Sets Measures of Central Tendency Standard Deviation and Normal - PDF document

Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Slide 3 / 241 Slide 4 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional Probability Permutations & Combinations Sets Measures of Central Tendency Standard Deviation and Normal Distribution Two-Way Frequency Tables Sampling and Experiments Return to Table of Contents Slide 5 / 241 Slide 6 / 241 Why do we need this? Goals and Objectives Students will be able to use characteristics of problems, including Being able to categorize and describe situations allows us to unions, intersections and complement, to describe events with analyze problems that we are presented with in their most basic appropriate set notation and Venn Diagrams. forms. Many different fields need to categorize elements they use or study. Businesses need to look at what they are offering, Biologists need to organize material they are studying and even you will need to categorize different options for your living situation, such as insurance, in the future.

Slide 7 / 241 Slide 8 / 241 Create a Venn Diagram to match the Vocabulary and Set Notation information. Sample Space - Set of all possible outcomes. U 4 Universe (U) - Set of all elements that need to be considered 7 A 2 B in the problem. 9 0 Empty Set ( ∅ ) - The set that has no elements. 10 8 1 Subset - a set that is a part of a larger set. 6 5 3 Sets are usually denoted with uppercase letters and listed with brackets. For example: A = {-5, -2, 0, 1, 5} A = {0, 2, 3, 7, 9} B = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Slide 9 / 241 Slide 10 / 241 Data Displays Data Displays Use a sample space that helps organize the data effectively. Venn Diagrams are one example of a sample space that helps 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 Tree Diagram for tossing a coin 3 times: Chart for rolling 2 dice (sums): a chart? Decide how to display the following H information. 6 7 8 9 10 11 12 H T 5 6 7 8 9 10 11 H H 4 5 6 7 8 9 10 T T 4 5 6 7 8 9 3 H 1. Survey results about what subject students like in school. 4 5 6 7 8 2 3 H T 2 3 4 5 6 7 T 1 2. The different ways you can deal two cards from a deck of cards. H T 5 6 1 2 3 4 T 3. Results that compare the number of men and women that like chocolate ice cream over vanilla ice cream. 4. A poll on which grocery store people prefer to go to. Slide 11 / 241 Slide 12 / 241 The Universe Empty Set The Universe (U) is all aspects that should be considered in a 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 Name the Universe (U) of the following: of people that answered "2 years old" would be ∅ . 1. Survey at a local college asking women what they are studying. An example of a subset would be the numbers 2, -6, and 13 in 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 / 241 Slide 14 / 241 Example 1 What is most likely the Universe of the following situation? A U = {men} U A B B U = {women} -3 C U = {people} 6 -2 17 7 D U = {people at a fitness club} -12 E U = {people exercising at home} 5 3 4 1 Women Men -1 0 15 C 6am aerobics 4pm 5pm cycling water 1. List the universe for this problem. 10am weight aerobics lifting 2. Name the different sets involved. 3. Find the subset that is in both A and B. 3pm nutrition 7pm weight lifting 4. Find the subset that is in all sets A, B and C. 2pm climbing 6pm swimming Slide 15 / 241 Slide 16 / 241 2 What is the most popular activity, or activities, at the club? 3 What are the most popular activities for both men and *Answer as many letters as necessary. women at the club? Men Women Women Men A 5 pm cycling A 6 am aerobics 6am aerobics 6am aerobics B 4 pm water aerobics 4pm B 4 pm water aerobics 4pm 5pm cycling water C 6 am aerobics 5pm cycling water 10am weight C 3 pm nutrition aerobics 10am weight aerobics lifting lifting D 10 am weight lifting D 5 pm cycling 3pm nutrition 7pm weight lifting 3pm nutrition E 7 pm weight lifting E 10 am weight lifting 7pm weight lifting F 3 pm nutrition 2pm climbing F 2 pm climbing 2pm climbing 6pm swimming 6pm swimming G 6 pm swimming G 6 pm swimming H 2 pm climbing H 7 pm weight lifting I Not enough information to tell I Not enough information to tell Slide 17 / 241 Slide 18 / 241 4 What is the best display for the sample space (or 5 What does the following set represent? {3, 6, 7} universe) of rolling an odd number on a single number A Set A cube? B Elements common to A and B C Elements common to A and C Answer C A S = {1, 2, 3, 4, 5, 6} A D The Universal set # 5 D 11 1 E A subset of set A 7 1 2 6 2 1 3 3 3 B 4 8 4 4 5 0 2 5 6 6 12 10 4 9 1 E 5 C S = {1, 3, 5} B 6 2 3

Slide 19 / 241 Slide 20 / 241 Unions 6 There are no elements of C that are not common to either set A or B, meaning that the set of numbers Unions (U) of two or more sets creates a set that includes everything belonging to ONLY set C is { ∅ }. in each set. C A Unions (U) are associated with "or." 5 True 11 7 C A 6 False 5 1 11 7 3 6 8 4 Examples: Shade in the areas! 1 0 2 3 A U B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} 8 4 12 0 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 / 241 Slide 22 / 241 Unions and Intersections Intersections Intersections (∩) of two or more sets indicates ONLY what is in BOTH sets. Intersections (∩) are associated with "and." Unions (U) and Intersections (∩) are often combined. C A Example: Shade in the areas! 5 C A 11 7 5 6 11 A ∩ B = {0, 3, 8} Find: 1 7 6 3 1. (A U C) ∩ B 1 8 4 B ∩ C = {3, 4, 2} 0 3 2 8 4 2. A ∩ B ∩ C 0 (said "B intersect C") 2 12 10 9 12 3. (A ∩ C) U (B ∩ C) 10 9 B **Shade the diagram as you go to help. B Way to remember the difference between "∩" and "U": The intersection symbol (∩) looks like a lowercase "n". The word "and" also has the lowercase "n" in it, so "∩" means "and". Slide 23 / 241 Slide 24 / 241 Examples Complements One last aspect of sets for this unit are Complements. Complements of a set are all elements of the Universe that are 1. If U = {all students in college} and A = {female students}, find ~A. NOT in the set. 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 / 241 Slide 26 / 241 Examples 7 Find the complement of C or (~C). A {3, 5, 6, 10, 12} B {3, 5, 6, 7, 9, 10, 12} You can also combine Complements with Intersections and C {1, 2, 3, 4, 5, 6, 8, 10, 12, 14} Unions. C D {7, 9} A 5 Find: 11 7 1. (A ∩ C) U ~B 6 1 U B 3 A 5 2. (A U B) ∩ ~C 8 4 6 0 2 12 3 10 3. C ∩ B U ~A 12 1 10 8 2 9 4 4. ~A U ~B 14 B 7 **Shade the diagram as you go to help. 15 11 9 13 C Slide 27 / 241 Slide 28 / 241 8 Find ~(A U B U C) 9 Find A U ~C A {3, 5, 6, 10, 12} A {7, 9} B {1, 2, 4, 8} B {1, 8} C {1, 3, 5, 6, 8, 10, 12, 14} C {1, 2, 4, 8, 12, 14} D {1, 3, 5, 6, 7, 8, 9, 10, 12, 14} D {3, 5, 6, 11, 13, 15} U U B B A A 5 5 6 6 12 12 3 3 10 10 1 1 8 8 2 2 4 4 14 14 7 7 15 15 11 11 9 9 13 13 C C Slide 29 / 241 Slide 30 / 241 10 Find ~B U A 11 Find ~(A ∩ B) A 12 A 45 B 27 B 30 C 45 C 18 D 63 D 12 A B A B 15 12 18 18 12 18 18 15 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