of men and women that like chocolate ice 4 5 6 7 8 9 10 T cream over vanilla ice cream. T 4 5 6 7 8 9 4. A poll on which grocery store people prefer to go to. 3 H 4 5 6 7 8 2 3 H T 2 3 4 5 6 7 T 1 H T 5 6 1 2 3 4 T Slide 11 / 203 Slide 12 / 203 Sets Sets The Universe (U) is all aspects that should be considered in a Teacher situation. The Universe (U) is basically the same as a sample The Empty Set ( # ) is the equivalent of zero when referring to sets. space also used in probability. For example, if you asked people at a college their age, the number of people that answered "2 years old" would be # . Name the Universe (U) of the following: 1. Survey at a local college asking women what they are An example of a subset would be the numbers 2, -6, and 13 in studying. the set of integers. 2. Calculating the probability that you would draw a red 10 out of a deck of cards. An outcome is a result of an experiment or survey. 3. Phone survey on who you will vote for in the U. S. Presidential race.
Slide 13 / 203 Slide 14 / 203 Sets Sets U A B Teacher Teacher 1 What is most likely the Universe of the following situation? -3 6 -2 Men Women 17 7 A U = {men} -12 B U = {women} 5 3 C U = {people} 6am aerobics 4 4pm 1 D U = {people at a fitness club} 5pm cycling water 10am weight aerobics E U = {people exercising at home} lifting -1 0 15 3pm nutrition C 7pm weight lifting 2pm climbing 1. List the universe for this problem. 6pm swimming 2. Name the different sets involved. 3. Find the subset that is in both A and B. 4. Find the subset that is in all sets A, B and C. Slide 15 / 203 Slide 16 / 203 Sets Sets 2 What is the most popular activity, or activities, at the club? 3 What are the most popular activities for both men and women at Teacher Teacher *Answer as many letters as necessary. the club? Women Men Men Women A 6 am aerobics A 5 pm cycling B 4 pm water aerobics B 4 pm water aerobics 6am aerobics C 3 pm nutrition 6am aerobics C 6 am aerobics 4pm 4pm 5pm cycling water D 5 pm cycling D 10 am weight lifting 5pm cycling water 10am weight 10am weight aerobics aerobics lifting E 10 am weight lifting lifting E 7 pm weight lifting 3pm nutrition 3pm nutrition F 2 pm climbing 7pm weight lifting F 3 pm nutrition 7pm weight lifting G 6 pm swimming G 6 pm swimming 2pm climbing 2pm climbing 6pm swimming 6pm swimming H 7 pm weight lifting H 2 pm climbing I Not enough information to tell I Not enough information to tell Slide 17 / 203 Slide 18 / 203 Sets Sets 4 What is the best display for the sample space (or universe ) of 5 What does the following set represent? {3, 6, 7} Teacher Teacher rolling an odd number on a single die? C A A S = {1, 2, 3, 4, 5, 6} A Set A 5 # 11 D 7 1 B Elements common to A and B 1 6 2 1 C Elements common to A and C 2 3 3 3 D The Universal set B 8 4 4 0 4 2 5 E A subset of set A 5 6 12 6 10 9 4 1 E B 5 C S = {1, 3, 5} 6 2 3
Slide 19 / 203 Slide 20 / 203 Sets Sets Teacher Unions (U) of two or more sets creates a set that includes 6 There are no elements of C that are not common to either set A or B, everything in each set. meaning that the set of numbers belonging to ONLY set C is { # }. C A Unions (U) are associated with "or." 5 11 C 7 True A 6 5 1 False 11 7 3 6 8 4 1 Examples: Shade in the areas! 0 2 3 A U B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} 8 4 0 12 2 10 9 B U C = {0, 2, 3, 4, 6, 7, 8, 9, 10, 12} 12 B (said "B union C") 10 9 B Slide 21 / 203 Slide 22 / 203 Sets Sets Unions (U) and Intersections (∩) are often combined. Intersections (∩) of two or more sets indicates ONLY what Teacher is in BOTH sets. C Intersections (∩) are associated with "and." A 5 11 C Find: A 7 Example: Shade in the areas! 5 6 1. (A U C) ∩ B 1 11 7 3 6 A ∩ B = {0, 3, 8} 1 8 4 2. A ∩ B ∩ C 0 2 3 8 4 B ∩ C = {3, 4, 2} 0 12 2 3. (A ∩ C) U (B ∩ C) 10 (said "B intersect C") 9 12 **Shade the diagram as you go to help. 10 B 9 B Slide 23 / 203 Slide 24 / 203 Sets Sets Teacher One last aspect of sets for this unit are Complements. More examples: Teacher Complements of a set are all elements of the Universe that are NOT in the set. 1. If U = {all students in college} and A = {female students}, find ~A. If U = {0, 1, 2, 3, 4, 5, 6} and A = {0, 1, 2, 3}, then the 2. If U = {a traditional deck of cards} and B = {Clubs and Diamonds, complement of A is {4, 5, 6} find ~B. There are several ways to denote a complement: 3. If U = {the students at your school} and C = {students that like math}, find ~C. ~A, A c , A' and not A In this unit, we will use "~A" or "not A"
Slide 25 / 203 Slide 26 / 203 Sets Sets Teacher Teacher You can also combine Complements with Intersections and 7 Find the complement of C or (~C). Unions . C A {3, 5, 6, 10, 12} A U 5 B A Find: B {3, 5, 6, 7, 9, 10, 12} 11 5 7 6 1. (A ∩ C) U ~B 6 C {1, 2, 3, 4, 5, 6, 8, 10, 12, 14} 12 3 1 10 D {7, 9} 3 2. (A U B) ∩ ~C 8 1 4 0 8 2 2 4 14 3. C ∩ B U ~A 7 12 15 10 9 11 9 4. ~A U ~B 13 B **Shade the diagram as you go to help. C Slide 27 / 203 Slide 28 / 203 Sets Sets Teacher Teacher 8 Find ~(A U B U C) 9 Find A U ~C U U B A B 5 A A {3, 5, 6, 10, 12} 5 6 A {7, 9} 6 12 3 B {1, 2, 4, 8} 10 12 3 B {1, 8} 10 C {1, 3, 5, 6, 8, 10, 12, 14} C {1, 2, 4, 8, 12, 14} 1 1 8 2 D {1, 3, 5, 6, 7, 8, 9, 10, 12, 14} 8 2 D {3, 5, 6, 11, 13, 15} 4 14 4 14 7 7 15 11 9 15 11 9 13 13 C C Slide 29 / 203 Slide 30 / 203 Sets Sets Teacher 10 Find ~B U A Teacher 11 Find ~(A B) ∩ A B A B A 12 A 45 B 27 B 30 15 12 18 18 15 C 45 12 18 18 C 18 D 63 D 12 U = The number of students in your grade U = The number of students in your grade A = the number of students that like English A = the number of students that like English B = the number of students that like Math B = the number of students that like Math
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