Algebra II. Sets. Slide 1 / 241 Slide 2 / 241. Slide 4 / 241. Slide 3 / 241. Slide 6 / 241. Slide 5 / 241. Probability and Statistics

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1 Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 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 Measures of Central Tendency Standard Deviation and Normal Distribution Two-Way Frequency Tables Sampling and Experiments Sets Return to Table of Contents Slide 5 / 241 Goals and Objectives Slide 6 / 241 Why do we need this? Students will be able to use characteristics of problems, including unions, intersections and complement, to describe events with appropriate set notation and Venn Diagrams. eing 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. usinesses need to look at what they are offering, iologists 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.

2 Slide 7 / 241 Vocabulary and Set Notation Slide 8 / 241 Create a Venn Diagram to match the information. Sample Space - Set of all possible outcomes. Universe (U) - Set of all elements that need to be considered in the problem. Empty Set ( ) - The set that has no elements. Subset - a set that is a part of a larger set. Sets are usually denoted with uppercase letters and listed with brackets. For example: A = {-5, -2, 0, 1, 5} U A A = {0, 2, 3, 7, 9} = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {0, 2, 3, 7, 9} = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Slide 8 () / 241 Create a Venn Diagram to match the information. U A 8 Teacher Notes 6 2 Move the circles and numbers around to 9 mirror the given 10information Slide 9 / 241 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. Tree Diagram for tossing a coin 3 times: H T H T H T H T H T H T H T Data Displays Chart for rolling 2 dice (sums): Slide 10 / 241 Data Displays Use a sample space that helps organize the data effectively. For example, would you be able to effectively display a coin toss in a Venn Diagram or on a chart? Decide how to display the following information. 1. Survey results about what subject students like in school. 2. The different ways you can deal two cards from a deck of cards. 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 10 () / 241 Data Displays Use a sample space that helps organize 1. Venn Diagram the data or chart effectively. 2. Chart 3. Venn Diagram or chart For example, would you 4. be Venn able Diagram to effectively or chart display a coin toss in a Venn Diagram or on a chart? Decide how to There display can be the different following answers. information. This question brings up other concerns such as how many people were asked and other parameters we will address in the unit. 1. Survey results about what subject students [This object is like a pull in tab] school. Teacher Notes 2. The different ways you can deal two cards from a deck of cards. 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.

3 Slide 11 / 241 The Universe Slide 11 () / 241 The Universe The Universe (U) is all aspects that should be considered in a situation. The Universe (U) is basically the same as a sample space also used in probability. Name the Universe (U) of the following: 1. Survey at a local college asking women what they are studying. 2. Calculating the probability that you would draw a red 10 out of a deck of cards. 3. Phone survey on who you will vote for in the U. S. Presidential race. The Universe (U) is all aspects that should be considered in a 1. Women that are enrolled as students at situation. The Universe (U) is basically the same as a sample that particular college. space also used in probability. 2. The deck of cards 3. People in the United States that not only Name the Universe picked up their (U) phone, of the but following: answered the question. 1. Survey at a local college asking women what they are studying. The term "Universe" is more often used in set theory while "sample 2. Calculating the probability space" is that used you with probability. would draw a red 10 out of a deck of cards. Teacher Notes 3. Phone survey on who you will vote for in the U. S. Presidential race. Slide 12 / 241 Empty Set Slide 13 / 241 Example The Empty Set ( ) is the equivalent of zero when referring to sets. For example, if you asked people at a college their age, the number of people that answered "2 years old" would be. An example of a subset would be the numbers 2, -6, and 13 in the set of integers. U A C An outcome is a result of an experiment or survey. 1. List the universe for this problem. 2. Name the different sets involved. 3. Find the subset that is in both A and. 4. Find the subset that is in all sets A, and C. Slide 13 () / 241 Slide 14 / 241 Example U A U = {-12, -3, -2, -1, 0, 1, 1 43, 4, 5, 6, 7, 15, 17} 2. A = {-3, -2, 1, 5} = {-3, 4, 5, 6} C = {0, 1, 4, 5, 15} -1 *3. A = {-3, 15} 0 *4. A C C = {5} 1. List the universe for *note: this the problem. notation and concept of intersection will be dealt with the next section of the unit. 2. Name the different sets involved. 3. Find the subset that is in both A and. 4. Find the subset that is in all sets A, and C. 7 1 What is most likely the Universe of the following situation? A U = {men} U = {women} C U = {people} D U = {people at a fitness club} E U = {people exercising at home} Men 5pm cycling 4pm water aerobics 7pm weight lifting 3pm nutrition 6pm swimming Women 6am aerobics 10am weight lifting 2pm climbing

4 Slide 14 () / What is most likely the Universe of the following situation? A U = {men} U = {women} C U = {people} D U = {people at a fitness club} E U = {people exercising at home} Men 5pm cycling 4pm water aerobics 7pm weight lifting 3pm nutrition Women 6am aerobics D 10am weight lifting Slide 15 / What is the most popular activity, or activities, at the club? * as many letters as necessary. Men A 6 am aerobics 4 pm water aerobics C 3 pm nutrition D 5 pm cycling E 10 am weight lifting F 2 pm climbing G 6 pm swimming H 7 pm weight lifting I Not enough information to tell 5pm cycling 4pm water aerobics 7pm weight lifting 3pm nutrition 6pm swimming Women 6am aerobics 10am weight lifting 2pm climbing 6pm swimming 2pm climbing Slide 15 () / What is the most popular activity, or activities, at the club? * as many letters as necessary. I or all of A through Men H. A 6 am aerobics You need actual 4 pm water aerobics numbers to tell 5pm what cyclingis C 3 pm nutrition most popular or a better D 5 pm cycling explanation of what the 7pm weight lifting E 10 am weight diagram lifting is about. F 2 pm climbing 6pm swimming G 6 pm swimming H 7 pm weight lifting I Not enough information to tell 4pm water aerobics 3pm nutrition Women 6am aerobics 10am weight lifting 2pm climbing Slide 16 / What are the most popular activities for both men and women at the club? Men Women A 5 pm cycling 4 pm water aerobics C 6 am aerobics D 10 am weight lifting E 7 pm weight lifting F 3 pm nutrition G 6 pm swimming H 2 pm climbing I Not enough information to tell 5pm cycling 4pm water aerobics 7pm weight lifting 3pm nutrition 6pm swimming 6am aerobics 10am weight lifting 2pm climbing Slide 16 () / What are the most popular activities for both men and women at the club? Men Women A 5 pm cycling 4 pm water aerobics 4 pm water aerobics C 6 am aerobics 3 pm nutrition 5pm cycling D 10 am weight lifting 7pm weight lifting E 7 pm weight lifting F 3 pm nutrition G 6 pm swimming H 2 pm climbing I Not enough information to tell 6pm swimming 4pm water aerobics 3pm nutrition 6am aerobics 10am weight lifting 2pm climbing Slide 17 / What is the best display for the sample space (or universe) of rolling an odd number on a single number cube? A S = {1, 2, 3, 4, 5, 6} C S = {1, 3, 5} D E #

5 Slide 18 / What does the following set represent? {3, 6, 7} Slide 18 () / What does the following set represent? {3, 6, 7} A Set A Elements common to A and C Elements common to A and C D The Universal set A E A subset of set A C A Set A Elements common to A and C Elements common to A and C D The Universal set E A subset of set A A 5 C C Slide 19 / There are no elements of C that are not common to either set A or, meaning that the set of numbers belonging to ONLY set C is { }. True False A C Slide 19 () / There are no elements of C that are not common to either set A or, meaning that the set of numbers belonging to ONLY set C is { }. True False A True C Slide 20 / 241 Unions Unions (U) of two or more sets creates a set that includes everything in each set. A C Unions (U) are associated with "or." Examples: Shade in the areas! A U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} U C = {0, 2, 3, 4, 6, 7, 8, 9, 10, 12} (said " union C") Slide 21 / 241 Intersections ( ) of two or more sets indicates ONLY what is in OTH sets. Intersections ( ) are associated with "and." Example: Shade in the areas! A = {0, 3, 8} C = {3, 4, 2} (said " intersect C") Intersections A 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". 10 C

6 Slide 22 / 241 Unions and Intersections Slide 22 () / 241 Unions and Intersections Unions (U) and Intersections ( ) are often combined. Unions (U) 1. and (A Intersections U C) = {0, ( ) 2, are 3, 4, often 8} combined. Find: 1. (A U C) 2. A C A C 2. A C = {3} A 5 11 Find: (A U C) 3. (A C) U ( C) = {2, 3, 4, 6, 7} A COrder of operations applies. 0 2 C 3. (A C) U ( C) **Shade the diagram as you go to help (A C) U ( C) **Shade the diagram as you go to help Slide 23 / 241 Complements Slide 24 / 241 Examples One last aspect of sets for this unit are Complements. Complements of a set are all elements of the Universe that are NOT in the set. If U = {0, 1, 2, 3, 4, 5, 6} and A = {0, 1, 2, 3}, then the complement of A is {4, 5, 6} There are several ways to denote a complement: ~A, A c, A' and not A 1. If U = {all students in college} and A = {female students}, find ~A. 2. If U = {a traditional deck of cards} and = {Clubs and Diamonds, find ~. 3. If U = {the students at your school} and C = {students that like math}, find ~C. In this unit, we will use "~A" or "not A" Slide 24 () / 241 Examples Slide 25 / 241 Examples 1. If U = {all students in college} 1. ~A = and {male A = students} {female students}, find ~A. 2. ~ = {Spades and Hearts} 2. If U = {a traditional deck 3. of ~C cards} = {Students and = that {Clubs do not and Diamonds, find ~. like math} 3. If U = {the students at your school} and C = {students that like math}, find ~C. You can also combine Complements with Intersections and Unions. A C 5 Find: (A C) U ~ (A U ) ~C C U ~A 4. ~A U ~ **Shade the diagram as you go to help

7 Slide 25 () / 241 Examples You can also combine Complements with Intersections and Unions. A C 1. (A C) U ~ = {1, 5, 6, 7, 11} 5 Find: (A C) U ~ (A U ) ~C = {0, 1, 5, 8, 9, 10, 11, 1 12} 3 3. C U ~A = {2, 4, 9, 10, 12} 2. (A U ) ~C ~A U ~ = { } 3. C U ~A ~A U ~ **Shade the diagram as you go to help. Slide 26 / Find the complement of C or (~C). A {3, 5, 6, 10, 12} {3, 5, 6, 7, 9, 10, 12} C {1, 2, 3, 4, 5, 6, 8, 10, 12, 14} D {7, 9} U A C Slide 26 () / 241 Slide 27 / Find the complement of C or (~C). A {3, 5, 6, 10, 12} {3, 5, 6, 7, 9, 10, 12} C {1, 2, 3, 4, 5, 6, 8, 10, 12, 14} D {7, 9} 8 Find ~(A U U C) A {7, 9} {1, 8} C {1, 2, 4, 8, 12, 14} D {3, 5, 6, 11, 13, 15} U A C U A C Slide 27 () / 241 Slide 28 / Find ~(A U U C) A {7, 9} {1, 8} C {1, 2, 4, 8, 12, 14} D {3, 5, 6, 11, 13, 15} A 9 Find A U ~C A {3, 5, 6, 10, 12} {1, 2, 4, 8} C {1, 3, 5, 6, 8, 10, 12, 14} D {1, 3, 5, 6, 7, 8, 9, 10, 12, 14} U A C U A C

8 Slide 28 () / 241 Slide 29 / Find A U ~C A {3, 5, 6, 10, 12} {1, 2, 4, 8} C {1, 3, 5, 6, 8, 10, 12, 14} D {1, 3, 5, 6, 7, 8, 9, 10, 12, 14} D 10 Find ~ U A A C 45 D 63 U A C A U = The number of students in your grade A = the number of students that like English = the number of students that like Math Slide 29 () / 241 Slide 30 / Find ~ U A 11 Find ~(A ) A C 45 D 63 A C, it indicates the number of students that do not like math only. (They either like English only(12), math and English (18) or neither subject (15). A C 18 D 12 A U = The number of students in your grade A = the number of students that like English = the number of students that like Math U = The number of students in your grade A = the number of students that like English = the number of students that like Math Slide 30 () / 241 Slide 31 / Find ~(A ) A C 18 D 12 A A, it indicates the number of students who do not like OTH math and English Independence and Conditional Probability U = The number of students in your grade A = the number of students that like English = the number of students that like Math Return to Table of Contents

9 Slide 32 / 241 Goals and Objectives Slide 33 / 241 Why do we need this? Students will be able to verify that two events are independent or dependent and calculate the conditional probability of the events. As well, students will be able to translate their results using everyday language. Deciding things such as the cost of insurance can get very complicated. These decisions need to be based on many different elements. For example, who should pay more for health care: a person who smokes or a person who does not smoke? What about car insurance: a female driver, age 45, that drives a brand new Camaro or a 17 year old male driving a used Honda Civic? Slide 34 / 241 Independence and Conditional Probability Slide 35 / 241 Independence and Conditional Probability Independent events (or mutually exclusive events) are events whose outcomes are not affected by the other event. For example, the fact that a heads was thrown on a fair coin is not affected by the fact that a 6 of hearts was drawn out of a traditional deck of cards. Dependent events are events whose outcomes are affected by another event. Three sixes taken out of a deck of cards and not replaced directly affects the probability that you will draw another 6 next. You can also relate this to everyday situations: 1. Are you independent of, or dependent on, your parents and guardians right now? 2. True or false: Smoking causes lung cancer. Is this a dependent or an independent event? 3. Is how you do on a test based on how others study? Slide 35 () / 241 Independence and Conditional Probability You can also relate this to everyday situations: Discuss the different 1. Are you independent of, or dependent scenarios on, involved. your parents and guardians right now? 2. True or false: Smoking causes lung cancer. Is this a dependent or an independent event? 3. Is how you do on a test based on how others study? Teacher Notes Slide 36 / 241 Independence and Conditional Probability Since many of these situations are based on specific circumstances, we can use probability to study them. The 45 year old female driving a Camaro may have a terrible driving record. Therefore, what she pays for insurance will be dependent on her previous driving and she gets an extremely high rate. The probability she will have another accident is high. While not every smoker will get lung cancer, the probability that people get lung cancer if they smoke is very high. Probability allows us to make predictions! And, therefore, choices.

10 Slide 36 () / 241 Independence and Conditional Probability Since many of these situations are based on specific circumstances, we can use probability to study them. Discuss the different scenarios involved. The 45 year old female driving a Camaro may have a terrible driving record. Therefore, what she pays for insurance will be dependent on her previous driving and she gets an extremely high rate. The probability she will have another accident is high. Teacher Notes Slide 37 / When renting two cars, you decide to choose one of the blue cars. Your friend then chooses one of the red cars. Are these independent events? Yes No While not every smoker will get lung cancer, the probability that people get lung cancer if they smoke is very high. Probability allows us to make predictions! And, therefore, choices. Slide 37 () / When renting two cars, you decide to choose one of the blue cars. Your friend then chooses one of the red cars. Are these independent events? Yes No Yes. The fact that you chose blue does not have to affect that your friend chose red. Slide 38 / You choose to rent two cars. You choose the only blue car. Your friend chooses a red car. These are independent events. True False Slide 38 () / You choose to rent two cars. You choose the only blue car. Your friend chooses a red car. These are independent events. True False False. ecause you chose the only blue car, your friend cannot choose blue. Slide 39 / The probability that you will get lung cancer if you smoke is the same as the probability of you being a smoker if you have lung cancer. True False

11 Slide 39 () / The probability that you will get lung cancer if you smoke is the same as the probability of you being a smoker if you have lung cancer. True False False. You would have a 10 per cent chance of getting lung cancer by the age of 75 if you did not stop smoking by Lung cancer is the leading cause of cancer death among both men and women in the United States, and 90 percent of lung cancer deaths among men and approximately 80 percent of lung cancer deaths among women are due to smoking Find the probability of drawing a 6 of clubs followed by a 5 of hearts without replacement. Slide 40 / 241 Review of General Probability 2. Calculate the probability of throwing 3 heads in a row. 3. There are 20 marbles in a bag. 10 are blue, 4 are red and 6 are white. You draw out two marbles, one at a time, and do not replace them. Calculate the probability of drawing a red marble first, and then a white marble. Slide 40 () / 241 Review of General Probability Slide 41 / 241 Review of Mutually Exclusive Events and the Addition Law of Probability 1. Find the probability of drawing a 6 of clubs followed by a 5 of hearts without replacement. 2. Calculate the probability of throwing 3 heads in a row. 3. There are 20 marbles in a bag. 10 are blue, 4 are red and 6 are white. You draw out two marbles, one at a time, and do not replace them. Calculate the probability of drawing a red marble first, and then a white marble. **Remind students that probability can be a decimal, [This fraction object is or a percentage. pull tab] Mutually Exclusive events (or disjoint events) are two events that have no outcomes in common. For example, rolling a number on a number cube and drawing a card out of a deck are mutually exclusive. Mutually exclusive events A and satisfy P(A ) =. Slide 42 / 241 Independence and Conditional Probability Slide 43 / 241 Addition Law of Probability Drawing a 6 and drawing a red card from a traditional deck of cards are not mutually exclusive events because two of the 6's are red. These are not mutually exclusive and known as overlapping events. Overlapping events A and satisfy P(A ). Using the Addition Law of Probability: if two events are mutually exclusive, then P(A U ) = P(A) + P() if two events are overlapping, then P(A U ) = P(A) + P() - P(A )

12 Slide 44 / 241 Independence and Conditional Probability Slide 44 () / 241 Independence and Conditional Probability Mutually Exclusive Overlapping Mutually Exclusive Overlapping P(A U ) = P(A) + P() P(A U ) = P(A) + P() - P(A ) Find the probability that Find the probability that you you roll a 6 on a green draw a face card or a red number cube or a 3 on a card. red number cube. P(A U ) = P(A) + P() P(A U ) = P(A) + P() - P(A ) Find the probability that Find the probability that you you roll a 6 on a green draw a face card or a red number cube or a 3 on a card. red number cube. Slide 45 / A bag of 30 marbles has 9 black, 7 white, 6 yellow and the rest are green. What is the probability, in a percentage, that you will draw out a white or a yellow? A 20% 35% C 43% D 57% Slide 45 () / A bag of 30 marbles has 9 black, 7 white, 6 yellow and the rest are green. What is the probability, in a percentage, that you will draw out a white or a yellow? A 20% 35% C C 43% D 57% Slide 46 / You draw two cards out of a deck of cards. As a decimal, what is the probability that you draw an Ace or a 7? A C 0.50 D 0.65 Slide 46 () / You draw two cards out of a deck of cards. As a decimal, what is the probability that you draw an Ace or a 7? A C 0.50 D 0.65 A

13 Slide 47 / Using the Venn Diagram, how many people like to ski or ride snowmobiles? A C 69 D Slide 47 () / Using the Venn Diagram, how many people like to ski or ride snowmobiles? A C 69 D D Just add up all three numbers. Or, you could still use 44 15the formula: People that like to ski. People that like to ride snowmobiles. People that like to ski. People that like to ride snowmobiles. Slide 48 / 241 Slide 48 () / In your English class of 32 students, 7 of them play soccer and 10 run cross country. Of those same students, four play both soccer and run cross country. Find the probability that one of the students, chosen at random, plays soccer or runs cross country. A 12.5% 40.6% C 53.1% D 65.6% Slide 49 / Events A and are NOT mutually exclusive. P(A) = 0.3, P() = 0.45 and P(A ) is Find P(A U ). A C 0.63 D 0.75 Slide 49 () / Events A and are NOT mutually exclusive. P(A) = 0.3, P() = 0.45 and P(A ) is Find P(A U ). A C 0.63 D 0.75 C = 0.63

14 Slide 50 / 241 Conditional Probability Conditional Probability is the probability of an event (), given that another (A) has already occurred. The notation for conditional probability is P( A) or P( given A). To calculate conditional probability, use: Slide 50 () / 241 Conditional Probability Conditional Probability is the probability e sure students of an event (), given that another (A) has already understand occurred. to The divide notation by for conditional probability is P( A) the or P( probability given A). of the first event. Teacher Notes To calculate conditional probability, use: These events are only independent if: These events are only independent if: Slide 51 / 241 To calculate P( A), we use what is given, or P(A) P(), if the events are independent and P(A) P( A) if the events are dependent. Independent Two cards are drawn one at a time, and are replaced. What is the probability of drawing two Aces? Conditional Probability Dependent Two cards are drawn one at a time, and are not replaced. What is the probability of drawing two Aces? Slide 52 / 241 In Venn Diagrams, obviously P(A ) is the intersection of A and. Use numbers from the diagram for calculations. P(A) = 30% A Venn Diagrams 20% 10% 50% P() = 60% click click click click P(A ) **Add probabilities in all of A to get P(A) and all of to get P(). click click click click Slide 53 / 241 Example Slide 53 () / 241 Example A bag contains plastic disks with numbers on them. You must choose two disks, one at a time without replacing them. The probability that the first disk is odd and the second disk even is The probability that the first disk is odd is What is the probability of drawing an even number on the second draw given that the first disk was odd? A bag contains plastic disks with numbers on them. You must choose two disks, one at a time without replacing them. The probability that the first disk is odd and the second disk even is The probability that the first disk is odd is What is the probability of drawing an even number on the second draw given that the first disk was odd?

15 Slide 54 / 241 Example Slide 54 () / 241 Example Using the Venn Diagram, find the probability that a student is taking music given that they are taking math. Students that take music Students that take math Using the Venn Diagram, find the probability that a student is taking music given that they are taking math. Students that take music Students that take math Slide 55 / 241 Example Slide 55 () / 241 Example On any given day, the probability that it will rain and be windy is 18%. At the same time, the probability that it will just rain is 68%. Find the probability that it is windy, Given that it is raining. On any given day, the probability that it will rain and be windy is 18%. At the same time, the probability that it will just rain is 68%. Find the probability that it is windy, Given that it is raining. Slide 56 / 241 Formula Slide 57 / 241 Example with Formula To decide if the events in a conditional probability situation are independent, use the following formula: Use the formula to decide if these two events are independent. Students that take music Students that take math

16 Slide 57 () / 241 Example with Formula Slide 58 / 241 Example with Formula Use the formula to decide if these two events are independent. On the other hand, think about rolling dice. Use the formula to decide if rolling a 5, and then a 6 is independent. Students that take music Students that take math Slide 58 () / 241 Example with Formula On the other hand, think about rolling dice. Use the formula to decide if rolling a 5, and then a 6 is independent. Slide 59 / In Colorado, the probability that a person owns skis is 65% and the probability that they own skis and a snowboard is 25%. Find the probability that a person owns a snowboard given that they already own skis. A 25% 38% C 65% D 78% Slide 59 () / In Colorado, the probability that a person owns skis is 65% and the probability that they own skis and a snowboard is 25%. Find the probability that a person owns a snowboard given that they already own skis. A 25% 38% C 65% D 78% Slide 60 / These days, 96.7% of Americans own a TV and 25.4% of Americans own a TV and a laptop. Find the probability that an American owns a laptop given that they own a TV. A 96.7% 81.2% C 71.3% D 26.2%

17 Slide 60 () / These days, 96.7% of Americans own a TV and 25.4% of Americans own a TV and a laptop. Find the probability that an American owns a laptop given that they own a TV. A 96.7% 81.2% C 71.3% D 26.2% D Slide 61 / Given the Venn Diagram, what is the probability that a person enjoys both weightlifting and yoga? What is the correct notation indicating that preference? A 10%, P(A U ) 10%, P(A ) Weightlifting Yoga C 75%, P(A U ) D 75%, P(A ) 30% 10% 35% 25% Preferences of activities at a local gym. Slide 61 () / Given the Venn Diagram, what is the probability that a person enjoys both weightlifting and yoga? What is the correct notation indicating that preference? A 10%, P(A U ) 10%, P(A ) Weightlifting Yoga C 75%, P(A U ) D 75%, P(A ) Slide 62 / Calculate the percentage of people that like yoga, given that they enjoy weightlifting. A 10% 25% C 30% D 33% Weightlifting Yoga 30% 10% 35% 25% 30% 10% 35% 25% Preferences of activities at a local gym. Preferences of activities at a local gym. Slide 62 () / Calculate the percentage of people that like yoga, given that they enjoy weightlifting. A 10% 25% C 30% D 33% Weightlifting Yoga 30% 10% 35% 25% Slide 63 / What percentage of gym members asked about their preferences did not like either weightlifting or yoga? A 10% 25% C 30% D 65% Weightlifting Yoga 30% 10% 35% 25% Preferences of activities at a local gym. Preferences of activities at a local gym.

18 Slide 63 () / What percentage of gym members asked about their preferences did not like either weightlifting or yoga? A 10% 25% C 30% D 65% Weightlifting Yoga 30% 10% 35% 25% Slide 64 / At some schools, the probability that students like math is 30%. At those same schools, the students that like math and choose to go to MIT is 20%. What percentage of these students go to MIT given that they like math? A 10% 20% C 30% D 67% Preferences of activities at a local gym. Slide 64 () / 241 Slide 65 / At some schools, the probability that students like math is 30%. At those same schools, the students that like math and choose to go to MIT is 20%. What percentage of these students go to MIT given that they like math? A 10% 20% C 30% D 67% D Permutations & Combinations Return to Table of Contents Slide 66 / 241 Goals and Objectives Slide 67 / 241 Why do we need this? Students will be able to calculate the number of possible outcomes using the fundamental counting principle, permutation formula and combination formula. Also, students will be able to calculate the probability of an event occurring when the permutation and combination formulas are involved. Deciding things such as what you want on your sandwich when you place your order. Do you want your sandwich on wheat, rye, or white bread? Do you want ham, pepperoni, turkey, chicken, salami, or meatballs? What type of cheese would you like: Provolone, American, Swiss, or Mozzarella? What type of condiments do you want to be used: mustard, mayonnaise, ketchup, oil, or vinegar? Lab - Fundamental Counting Principle

19 Slide 68 / 241 Fundamental Counting Principle Fundamental Counting Principle: If event M can occur in m ways & is followed by event N that can occur in n ways, then the event M followed by the event N can occur in M N ways. - Ex: If a number cube is rolled & a coin is tossed, then there are 6 2, or 12 possible outcomes. Example: Slide 69 / 241 Fundamental Counting Principle A manager assigns different codes to all the tables in a restaurant to make it easier for the wait staff to identify them. Each code consists of a vowel, A, E, I, O or U, followed by 2 digits from 0 through 9. How many codes could the manager assign using this method? Example: Slide 69 () / 241 Fundamental Counting Principle A manager assigns different codes to all the tables in a restaurant to make it easier for the wait staff to identify them. Each code consists of a vowel, A, E, I, O or U, followed by 2 digits from 0 through 9. How many codes could the manager assign using this method? Slide 70 / A flea market vendor sells new & used books for adults & teens. Today, she has fantasy novels & poetry collections to choose from. Determine the number of categories for the books being sold. A x 10 x 10 = 500 codes C 4 D 2 Slide 70 () / A flea market vendor sells new & used books for adults & teens. Today, she has fantasy novels & poetry collections to choose from. Determine the number of categories for the books being sold. A 16 8 C 4 D 2 Slide 71 / At a restaurant, there are 10 beverages, 5 salad choices, 6 main courses, and 3 desserts. How many possible meals can be made? A C 300 D 900

20 Slide 71 () / At a restaurant, there are 10 beverages, 5 salad choices, 6 main courses, and 3 desserts. How many possible meals can be made? A C 300 D 900 D Slide 72 / A telephone number in a single area code is composed of 7 digits from 0 to 9. Determine the amount of phone numbers available in the 856 area code if the first digit cannot be 0 or 1. A 483, ,800 C 8,000,000 D 10,000,000 Slide 72 () / A telephone number in a single area code is composed of 7 digits from 0 to 9. Determine the amount of phone numbers available in the 856 area code if the first digit cannot be 0 or 1. A 483, ,800 C 8,000,000 D 10,000,000 C Slide 73 / In the state of New Jersey, random license plates are created by selecting 3 letters followed by 2 numbers 0 through 9 & 1 letter at the end. How many license plates are possible? A 45,697,600 37,015,056 C 32,292,000 D 6,760,000 Slide 73 () / In the state of New Jersey, random license plates are created by selecting 3 letters followed by 2 numbers 0 through 9 & 1 letter at the end. How many license plates are possible? A 45,697,600 Slide 74 / 241 Permutations Factorial: n! means the product of all counting numbers beginning w/ n & counting backwards to 1. 0! has a value of 1. Example: 4! = 4 x 3 x 2 x 1 = 24 37,015,056 C 32,292,000 D 6,760,000 A Permutation: an arrangement or listing of objects when no repetition is allowed and order matters. Example: AC and AC are different permutations of the letters A,, and C Formula for finding the number of permutations of n objects taken r at a time np r = n! (n - r)!

21 Slide 75 / 241 Permutations Slide 75 () / 241 Permutations Example: How many ways can you arrange the letters A,, C and D? Example: How many ways can you arrange the letters A,, C and D? 4 x 3 x 2 x 1 = 4! = 24 ways Example: Slide 76 / 241 Permutations There are 12 players on a softball team. In how many ways can the manager select 3 players for 1st base, 2nd base, and 3rd base? n = 12, r = 3 click 12P 3 = click 12! (12-3)! = 12! 9! 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 click Slide 77 / In how many ways can the letters in the word "WEIGHT" be arranged? A C 720 D 46, x 11 x 10 = 1,320 ways click Slide 77 () / In how many ways can the letters in the word "WEIGHT" be arranged? A C 720 D 46,656 C Slide 78 / How many different 4 letter arrangements can be formed from the letters in the word "DECAGON" A C 5,040 D 823,543

22 Slide 78 () / How many different 4 letter arrangements can be formed from the letters in the word "DECAGON" A 210 Slide 79 / There are 15 players on a basketball team. In how many ways can the coach select the 5 starting players? A C 5, ,360 C 12,454,041,600 D 823,543 D 1,307,674,368,000 Slide 79 () / There are 15 players on a basketball team. In how many ways can the coach select the 5 starting players? A 120 Slide 80 / A certain marathon had 50 people running. Prizes are awarded to the runners who finish in 1st, 2nd, and 3rd place. How many different possible outcomes are there for the first 3 runners to cross the finish line? 360,360 C 12,454,041,600 D 1,307,674,368,000 A 254,251,200 5,527,200 C 125,000 D 117,600 Slide 80 () / A certain marathon had 50 people running. Prizes are awarded to the runners who finish in 1st, 2nd, and 3rd place. How many different possible outcomes are there for the first 3 runners to cross the finish line? A 254,251,200 5,527,200 D Slide 81 / 241 Combinations Combination: an arrangement or listing of objects when no repetition is allowed and order does not matter. Example: AC and AC are the same combination of the letters A,, C, and D Example: AC and AD are different combinations of the letters A,, C, and D C 125,000 D 117,600 Formula for finding the number of combinations of n objects taken r at a time nc r = n! (n - r)! r!

23 Example: Slide 82 / 241 Combinations How many possible fruit salads can be made from 4 different kinds of fruit when you have 9 fruits to choose from? n = 9, r = 4 click 9C 4 = click 9! = 9! (9-4)! 4! 5! 4! 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 5 x 4 x 3 x 2 x 1 x 4 x 3 x 2 x 1 click 9 x 8 x 7 x 6 4 x 3 x 2 x 1 click = 126 fruit salads Slide 83 / 241 Permutation vs. Combination In a permutation, the order matters. In a combination, the order does not matter. Way to remember: "P" and "M" are really close in the alphabet (see underlined words above). "C" and "D" are really close in the alphabet (see underlined words above). Slide 84 / Determine if this question is asking for a permutation or a combination. How many 3 person committees are possible when selected from a pool of 10 people? A Permutation Combination Slide 84 () / Determine if this question is asking for a permutation or a combination. How many 3 person committees are possible when selected from a pool of 10 people? A Permutation Combination Slide 85 / How many possible 3 person committees are possible when selected from a pool of 10 people? Slide 85 () / How many possible 3 person committees are possible when selected from a pool of 10 people? A 45 A C C 120 C D 720 D 720

24 Slide 86 / Determine if this question is asking for a permutation or a combination. How many 4 person committees are possible when selected from a pool of 9 people consisting of a president, vice-president, secretary, and treasurer? A Permutation Combination Slide 86 () / Determine if this question is asking for a permutation or a combination. How many 4 person committees are possible when selected from a pool of 9 people consisting of a president, vice-president, secretary, and treasurer? A Permutation Combination A Slide 87 / How many 4 person committees are possible when selected from a pool of 9 people consisting of a president, vice-president, secretary, and treasurer? A 126 Slide 87 () / How many 4 person committees are possible when selected from a pool of 9 people consisting of a president, vice-president, secretary, and treasurer? A C 756 D 3,024 C 756 D 3,024 D Slide 88 / Determine if the question below is asking for a permutation or a combination: How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A Permutation Combination Slide 88 () / Determine if the question below is asking for a permutation or a combination: How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A Permutation Combination

25 Slide 89 / How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A 311,875,200 Slide 89 () / How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A 311,875, ,784,560 C 5,197, ,784,560 C 5,197,920 D D 2,598,960 D 2,598,960 Slide 90 / 241 Probability Involving Permutations & Combinations Some questions will ask you to calculate the probability of an event, or multiple events, that use the counting techniques of permutations & combinations. When this occurs, calculate the number of outcomes of your event(s) and sample space to create your probability fraction. Slide 91 / 241 Probability Involving Permutations & Combinations Example: Consider all of the 5-digit numbers that can be made with the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that the number is between 20,000 and 30,000. Is this event a permutation or a combination? Explain how you know. Permutation: with numbers, the order matters click How many outcomes are possible in the sample space? 5 x 4 x 3 x 2 x 1 = 5! = 120 outcomes click Slide 92 / 241 Probability Involving Permutations & Combinations Example: Consider all of the 5-digit numbers that can be made with the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that the number is between 20,000 and 30,000. Slide 93 / 241 Probability Involving Permutations & Combinations Example: When playing a game of poker, each player is dealt 5 cards from a standard deck of 52. A pair is when 2 cards are the same. What is the probability of getting a pair of Kings? How many outcomes are possible in the event? 4 x 3 x 2 x 1 = 4! = 24; 1st number can only be a 2, so the only last 4 digits can vary. click What is the probability that the number is between 20,000 & 30,000? 24/120 = 1/5 = 20% click Is this event a permutation or a combination? Explain how you know. Combination: the order in which you get the cards doesn't matter click How many hands of cards can be dealt (the sample space)? 52C 5 = click 52! = 2,598,960 hands (52-5)! 5!

26 Example: Slide 94 / 241 Probability Involving Permutations & Combinations When playing a game of poker, each player is dealt 5 cards from a standard deck of 52. A pair is when 2 cards are the same. What is the probability of getting a pair of Kings? How many outcomes are possible in the event? 4C 2 x 48C 3 = 6 x 17,296 = 103,776 There are 4 kings in a standard deck and you need 2 of them for the pair in your hand. For the remaining 48 cards, you can be dealt any 3 of them. click What is the probability that you are dealt the pair of Kings? 103,776/2,598,960 = 3.99% click Slide 95 / A committee of 3 students is to be chosen from a group of 6 students. Jason, Lily & Marlene are students in the group. What is the probability that all 3 of them will be chosen for the committee? A 1/120 1/60 C 1/20 D 1/10 Slide 95 () / A committee of 3 students is to be chosen from a group of 6 students. Jason, Lily & Marlene are students in the group. What is the probability that all 3 of them will be chosen for the committee? A 1/120 1/60 C 1/20 D 1/10 C Slide 96 / If the letters in the word DECAGON are arranged at random, find the probability that the first letter is a G. A 1/7 1/42 C 1/840 D 1/5040 Slide 96 () / If the letters in the word DECAGON are arranged at random, find the probability that the first letter is a G. A 1/7 1/42 C 1/840 D 1/5040 A Slide 97 / If a 3 digit number is formed from the numbers 1, 2, 3, 4, 5, 6, 7, and 8, with no repetitions, what is the probability that the number will be between 100 and 400? A 5/8 1/2 C 3/8 D 1/4

27 Slide 97 () / If a 3 digit number is formed from the numbers 1, 2, 3, 4, 5, 6, 7, and 8, with no repetitions, what is the probability that the number will be between 100 and 400? A 5/8 1/2 C 3/8 D 1/4 C Slide 98 / When playing a game of poker, each player is dealt 5 cards from a standard deck of 52. A three of a kind is when 3 cards are the same. What is the probability of getting dealt 3 Jacks? A 0.017% 0.17% C 1.7% D 17% Slide 98 () / 241 Slide 99 / When playing a game of poker, each player is dealt 5 cards from a standard deck of 52. A three of a kind is when 3 cards are the same. What is the probability of getting dealt 3 Jacks? A 0.017% 0.17% C 1.7% Measures of Central Tendency D 17% Return to Table of Contents Slide 100 / 241 Goals and Objectives Slide 101 / 241 Why do we need this? After reviewing mean, median, mode, range and outliers, students will be able to calculate Interquartile Range and Standard Deviation of two or more data sets. Data and how it is manipulated can be misused by the media. Consumers need to be able to interpret and understand the different ways to calculate tendencies. For example, having a mean average of 85% on an exam is very different than reporting a mode of 35%. Can these numbers appear for the same test? These are both ways to report measures of central tendency.

28 Slide 102 / 241 Review Slide 103 / 241 Review: Example Mean: the average of a set of numbers. Add up the numbers and divide by the number of numbers. Median: The number in the middle of the set of data when it is put in order. If two numbers are in the middle, take the average of those two numbers. Example: Find the mean, median and mode of the following set of test scores: 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 Mode: The number that appears most frequently in the data set. Slide 103 () / 241 Review: Example Slide 104 / 241 Review Let students try this on their own. Go Example: Find the mean, through median answers and and mode have of them the following set of test scores: check their work. Mean: 78, 79, 81, 45, 71, 71, 95, Median: 92, 95, 71, 85, 93, 78, 98, 96 45, 71, 71, 71, 78, 78, 79, 81, 85, 92, 93, 95, 95, 96, 98 Range: The difference between the highest and the lowest numbers in the set of data. Outliers: Numbers that are significantly larger or smaller than the rest of the numbers. Mode: 71 Slide 105 / 241 Review: Example Find the range and identify any outliers of the following test scores: 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 Slide 105 () / 241 Review: Example Find the range Range: and identify any = 53outliers of the following test scores: Outlier: 45 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 Show students what happens to the mean if 45 was not in the set of data: Have students attempt on their own. Go over results.

29 Slide 106 / 241 Review Slide 107 / 241 Interquartile Range The Spread of a set of data is used to describe the variability of the information. This looks at how different the numbers are. Interquartile Range is the difference of the value of quartile 3 and quartile 1. *We will review quartiles in the next slide. Standard Deviation is a measure of how close all of the data is to the mean. Remember making box plots in Algebra 1? lowest number Quartile 1 Median of lower half of data Quartile 2 Median of data Quartile 3 Median of upper half of data highest number Slide 107 () / 241 Interquartile Range Slide 108 / 241 Interquartile Range Remember making box plots in Algebra 1? Refer to statistics unit in Algebra 1 if students need more of highest a review. lowest number Teacher Notes Quartile 1 Median of lower half of data Quartile 2 Median of data number Quartile 3 Median of upper half of [This dataobject is a pull tab] Interquartile Range is the difference between Q2 and Q1, or Q3 - Q1. Find all three quartiles and calculate the interquartile range for the following test scores. 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 Slide 108 () / 241 Slide 109 / 241 Interquartile Range Interquartile Range is the difference between Q2 and Q1, or Q3 - Q1. 45, 71, 71, 71, 78, 78, 79, 81, 85, 92, 93, 95, 95, 96, 98 Q1 Q2 Q3 Find all three quartiles and calculate the interquartile range for the following test scores. Interquartile range: = 24 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96

30 Slide 110 / 241 Slide 110 () / 241 Slide 111 / 241 Standard Deviation Find the standard deviation (σ) for this set of test scores. Remember, we already found the mean % 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 Slide 111 () / 241 Standard Deviation Find the standard deviation (σ) = for (-36.87) this 2 = set of test scores. Remember, we already 71 found the = (-10.87) mean 2 = % 78, 79, 81, 45, 71, 71, 95, , = 95, (13.13) 71, 2 = 85, , 78, 98, = (-3.87) 2 = = (-2.87) 2 = = (-0.87) 2 = = (-10.87) 2 = = (10.13) 2 = = (13.13) 2 = = (-10.87) 2 = = (3.13) 2 = = (11.13) 2 = = (-3.87) 2 = = (16.13) 2 = = (14.13) [This object = is a pull tab] Slide 112 / 241 Standard Deviation Slide 112 () / 241 Standard Deviation What do you think would happen to the standard deviation if we eliminated the outlier of 45? What do you think would happen to the standard deviation if we eliminated the outlier of 45? Notice how the standard deviation went down.

31 Slide 113 / 241 Standard Deviation Find the standard deviation of the following set of numbers: 6.7, 7.1, 6.5, 7.2, 6.23, 6.9 Slide 113 () / 241 Standard Deviation Have students make a prediction of how big or small the standard Find the standard deviation of deviation the following will be. set of numbers: 6.7, 7.1, 6.5, 7.2, 6.23, = (-0.7) 2 = = (0.33) 2 = = (-0.27) 2 = = (0.43) 2 = = (-0.54) 2 = = (0.13) 2 = Slide 114 / 241 Standard Deviation Discuss the standard deviations of both sets that we just calculated. How do each reflect the spread of the data? Slide 114 () / 241 Standard Deviation Discuss the standard deviations of both sets that we just calculated. How do each reflect the spread of the data? 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, , 7.1, 6.5, 7.2, 6.23, 6.9 The smaller 6.7, the 7.1, standard 78, 79, 81, 45, 71, 71, 6.5, 7.2, 6.23, 6.9 deviation, the closer the data 95, 92, 95, 71, 85, 93, is to the mean. 78, 98, 96 Slide 115 / Find the Interquartile Range of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48 Slide 115 () / Find the Interquartile Range of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 36, 36, 49, 37, 48 48, 49, 50 Q1 Q2 Q3 IQR = = 22

32 Slide 116 / Find the Standard Deviation of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48 Slide 116 () / Find the Standard Deviation of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48 Slide 117 / Find the Interquartile Range for the following set of data: 1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6 Slide 117 () / Find the Interquartile Range for the following set of data: 1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, , 1.6, 2.3, 2.4, 2.6, 3.4, 4.6, Q1 2.5 Q2 4 Q3 IQR = = 2.05 Slide 118 / Find the Standard Deviation of the following set of data: Slide 118 () / Find the Standard Deviation of the following set of data: 1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, , 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6

33 Slide 119 / What does an IQR of 10 and a Standard Deviation of 2.1 say about a set of data? Assume that the numbers are in the 100's. A The spread is small. The spread is large Slide 119 () / What does an IQR of 10 and a Standard Deviation of 2.1 say about a set of data? Assume that the numbers are in the 100's. A The spread is small. The spread is large The spread would be small, meaning that the data values are fairly close together. Slide 120 / 241 Calculators When sets of information are very large, calculators can be very helpful. We will reference operations on a TI-84 for this exercise. Please refer to the manuals of other calculators for alternative directions. Slide 121 / 241 Calculators Input the following sets into your calculator: L 1: 1.3, 1.5, 1.7, 1.9, 0.9, 1.3, 1.4, 0.8, 2.1, 1.5, 1.6, 1.7, 1.4, 1.9, 1.3 L 2: 1.2, 5.7, 0.1, 7.9, 2.0, 0.2, 9.8, 2.1, 4.6, 9.2, 1.1, 4.6, 7.2, 6.4, 9.1 To find L 1 and L 2, go to STAT and then 1: Edit. Slide 122 / 241 Calculators Slide 122 () / 241 Calculators Now, calculate the Standard Deviation of each set. 1. Go to calculation screen ( 2nd, Quit ). 2. Push 2nd, Stat. 3. Go to Math. 4. Find 7: stddev(). 5. Type stddev(l 1) and then stddev(l 2). Now, calculate the Standard Deviation of each set. 1. Go to calculation screen Students ( 2nd should, Quit get ) for L 1 2. Push 2nd, Stat and for L 2. The next page 3. Go to Math. you will hold a discussion on the 4. Find 7: stddev(). difference. 5. Type stddev(l 1) and then stddev(l 2). Teacher Notes

34 Slide 123 / 241 Calculators Slide 123 () / 241 Calculators L 1: 1.3, 1.5, 1.7, 1.9, 0.9, 1.3, 1.4, 0.8, 2.1, 1.5, 1.6, 1.7, 1.4, 1.9, 1.3 L 2: 1.2, 5.7, 0.1, 7.9, 2.0, 0.2, 9.8, 2.1, 4.6, 9.2, 1.1, 4.6, 7.2, 6.4, 9.1 For L 1, the standard deviation is With L 2, the standard deviation is Why is there such a large difference between the two numbers? What does it say about the data? Discuss the differences of L 1: 1.3, 1.5, 1.7, 1.9, 0.9, 1.3, 1.4, 0.8, 2.1, 1.5, 1.6, 1.7, 1.4, 1.9, 1.3 Standard Deviations. L 2: 1.2, 5.7, 0.1, 7.9, 2.0, 0.2, Emphasize 9.8, 2.1, 4.6, that 9.2, the 1.1, closer 4.6, the 7.2, 6.4, 9.1 data is to zero, the closer the data is to each other. Therefore, the smaller the spread. Teacher Notes For L 1, the standard deviation is [This object With is a pull Ltab] 2, the standard deviation is Why is there such a large difference between the two numbers? What does it say about the data? Slide 124 / 241 Standard Deviation Slide 124 () / 241 Standard Deviation Standard Deviation (σ) is a number that represents how different the data is from the mean. The smaller the standard deviation, the closer the data is to the mean. The higher the standard deviation, the further the data is from the mean. Standard Deviation (σ) is a number that represents how different the data is from the mean. The smaller the standard deviation, the closer the data is to the Stress mean. that The standard higher deviation the standard is deviation, the further the in data the is same from unit the of mean. measure that the data is. Teacher Notes Slide 125 / Using a calculator, find the standard deviation of the following set of data. 13.4, 14.6, 17.2, 21.3, 14.5, 17.8, 13.3, 22, 16.7, 17.3, 17.6, 15, 21, 16.8, 19.2, 15.3 Slide 125 () / Using a calculator, find the standard deviation of the following set of data. 13.4, 14.6, 17.2, 21.3, 14.5, 17.8, 13.3, 22, 16.7, 17.3, 17.6, 15, 21, 16.8, 19.2, 15.3 σ = 2.72

35 Slide 126 / Find the standard deviation of the following set of data. Slide 126 () / Find the standard deviation of the following set of data. 12, 7, 14, 16, 25, 17, 8, 20, 20, 30, 12, 14, 17, 17, 25, 6, 6, 22, 25, 26, 24, 16, 17, 19 12, 7, 14, 16, 25, 17, 8, 20, 20, 30, 12, 14, 17, 17, 25, 6, 6, 22, 25, σ 26, = , 16, 17, 19 Slide 127 / Which set of numbers will have the smallest standard deviation? A 1.1, 3.6, 12.2, 4.5, 19.3, 2.3, 4.3, 5.6, 13.7, 12.1, , 2.3, 1.5, 1.6, 2.1, 2.1, 1.7, 2.4, 1.5, 1.6, 1.6, 2.0 C 1, 2, 7, 3, 9, 20, 4, 3, 12, 5, 6, 6, 5, 12, 13, 17, 20 Slide 127 () / Which set of numbers will have the smallest standard deviation? A 1.1, 3.6, 12.2, 4.5, 19.3, 2.3, 4.3, 5.6, 13.7, 12.1, , 2.3, 1.5, 1.6, 2.1, 2.1, 1.7, 2.4, 1.5, 1.6, 1.6, 2.0 C 1, 2, 7, 3, 9, 20, 4, 3, 12, 5, 6, 6, 5, 12, 13, 17, 20 Slide 128 / 241 Slide 129 / 241 Goals and Objectives Standard Deviation and Normal Distribution Students will be able to calculate the standard deviation of a data set and analyze a a normal distribution. Return to Table of Contents

36 Slide 130 / 241 Why do we need this? In short, the standard deviation of data represents how close the data is to its mean. It is used to report such things as results from political polls and data from medical experiments. We need to understand how these numbers are calculated to make informed decisions. Slide 131 / 241 Graphs Check out the following graphs. What do they have in common? Calories in French Fries Diastolic lood Pressure Intervals of Peaks of Heartbeats Slide 131 () / 241 Graphs Check out the following graphs. What do they have in common? Calories in French Fries Diastolic lood Pressure Slide 132 / 241 Normal Distribution Many different aspects of life, when measured and graphed, fit this type of distribution. Imagine a what the graph of height for humans, weight for bears or size of homes would look like. Most of the data would be around the same number (the mean), yet there would be some that would be larger or smaller. Finally, you would have the extremes that would be rare. This is called a Normal Distribution. Teacher Notes Intervals of Peaks of Heartbeats Discuss both the fact that they are all "curved" the same and that the graphs are of different aspects of life. Slide 133 / 241 Normal Distribution Normal Distributions are very useful when analyzing data. It allows you to calculate the probability that an event will happen as well as a percentile ranking of scores. Consider the following examples... Slide 134 / 241 Normal Distribution A tennis ball manufacturer measures the height their tennis balls bounce after dropping them from 5 feet off of the ground. The balls will not bounce the same height each time, but should be very close. A graph of this, after many trials, would begin to resemble a normal distribution. From here, you can calculate a mean height of the ball and use that to test other tennis balls from the factory to make sure that the quality is consistent. The blue shaded area would represent the range of acceptable heights.

37 Slide 135 / 241 A particular engineering school at a university prides itself on producing high quality engineers. Each class coming through has to take an introductory physics class. The professor uses a normal distribution to calculate grades such that only the top 5% of students get As. This ensures the course is challenging and that the best are the ones that continue on. *note: graph does not represent top 5% with As. Normal Distribution Slide 136 / 241 Normal Distribution Using the mean and standard deviation takes into account different spreads of the graph. In fact, knowing the standard deviation of a study can tell you how reliable the study is. Small standard deviations indicate that the mean is a good representation of the information. Large standard deviations tell you that the data was actually very spread out and the mean may not be reliable. Small σ = small spread Large σ = large spread Slide 137 / 241 Normal Curve Slide 137 () / 241 Normal Curve Normal Curves are created using the mean (μ or x) of the data and the standard deviation (σ). mean Normal Curves are created using the mean (μ or x) of the data and the standard deviation (σ). mean Take the time to point out the different parts of the graph. Teacher Notes Slide 138 / 241 Normal Distribution Much of the time, instead of having more complicated numbers for μ ± σ, we write 0, ±1 or ±2 representing the number of standard deviations from the mean as shown below. These numbers are also known as z-scores. mean Slide 138 () / 241 Normal Distribution Much of the time, instead of having more complicated numbers for μ ± σ, we write 0, ±1 or ±2 representing the number of standard deviations from the mean as shown below. These numbers are also known as z-scores. Teacher Notes mean Take the time to point out the different parts of the graph

38 Slide 139 / 241 Normal Distribution Slide 140 / 241 Normal Distribution In normal distributions, the area under the curve is what is used to calculate percentages or probabilities. These numbers follow what is called the Empirical Rule and is the same for each distribution. Mean 68% of all data will fall within 1 standard deviation of the mean. 95% of all data falls within 2 standard deviations of the mean. 99.7% of all data falls within 3 standard deviations of the mean. The graph on the next page is an excellent illustration of this. Slide 140 () / 241 Normal Distribution Slide 141 / 241 Normal Distribution Each graph can be used differently even though there is a uniformity about their calculations. Teacher Notes Point Mean out the different areas of the graph. Take into account the different notations and realize that students will struggle with the amount of information Slide 142 / 241 Chart for Examples Slide 142 () / 241 Chart for Examples Use this chart to answer the following questions. Use this chart to answer the following questions. Teacher Notes This chart is particularly useful because it is divided into half standard deviations. It would be a typical graph given on a standardized test if a calculator was not allowed. It is also easy to memorize.

39 Slide 143 / 241 Examples Slide 143 () / 241 Examples a) John usually scores an average of 82% on his math tests with a standard deviation of 5%. What is the probability that John will get an between an 82% and an 87% on his next test? b) At ig Mama's Gym, there is a special weight loss program that is a big hit. And, it works! At the start of the program 95.4% of all members, centered about the mean, weighed between 180 and 260 pounds. Find the average weight and the standard deviation of the data. a) John usually a) scores 87% - 82% an = average 5% which of is 82% 1σ. The on his math tests with a area between 82% (the mean) and standard deviation 87%(1σ) of 5%. is 34.1%. What Therefore, is the probability he has that John will get an between an a 82% 34.1% and chance 87% of getting on his between next an test? 82% and an 87% on his next test. b) 95.4% tells you that there is a b) At ig Mama's spread Gym, of 2 standard there is deviations a special from weight loss the mean. The mean weight is (180 + program that is a big hit. And, it works! At the start 260)/2 = 220 pounds. The standard of the program deviation 95.4% is of (260 all members, - 220)/2 = 20 centered about the mean, pounds. weighed between 180 and 260 pounds. Find the average weight and the standard deviation of the data. Slide 144 / 241 Example Slide 144 () / 241 Example c) A machine at Superfoods Food Factory puts a mean of 44 oz of mayonnaise in their bottles. The machine has a standard of deviation of 0.5 ounces. While filling 1000 bottles with mayonnaise, about how many times will the machine fill a bottle with 45 or more ounces? c) A machine at Superfoods Food c) With Factory a standard puts a deviation mean of 44 oz of mayonnaise in their bottles. The of 0.5 machine ounces, has 45 a ounces standard is of deviation of 0.5 ounces. While 2 filling standard 1000 deviations bottles with away mayonnaise, about how many times will the from machine the mean fill a bottle of 44. with Add 45 or more ounces? 1.7% + 0.5% + 0.1% to get 2.3%. Find 2.3% of It will fill approximately 23 bottles with 45 or more ounces. Slide 145 / 241 Example d) Scores on the final exam in Mr. Dahlberg's Precalculus classes are normally distributed. He calculates a mean to be 71% with a standard deviation of 7. What is the probability that a student in his classes will get between an 85 and a 92 on the final exam? Slide 145 () / 241 Example d) Scores on the final exam in Mr. Dahlberg's Precalculus classes are normally distributed. He calculates a mean to be 71% with a standard deviation of 7. What is the probability that a student in his classes will get between an and and a are on the between final exam? 2 and 3 standard deviations away from the mean. Add together 1.7% and 0.5% to get a probability of 2.2%.

40 Slide 146 / attery lifetime is normally distributed for large samples. The mean lifetime is 500 days and the standard deviation is 61 days. To the nearest percent, what percent of batteries have lifetimes longer than 561 days? Slide 146 () / attery lifetime is normally distributed for large samples. The mean lifetime is 500 days and the standard deviation is 61 days. To the nearest percent, what percent of batteries have lifetimes longer than 561 days? 15.9% or 16% Slide 147 / A normal distribution of a group the ages of 340 students has a mean age of 15.4 years with a standard deviation of 0.6 years. How many students are younger than 16 years? Express your answer to the nearest student. Slide 147 () / A normal distribution of a group the ages of 340 students has a mean age of 15.4 years with a standard deviation of 0.6 years. How many students are younger than 16 years? Express your answer to the nearest student x 340 = students Slide 148 / 241 Slide 148 () / Which of the following curves represents a mean of 85 and a standard deviation of 6? A C D Which of the following curves represents a mean of 85 and a standard deviation of 6? A C D A

41 Slide 149 / Given a mean of 27 and a standard deviation of 3 on a data set that is normally distributed, what is the number that is +2 from the mean? Slide 149 () / Given a mean of 27 and a standard deviation of 3 on a data set that is normally distributed, what is the number that is +2 from the mean? 33 Slide 150 / Given a mean of 27 and a standard deviation of 3 on a data set that is normally distributed, what is the number that is -3 from the mean? Slide 150 () / Given a mean of 27 and a standard deviation of 3 on a data set that is normally distributed, what is the number that is -3 from the mean? 18 Slide 151 / A set of information collected by the Department of Wildlife is normally distributed with a mean of 270 and a standard deviation of 12. What percent of the data falls between 246 and 258? Slide 151 () / A set of information collected by the Department of Wildlife is normally distributed with a mean of 270 and a standard deviation of 12. What percent of the data falls between 246 and 258? 13.6%

42 The graph we have been using to the right helps us find values that are multiples of 0.5 away from the mean. ut what about numbers that are in between? For those, we use a formula for the z-score and a table of values. Slide 152 / 241 Z-Score z-score = Slide 153 / 241 Z-Score A table of z-scores is shown on the next 2 slides. Each score is associated with the amount of area under the normal curve from the score to the left. Slide 154 / 241 Z-Scores: Negative Slide 155 / 241 Z-Scores: Positive Slide 156 / 241 Z-Score Slide 157 / 241 Z-Score Z-scores are what is used to calculate all of the percentile values that are reported for standardized tests. Remember how you are given a result of, say, the 94th percentile? This means that you have done better than 94% of the students who have taken the test. Welcome to a major use of z- scores, normal distribution and standard deviation! Example: On a test, your score was 83%. The mean of all of the tests was 79, the data was normally distributed and the standard deviation was Find your z-score and then use the table to calculate the percentile. z-score = z-score =

43 Slide 157 () / 241 Z-Score Example: On a test, your score was 83%. The mean of all of the tests was 79, the data was normally distributed and the standard deviation was Find your z-score and then use the table to calculate the percentile. From the table, 0.94 is z-score = associated with That means your score was in the 82nd percentile. Slide 158 / 241 Z-Score Your friend took the same test and got a score of 92%. Find your friend's z-score and calculate their percentile. Slide 158 () / 241 Z-Score Your friend took the same test and got a score of 92%. Find your friend's z-score and calculate their percentile. Slide 159 / Find the z-score for a 29 if the mean was 34 and the standard deviation is 2.3. From the table, 3.06 is associated with That means your friend's score was in the 99th percentile. Slide 159 () / Find the z-score for a 29 if the mean was 34 and the standard deviation is Slide 160 / Which is the z-score and percent of area under the curve for a score of 520 in a normally distributed set of data with a mean of 565 and a standard deviation of A -1.86, 1.95% -1.86, 3.14% C 1.86, 31.4% D 1.86, 97.5%

44 Slide 160 () / Which is the z-score and percent of area under the curve for a score of 520 in a normally distributed set of data with a mean of 565 and a standard deviation of A -1.86, 1.95% -1.86, 3.14% C 1.86, 31.4% D 1.86, 97.5% Slide 161 / A value has a z-score of The mean for the data is 73 and the standard deviation is What was the original value? Slide 161 () / A value has a z-score of The mean for the data is 73 and the standard deviation is What was the original value? Slide 162 / A student calculated a z-score of What percentile does this score fall in? Solving for x = Slide 162 () / A student calculated a z-score of What percentile does this score fall in? Slide 163 / Find the z-score of 10 if the data set is: The percentage associated with is 10.56%. Therefore, the score is in the 10th percentile.

45 Slide 163 () / 241 Slide 164 / Find the z-score of 10 if the data set is: z-score = 0.48 Two-Way Frequency Tables Return to Table of Contents Slide 165 / 241 Goals and Objectives Slide 166 / 241 Why do we need this? Students will be able to recognize trends with and interpret different association of data in a two-way frequency table. All of us are marketed to on a regular basis. Television, the Internet and magazines are different ways that businesses get us to buy their product or use their service. It is vital to be able to interpret information that is given to us and make smart choices. Slide 167 / 241 Remember from Algebra 1... Slide 167 () / 241 Remember from Algebra 1... Frequency Table Stem-and-Leaf Plot ox-and-whisker Plot Frequency Table Stem-and-Leaf Plot ox-and-whisker Plot llll llll llll l llll llll lll llll lll llll l lll Ages of people at the gym Stem Leaf Ages of Professors at a College *These are all ways to display a collection of data. llll llll llll l llll llll lll llll lll llll l lll Ages of people at the gym Teacher Notes Stem Discuss each type of graph. If Leaf students need review, see the statistics unit in Algebra Ages of Professors at a College *These are all ways to display a collection of data.

46 Slide 168 / 241 Slide 168 () / 241 Slide 169 / 241 Remember from Algebra 1... Slide 169 () / 241 Remember from Algebra 1... Line Plots Scatter Plots Line Plots Teacher Notes Discuss each type of graph. If students need review, see the statistics unit in Scatter Algebra Plots1. Slide 170 / 241 Two-Way Frequency Tables Slide 171 / 241 Two-Way Frequency Tables In this section, we are going to study Two-Way Frequency Tables. These displays allow us to study situations that have more than one variable such as how many men and women that exercise regularly. The chart below shows a survey of 100 people. Two-Way Frequency Tables connect the collection of data with probability. Using these tables, we can calculate three different frequencies that are very useful when discussing results: 1. Joint Relative Frequency 2. Marginal Relative Frequency 3. Conditional Relative Frequency

47 Slide 172 / 241 Two-Way Frequency Tables The yellow boxes represent Joint Relative Frequency and the pink boxes represent Marginal Relative Frequency. Slide 172 () / 241 Two-Way Frequency Tables The yellow boxes represent Joint Relative Frequency and the pink boxes represent Marginal Relative Frequency. Joint Relative Frequency is found by dividing the number in that category by the total observations or outcomes. Marginal Relative Frequency is found by totaling the rows and columns. Teacher Notes Joint Relative Frequency Show is found students by dividing this chart the number if they in that category by the total observations need a visual or outcomes. representation. Marginal Relative Frequency is found by totaling the rows and columns. Slide 173 / 241 Two-Way Frequency Tables Slide 173 () / 241 Two-Way Frequency Tables These relative frequencies directly translate into quantitative statements. Such statements mirror those that are reported in the media. Teacher Notes These relative frequencies Stress directly that translate it more into meaningful quantitative statements. Such statements to state mirror that those "18% that of are men reported in the media. exercised" verses "18 men exercised." 18% of the men surveyed exercise regularly. 22% of the women surveyed did not exercise regularly. 54% of the people surveyed were women. 18% of the men surveyed exercise regularly. 22% of the women surveyed did not exercise regularly. 54% of the people surveyed were women. Slide 174 / 241 Two-Way Frequency Tables Slide 174 () / 241 Two-Way Frequency Tables A teacher asked their class if they had been to an amusement park before or not. Out of 36 students, there were 22 boys and 14 girls. 16 of the boys and 10 of the girls answered that they had been to an amusement park before. Create a relative frequency table from the data collected. A teacher asked their class if they had been to an amusement park before or not. Out of 36 students, there were 22 boys and 14 girls. 16 of the boys and 10 of the girls answered that they had been to an amusement park before. Create a relative frequency table from the data collected.

48 Slide 175 / 241 Two-Way Frequency Tables Together, write some quantitative statements about the information. Slide 175 () / 241 Two-Way Frequency Tables Together, write some quantitative statements about the information. 44% of the boys have been to an amusement park. 28% of the girls have been to an amusement park. 17% of the boys have not been to an amusement park. 11% of the girls have not been to an amusement park. 72% of the students have been to an amusement park. 28% of the students have not been to an amusement park. 61% of the students were boys. 39% of the students [This object were is a pull girls. tab] Slide 176 / 241 Two-Way Frequency Tables Slide 176 () / 241 Two-Way Frequency Tables At a vet clinic over the month of July, the vets saw a total of 150 animals. Out of those animals, 105 were dogs and 45 were cats. 70 of the dogs that were seen needed blood work. 20 of the cats needed blood work. Create a relative frequency table for the information. You will use your table to answer some questions. At a vet clinic over the month of July, the vets saw a total of 150 animals. Out of those animals, 105 were dogs and 45 were cats. 70 of the dogs that were seen needed blood work. 20 of the cats needed blood work. Create a relative frequency table for the information. You will use your table to answer some questions. Slide 177 / From the relative frequency table you created, find the joint relative frequency for the dogs that did not need blood work. Slide 177 () / From the relative frequency table you created, find the joint relative frequency for the dogs that did not need blood work or 23%

49 Slide 178 / What is the marginal relative frequency of cats that came to the clinic? Slide 178 () / What is the marginal relative frequency of cats that came to the clinic? 0.3 Slide 179 / What is the percentage of dogs that came in that needed blood work? Slide 179 () / What is the percentage of dogs that came in that needed blood work? 47% Slide 180 / Find the marginal relative frequency for the number of animals which came in and needed blood work? Slide 180 () / Find the marginal relative frequency for the number of animals which came in and needed blood work? 0.72

50 Slide 181 / 241 Two-Way Frequency Tables From these frequencies, you can also find a useful comparison called Conditional Relative Frequency which is directly correlated to Conditional Probability. To find Conditional Relative Frequency, divide the joint relative frequency by the appropriate marginal relative frequency. Slide 182 / 241 Conditional Relative Frequency and Conditional Probability Conditional Relative Frequency and Conditional Probability go hand in hand. In fact how statistics are reported usually involves some probability. For example, use the table to find the probability that if a cat was brought in to the clinic, it would not need blood work. Cats that did not need blood work. Cats that came in % Slide 183 / 241 Two-Way Frequency Tables Using the table, find the probability that if a pet was brought into the clinic that needed blood work, it would be a dog. Slide 183 () / 241 Two-Way Frequency Tables Using the table, find the probability that if a pet was brought into the clinic 0.47that needed blood work, it would be a dog = 78% This means that there is a 78% chance that your dog would need blood work if you brought it in. Slide 184 / 241 Two-Way Frequency Tables Slide 184 () / 241 Two-Way Frequency Tables Using the table, find the probability that if you brought in a cat, it would NOT need blood work? Using the table, find the probability that 0.17if you brought in a cat, it would NOT need blood 0.40 work? = 43%

51 Slide 185 / From the table, find the probability that a girl has gone to an amusement park. Slide 185 () / From the table, find the probability that a girl has gone to an amusement park. 28% Slide 186 / Find the conditional probability that out of the girls, the person has been to an amusement park. Slide 186 () / Find the conditional probability that out of the girls, the person has been to an amusement park = 72% Slide 187 / What is the probability that if a person has been to an amusement park, it was a boy? Slide 187 () / What is the probability that if a person has been to an amusement park, it was a boy? 61%

52 Slide 188 / Find the probability that out of the people that have not gone to an amusement park, it would be a girl. Slide 188 () / Find the probability that out of the people that have not gone to an amusement park, it would be a girl. 39% Slide 189 / 241 Two-Way Frequency Tables Information summarized like this can easily be analyzed when studying certain situations. Slide 189 () / 241 Two-Way Frequency Tables Information summarized like this can easily be analyzed when studying certain situations. At the same vet clinic during July, 42 of the same dogs that came in needed an x-ray. 10 of the cats needed an x-ray. Create a frequency table that dispays this information. Find joint and marginal relative frequencies. At the same vet clinic during July, 42 of the same dogs that came in needed an x-ray. 10 of the cats needed an x-ray. Create a frequency table that dispays this information. Find joint and marginal relative frequencies. Slide 190 / 241 Two-Way Frequency Tables Slide 190 () / 241 Two-Way Frequency Tables Find the probability that: a) if you brought in a dog, it would need an x-ray, b) if you brought in a cat, it would need an x-ray. Find the probability that: a) if you brought in a dog, Dog it would and x-ray need = an 0.28 x-ray, b) if you brought in a cat, it would need an x-ray = 40% Cat and x-ray = 0.07 = 23% 0.30

53 Slide 191 / 241 Two-Way Frequency Tables Slide 191 () / 241 Two-Way Frequency Tables Out of all of the animals x-rayed, calculate the percentages that were a) dogs and b) cats. X-rays that were dogs = X-rays that were cats = = 80% = 20% Out of all of the animals x-rayed, calculate the percentages that were a) dogs and b) cats. Slide 192 / 241 Two-Way Frequency Tables Using the information from both tables, what trends can you find in the data? Use quantitative statements to justify your answers. Slide 192 () / 241 Two-Way Frequency Tables Using the information from both tables, what trends can you find in the data? One Use possible quantitative example: statements During to justify your answers. July, dogs need more medical attention. 80% of the x-rays taken and 78% of the blood work was from dogs. Slide 193 / 241 Two-Way Frequency Tables At USA High School, 300 seniors went on to a 4-year college or university. A survey collected the following data on whether they chose an in-state or an out-of-state school. Use this information to answer the following questions. Slide 194 / ased on the data, which of the following is a plausible quantitative statement? A 58% of the students that chose an in-state college or university are female. 56% of the students that chose an out-of-state college or university are female. C 73% of females chose an in-state college or university.

54 Slide 194 () / ased on the data, which of the following is a plausible quantitative statement? A 58% of the students that chose an in-state college or university are female. A 56% of the students that chose an out-of-state college or university are female. C 73% of females chose an in-state college or university. Slide 195 / ased on the data, which of the following would be a plausible quantitative statement from the information displayed below? A 27% of the females surveyed chose an out-of-state college or university. 45% of the females surveyed chose an out-of-state college or university. C 18% of the females surveyed chose an out-of-state college or university. Slide 195 () / ased on the data, which of the following would be a plausible quantitative statement from the information displayed A 27% of below? the females surveyed chose an out-of-state college or university. C 45% of the females surveyed chose an out-of-state college or university. C 18% of the females surveyed chose an out-of-state college or university. Slide 196 / The marginal relative frequency of in-state students is: A C 0.45 D 0.22 Slide 196 () / The marginal relative frequency of in-state students is: A C 0.45 D 0.22 Slide 197 / The joint relative frequency that a female would choose an out-of-state college or university is: A C 0.22 D 0.10

55 Slide 197 () / The joint relative frequency that a female would choose an out-of-state college or university is: A D C 0.22 D 0.10 Slide 198 / 241 Sampling and Experiments Return to Table of Contents Slide 199 / 241 Goals and Objectives Students will be able to recognize appropriate uses and models for statistics, justify their results using data or experimentation, and calculate a margin of error for sets of information. Slide 200 / 241 Why do we need this? Everyone needs to learn appropriate ways to interpret statistical analyses. Just because someone comes up with a survey and publicizes their results, does not mean that the survey has validity. In today's society, we need to have educated opinions and to question what we are told in the media. Slide 201 / 241 Sampling Slide 202 / 241 Sampling Sampling is a method of getting information about a large population without having to test or ask each element of the population. How many of you have gotten a phone call requesting that you answer survey questions? Such sampling allows the company or agency to get an idea of what people think or, especially, how they will vote. y choosing a certain number of elements to be a sample, you can efficiently gather results and make a quantitative statement about the entire population. This method is used in many different situations. Some examples include: a) quality control in a parts factory or in food production, b) experimentation with different medical treatments, and c) predicting who or what people vote for.

56 Slide 203 / 241 Sampling There are several aspects of sampling that deserve attention: 1) randomization and bias, 2) sample size, and 3) margin of error. Slide 204 / Which of the following samples would most accurately represent the way people would vote on lowering the drinking age to 18? A Polling 100 random students at all college campuses. Asking 10 mothers at a Mother's Against Drunk Driving meeting. C Phoning 1000 random households between 10 am and 1 pm. D Phoning 10,000 random households between 5 pm and 9 pm. Slide 204 () / Which of the following samples would most accurately represent the way people would vote on lowering the drinking age to 18? A Polling 100 random students at all college campuses. Asking 10 mothers D at a Mother's Against Drunk Driving meeting. Discuss why the other answers C Phoning 1000 random would not households represent the between feelings 10 am and 1 pm. of the entire population. D Phoning 10,000 random households between 5 pm and 9 pm. Slide 205 / Which of the following samples would most accurately represent what the most popular clothing stores are in the U.S.? A Polling 2000 shoppers in a mall. As they come off of the course, asking 2000 golfers at popular clubs where they like to shop for clothes. C Ask 2000 random females between the ages of 12 and 1 D Question 2000 fishermen at a fishing convention. Slide 205 () / Which of the following samples would most accurately represent what the most popular clothing stores are in the U.S.? A Polling 2000 shoppers in a mall. As they come off of the course, asking 2000 golfers at popular clubs where they like to shop for clothes. C Ask 2000 random females between the ages of 12 and 1 D Question 2000 Depending fishermen on at the a goal fishing of the convention. survey, all answers may be viable. Discuss what company would value each sample. Slide 206 / 241 Sampling As in the last examples, different samples may get different results. Knowing the purpose of the sampling is very important. If the sample size is too small, if it is not randomized or if the method of obtaining samples is not well thought out, you will get biased results.

57 Slide 207 / 241 Sampling - ias ias comes from how a question is asked as well as who is being asked. Surveys or statistics that are biased do not return valid results. For example, if you ask men at an electrician's convention which purse they prefer, would you get valid answers? It is important for questions or surveys to be unbiased. That way, the results mean something. Slide 208 / 241 Sampling - Sample Size Once a sampling method has been well thought out and proven not to be biased, one must consider sample size. As a rule, small sample sizes will result in a large variation while larger sample sizes result in less variation. Slide 209 / 241 Sampling - Example Slide 209 () / 241 Sampling - Example For example: You are a quality control manager at an ice cream factory. Out of the 4,000 gallons of ice cream they produce in one day, you choose 4 of those gallons to pull off of the line to check for quality. a) Is this enough? b) If not, decide on a range of values that would be sufficient. For example: You are a quality control manager at an ice cream factory. Out of the 4,000 gallons of ice cream they produce in one day, you As choose a class, 4 of decide those what gallons would to pull off of the line to check for quality. be a reasonable amount to pull off of the line to test quality. a) Is this enough? Pose the question about what b) If not, decide on a range "sufficient" of values means. that would be sufficient. Teacher Notes Slide 210 / 241 Margin of Error One way businesses and organizations calculate the answer to "what is sufficient" is to decide on the margin of error that they want to be within. We have all seen margins of error reported in polls. Although, they are usually an add-on at the end. Slide 211 / 241 Margin of Error What this means is that Obama actually had a range of votes from 47.8% to 56.2% and Romney had a range that was from 36.8% to 45.2%. If the numbers were looked at a bit differently, it could be a much closer race and lead to reports such as this: ** **

58 Slide 212 / 241 Margin of Error Slide 213 / 241 Margin of Error **The margin of error represents an interval that would contain the true population parameter and usually has a 95% confidence level which is two standard deviations. In its simplest form, we can use the margin of error to calculate a sample size as well as use the sample size to calculate the margin of error. This is generally used for surveys that are going to be conducted in the future. At the ice cream factory, calculate the margin of error if you used a sample of 4 gallons out of the To do this, use the formula: M = margin of error n = sample size What is the margin of error if you used a sample of 400? Slide 213 () / 241 Margin of Error Slide 214 / 241 Margin of Error 4 gallon sample: 400 gallon sample: At the ice cream factory, calculate the margin of error if you used a sample of 4 gallons out of the Likewise, you can use a desired margin of error to find your sample size. Say that you wanted to have a margin of error of 3% when testing quality of the ice cream. Calculate the sample size: What is the margin of error if you used a sample of 400? Is your answer reasonable in this situation? Slide 214 () / 241 Margin of Error Slide 215 / What is the margin of error for a sample size of 30? Likewise, you can use a desired margin of error to find your sample size. Say that you wanted to have a margin of error of 3% when testing quality of the ice cream. Calculate the sample size: 1 Is your answer reasonable in this situation? UT..is this reasonable?

59 Slide 215 () / What is the margin of error for a sample size of 30? ±18% Slide 216 / What is the actual range on a survey that reported 24% of the population smoked with a margin of error of 3.2%? A 20% - 27% 20.8% % C 3.2% - 24% D 24% % Slide 216 () / What is the actual range on a survey that reported 24% of the population smoked with a margin of error of 3.2%? A 20% - 27% 20.8% % C 3.2% - 24% D 24% % Slide 217 / In a survey of 25 people, 4 of those surveyed has locked their keys in their car. Find the margin of error and the interval of the true population parameter. A 5%, 20% to 30% 15%, 1% to 31% C 20%, 0% to 36% D 25%, 0% to 50% Slide 217 () / In a survey of 25 people, 4 of those surveyed has locked their keys in their car. Find the margin of error and the interval of the true population parameter. A 5%, 20% to 30% 15%, 1% to 31% C 20%, 0% to 36% D 25%, 0% to 50% C To find the percentage of people that locked their keys in the car take 4/25 = 16%. M = ±20% and the interval cannot go below 0%. Slide 218 / Find the sample size needed to achieve a margin of error of ±1%.

60 Slide 218 () / Find the sample size needed to achieve a margin of error of ±1%. 10,000 Slide 219 / Quality control reported that 2% of the toy cars being manufactured in March of 2010 were defective. If their margin of error was ±0.25% and they manufactured 5000 toy cars, what is the largest number of cars that could be defective (with a 95% confidence level)? Slide 219 () / Quality control reported that 2% of the toy cars being manufactured in March of 2010 were defective. If their margin of error was ±0.25% and they manufactured 5000 toy cars, what is the largest 112 cars number of cars that could be defective (with a 95% confidence level)? The interval of confidence is 1.75% to 2.25%. Therefore, that 2.25% of 5,000, which is 112. Slide 220 / 241 Margin of Error The margin of error calculated by the formula is a very simplified, general method. It will give you the largest possible margin of error and is a good estimate of the numbers you are looking for, but is not as accurate as it could be. The following formulas are used to calculate margin of error a bit more accurately. We will use only the one above and the second below. p = proportion in a decimal n = sample size σ = standard deviation n = sample size Slide 221 / 241 Margin of Error Slide 221 () / 241 Margin of Error Margins of error can also be calculated via simulation models for random sampling. Someone has made a claim that 45% of the students at USA High School have Smart Phones. A student from that school took a survey during one of her classes and found that out of an English class of 30 students, 11 students had a Smart Phone, which is approximately 37% of her class. Can this original claim be true? Margins of error can also be calculated via simulation models for random sampling. Lead students in a short discussion. They should come Someone has made a claim to that the 45% conclusion of the students that, yes, at the USA High School have Smart Phones. claim A student could be from true, that but school the took a survey during one of her classes sample and size found was that too small out of to an English class of 30 students, 11 students guarantee had a it. Smart Phone, which is approximately 37% of her class. Can this original claim be true? Teacher Notes

61 Slide 222 / 241 Margin of Error Since the sample size was too small to support the claim, we can use a simulation model to find a margin of error. Then, if the original claim falls inside of the confidence interval, we can support that claim. Slide 223 / 241 Margin of Error Margins of error can also be used to decide if a results of particular experiment are relevant. For example, if 45% of a sample population voted for Jane Doe with a margin of error of ±3%, you could predict that a second or third survey would return results that are in the confidence interval of 42% to 48%. To generalize, we can make a claim (or hypothesis) about a particular event by taking a survey and computing results. From those results, we can make further claims that can be proven or disproved based on the results falling within the original confidence interval. If the expectation a particular hypothesis does not fall within the interval, the hypothesis could be rejected. Slide 224 / 241 Sampling and Experiments Lab Slide 224 () / 241 Sampling and Experiments Lab Simulations are used if studies are not cost effective, tests are not ethical or if the calculating the probability is too difficult. Let's perform a simulation now. Click on the lab link below to get started. Lab - Sampling and Experiments Simulations are used if studies are not cost effective, tests are not ethical or if the calculating the probability is too difficult. Let's perform a simulation now. Click on the lab link below to get started. The next 4 slides can be used Lab - Sampling to complete and Experiments the lab as a class. If the students are working on this in small groups and the slides are not needed, click the link in the bottom left corner to skip over the teacher slides. Teacher Notes Slide 225 / 241 Sampling and Experiments Lab Lab: Teacher Slides - Part 1: Flipping a Coin Take out a coin or get one from your teacher. Everyone flip the coin 10 times and record whether you get heads or tails. Write this information on the board in the table below. Name(s) Heads tally Tails tally Slide 226 / 241 Sampling and Experiments Lab Mini-Lab: Teacher Slides - Part 1: Flipping a Coin How many heads did the class get? How many tails? What is your experimental probability for each? What is the longest streak of heads or tails?

62 Slide 227 / 241 Sampling and Experiments Lab Mini-Lab: Teacher Slides - Part 2: Simulation of Flipping a Coin using a Graphing Calculator The most efficient way of doing simulations is with a calculator or a computer. Let's "flip a coin" again, but use our calculator. 1. Make the following selections on your calculator. Math PR randint(beginning value, ending value, how many times) To flip the coin 10 times, use randint(0, 1, 10). Enter this into your calculator & press "Enter". 2. If you want to store it in your calculator, To view the list that you stored, press Stat Enter your list into the table on the Lab WS. Slide 229 / 241 Sto 2nd L 1 Edit Calculator Simulations Go back to one of our earlier problems... Someone has made a claim that 45% of the students at USA High School have Smart Phones. A student from that school took a survey during one of her classes and found that, out of an English class of 30 students, 11 students had a Smart Phone, which is approximately 37% of her class. Mathematically prove that this original claim is true. A sample size of 30 is way too small to make a decision, so let's use simulations to develop a mean and a margin of error for this problem. Get out your calculator again. Slide 228 / 241 Sampling and Experiments Lab Mini-Lab: Teacher Slides - Part 2: Simulation of Flipping a Coin using a Graphing Calculator 3. Quit out of your List by pressing 2nd Quit 4. To flip the coin 10 times, use randint(0, 1, 10). To flip it another 10 times hit 2nd Enter. 5. Write down the results that you found for 2nd round of flipping the coins in the space below. Slide 230 / 241 Calculator Simulations Since 37% of the students in the class had a Smart Phone, assign the numbers 1 to 37 as students having a Smart Phone. Therefore, 38 to 100 will represent students not having a Smart Phone. In your calculator, do randint(1, 100, 100). Store it in L 1. Stat Calc 1-Var Stats Enter x = mean σx = standard deviation Q 1 = 1st Quartile Med = Median Q 3 = 3rd Quartile Compare everyone's mean! Slide 231 / 241 Calculator Simulations Slide 231 () / 241 Calculator Simulations Our calculator gives us enough information that we can easily calculate a more accurate margin of error. From your data, find the confidence interval. Use: Confidence Interval = mean ± Margin of error. For example, the data from the last question generated a mean of 53.5 and a standard deviation of There were 100 random integers generated, so n = 100. Calculate the Confidence interval. Our calculator gives us enough information that we can easily calculate a more accurate margin of error. From your data, find the confidence interval. Use: Confidence Interval = mean ± Margin of error. For example, the data from the last question generated a mean of 53.5 and a standard deviation of There [This object were is a pull 100 tab] random integers generated, so n = 100. Calculate the Confidence interval.

63 Slide 232 / 241 Calculator Simulations Slide 232 () / 241 Calculator Simulations Now, this is just one simulation. Does it match what everyone in the class got? To be more accurate, you repeat your simulation several times and generate a mean of the means and a mean of the standard deviation. Our class just repeated the simulation several times as we all have random numbers that were generated. Let's calculate a class mean and class standard deviation to then find a more accurate interval. Now, this is just one simulation. Does it match what everyone in the class got? To be more accurate, you repeat your simulation several times and generate a mean of the means and a mean of the standard deviation. Our class Use just the repeated numbers the that simulation the class several times as we all have random generated numbers as that a whole. were generated. s Let's calculate a class mean and will class vary. standard deviation to then find a more accurate interval. Teacher Notes Slide 233 / 241 Calculator Simulations Slide 233 () / 241 Calculator Simulations Finally, does the claim that 45% of student fall within your class's margin of error? If so, it can be verified. If not, the claim may not be valid. Finally, does the claim that 45% of student fall within your class's margin of error? If so, it can be verified. If not, the claim may not be valid. This seems like a long, confusing process to many students. Make sure you take it slowly. Teacher Notes Slide 234 / 241 Calculator Simulations Try it again... A mom's local MADD (Mother's Against Drunk Driving) group stated that more than 30% of today's students have drank and drove in the last month. A group of students believe that number is overblown and way too high. In school, a teacher surveyed her class and found out that 5% of her students anonymously admitted to drinking and driving in the last month. Create and execute a simulation 10 times to prove or disprove MADD's claim. Steps are on the next page... Slide 234 () / 241 Calculator Simulations Try it again... A mom's local MADD (Mother's Against Drunk Driving) group stated that more than 30% of today's students have drank and drove in the last month. A group of students believe that number is overblown and way too high. Have students work on this one individually. s will In school, a teacher surveyed vary. her It class is the and process found that out that is 5% of her students anonymously admitted important. to drinking and driving in the last month. Teacher Notes Create and execute a simulation 10 times to prove or disprove MADD's claim. Steps are on the next page...

64 Slide 235 / 241 Calculator Simulations Slide 236 / 241 Calculator Simulations 1. Decide on parameters and input into calculator. randint(beginning value, ending value, how many times) Store in L Find mean and σ. Record. 3. Repeat 9 more times. 4. Find the average mean and average standard deviation. 5. Use to find the margin of error. This method is also useful in determining if there is a true difference between claims such as in treatments or products. To do this, use the difference in claims, develop a mean and a margin of error from simulations and then decide if the difference of Zero (0) falls within your confidence interval. What does a difference of 0 signify? 6. Generate the confidence interval. 7. Determine if the amount in question falls in your interval. Slide 236 () / 241 Calculator Simulations This method is also useful in determining if there is a true difference between claims such as in treatments or products. To do this, use the difference in claims, develop a mean and a margin of error from simulations Zero indicates and then that decide there if the is no difference of Zero (0) falls within significant your difference confidence between interval. treatments, therefore, neither treatment is better than the other. What does a difference of 0 signify? For example... Slide 237 / 241 Calculator Simulations Two different different dog foods claim to make a dog's teeth and gums healthier. Dog Food A claims that 25% of the dog's tested had better dental health and Dog Food claimed that 34% of the dogs had better dental health. Currently, Dog Food is advertising itself as the top brand of food. Create and execute a simulation to prove or disprove the claim. Slide 237 () / 241 Calculator Simulations For example... Two different different s dog foods will claim vary. to Since make a dog's teeth and gums healthier. Dog students Food are A claims using that 25% of the dog's tested had better dental calculators, health and have Dog them Food claimed that 34% of the dogs had better increase dental the health. amount Currently, of Dog Food is advertising itself as numbers the top randomly brand of food. generated Create and execute a simulation to to prove 300, 500 or disprove and 700. the What claim. happens to the mean and standard deviation? Teacher Notes Slide 238 / 241 Calculator Simulations This same type of simulation can be used to estimate how long it will take to collect a certain amount of objects. For example, Johnny Jim's, a sandwich shop is giving out 5 different cards every time someone orders a Macho Sandwich. Once you collect all 5 cards, you get a free sandwich. Create and execute a simulation that gives you an interval for how many sandwiches you should expect to buy before getting a free one.