Algebra II. Slide 1 / 241. Slide 2 / 241. Slide 3 / 241. Probability and Statistics. Table of Contents click on the topic to go to that section

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1 Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics Table of Contents click on the topic to go to that section Slide 3 / 241 Sets Independence and Conditional Probability Permutations & Combinations Measures of Central Tendency Standard Deviation and Normal Distribution Two-Way Frequency Tables Sampling and Experiments

2 Slide 4 / 241 Sets Return to Table of Contents Slide 5 / 241 Goals and Objectives Students will be able to use characteristics of problems, including unions, intersections and complement, to describe events with appropriate set notation and Venn Diagrams. Why do we need this? Slide 6 / 241 Being able to categorize and describe situations allows us to analyze problems that we are presented with in their most basic forms. Many different fields need to categorize elements they use or study. Businesses need to look at what they are offering, Biologists need to organize material they are studying and even you will need to categorize different options for your living situation, such as insurance, in the future.

3 Vocabulary and Set Notation Slide 7 / 241 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} Create a Venn Diagram to match the information. Slide 8 / 241 U 7 A B A = {0, 2, 3, 7, 9} B = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Create a Venn Diagram to match the information. Slide 8 () / 241 U A A = {0, 2, 3, 7, 9} B = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 8 Teacher Notes 6 2 B Move the circles and numbers around to 9 mirror the given 10information

4 Data Displays 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: Chart for rolling 2 dice (sums): H H T H T H T T H T H T H T Data Displays Slide 10 / 241 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. Data Displays Slide 10 () / 241 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.

5 The Universe Slide 11 / 241 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 Slide 11 () / 241 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. Empty Set Slide 12 / 241 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. An outcome is a result of an experiment or survey.

6 Example Slide 13 / 241 U A B C List the universe for this problem. 2. Name the different sets involved. 3. Find the subset that is in both A and B. 4. Find the subset that is in all sets A, B and C. Example Slide 13 () / 241 U A B U = {-12, -3, -2, -1, 0, 1, 1 43, 4, 5, 6, 7, 15, 17} 2. A = {-3, -2, 1, 5} B = {-3, 4, 5, 6} C = {0, 1, 4, 5, 15} -1 *3. A B = {-3, 15} 0 *4. A B 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 B. 4. Find the subset that is in all sets A, B and C. 7 1 What is most likely the Universe of the following situation? A U = {men} B U = {women} C U = {people} D U = {people at a fitness club} E U = {people exercising at home} Slide 14 / 241 Men Women 5pm cycling 4pm water aerobics 7pm weight lifting 3pm nutrition 6am aerobics 10am weight lifting 6pm swimming 2pm climbing

7 1 What is most likely the Universe of the following situation? A U = {men} B U = {women} C U = {people} D U = {people at a fitness club} E U = {people exercising at home} Slide 14 () / 241 Men 5pm cycling 4pm water aerobics 7pm weight lifting 3pm nutrition Women 6am aerobics D 10am weight lifting 6pm swimming 2pm climbing 2 What is the most popular activity, or activities, at the club? * as many letters as necessary. Men A 6 am aerobics B 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 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 B 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 15 () / 241

8 3 What are the most popular activities for both men and women at the club? Men Women A 5 pm cycling B 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 B 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 16 () / What is the best display for the sample space (or universe) of rolling an odd number on a single number cube? Slide 17 / 241 A S = {1, 2, 3, 4, 5, 6} B D # C S = {1, 3, 5} E

9 5 What does the following set represent? {3, 6, 7} Slide 18 / 241 A Set A B Elements common to A and B C Elements common to A and C D The Universal set A E A subset of set A C B What does the following set represent? {3, 6, 7} Slide 18 () / 241 A Set A B Elements common to A and B C Elements common to A and C D The Universal set E A subset of set A A 5 C C B 6 There are no elements of C that are not common to either set A or B, meaning that the set of numbers belonging to ONLY set C is { }. Slide 19 / 241 True False A C B

10 6 There are no elements of C that are not common to either set A or B, meaning that the set of numbers belonging to ONLY set C is { }. True False A True C Slide 19 () / 241 B Unions Slide 20 / 241 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 B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B U C = {0, 2, 3, 4, 6, 7, 8, 9, 10, 12} (said "B union C") B Intersections Slide 21 / 241 Intersections ( ) of two or more sets indicates ONLY what is in BOTH sets. Intersections ( ) are associated with "and." Example: Shade in the areas! A B = {0, 3, 8} B C = {3, 4, 2} (said "B intersect C") A B C 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".

11 Unions and Intersections Slide 22 / 241 Unions (U) and Intersections ( ) are often combined. Find: 1. (A U C) B 2. A B C A C 3. (A C) U (B C) **Shade the diagram as you go to help. B Unions and Intersections Slide 22 () / 241 Unions (U) 1. and (A Intersections U C) B = {0, ( ) 2, are 3, 4, often 8} combined. 2. A B C = {3} A 5 11 Find: (A U C) 3. B (A C) U (B C) = {2, 3, 4, 6, 7} A B COrder of operations applies. 0 2 C 3. (A C) U (B C) **Shade the diagram as you go to help. B Complements Slide 23 / 241 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 In this unit, we will use "~A" or "not A"

12 Examples Slide 24 / If U = {all students in college} and A = {female students}, find ~A. 2. If U = {a traditional deck of cards} and B = {Clubs and Diamonds, find ~B. 3. If U = {the students at your school} and C = {students that like math}, find ~C. Examples Slide 24 () / If U = {all students in college} 1. ~A = and {male A = students} {female students}, find ~A. 2. ~B = {Spades and Hearts} 2. If U = {a traditional deck 3. of ~C cards} = {Students and B = that {Clubs do not and Diamonds, find ~B. like math} 3. If U = {the students at your school} and C = {students that like math}, find ~C. Examples Slide 25 / 241 You can also combine Complements with Intersections and Unions. A C 5 Find: (A C) U ~B (A U B) ~C C B U ~A 4. ~A U ~B **Shade the diagram as you go to help. B

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

14 8 Find ~(A U B U C) A {7, 9} B {1, 8} C {1, 2, 4, 8, 12, 14} D {3, 5, 6, 11, 13, 15} Slide 27 / 241 U A B C Find ~(A U B U C) A {7, 9} B {1, 8} C {1, 2, 4, 8, 12, 14} D {3, 5, 6, 11, 13, 15} A Slide 27 () / 241 U A B C Find A U ~C A {3, 5, 6, 10, 12} B {1, 2, 4, 8} C {1, 3, 5, 6, 8, 10, 12, 14} D {1, 3, 5, 6, 7, 8, 9, 10, 12, 14} Slide 28 / 241 U A B C 7 9

15 9 Find A U ~C A {3, 5, 6, 10, 12} B {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 Slide 28 () / 241 U A B C Find ~B U A A 12 B 27 C 45 D 63 Slide 29 / 241 A B U = The number of students in your grade A = the number of students that like English B = the number of students that like Math 10 Find ~B U A Slide 29 () / 241 A 12 B 27 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 B(15) U = The number of students in your grade A = the number of students that like English B = the number of students that like Math

16 11 Find ~(A B) Slide 30 / 241 A 45 B 30 C 18 D 12 A B U = The number of students in your grade A = the number of students that like English B = the number of students that like Math 11 Find ~(A B) Slide 30 () / 241 A 45 B 30 C 18 D 12 A A, it indicates the number of students who do not like BOTH math and English. B U = The number of students in your grade A = the number of students that like English B = the number of students that like Math Slide 31 / 241 Independence and Conditional Probability Return to Table of Contents

17 Goals and Objectives Slide 32 / 241 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. Why do we need this? Slide 33 / 241 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? Independence and Conditional Probability Slide 34 / 241 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.

18 Independence and Conditional Probability Slide 35 / 241 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? Independence and Conditional Probability Slide 35 () / 241 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 Independence and Conditional Probability Slide 36 / 241 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.

19 Independence and Conditional Probability Slide 36 () / 241 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. While not every smoker will get lung cancer, the probability that people get lung cancer if they smoke is very high. Teacher Notes Probability allows us to make predictions! And, therefore, choices. 12 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 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 37 () / 241

20 13 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. Because you chose the only blue car, your friend cannot choose blue. Slide 38 () / 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 Slide 39 / 241

21 14 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. Slide 39 () / 241 Review of General Probability Slide 40 / 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. Review of General Probability Slide 40 () / 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]

22 Review of Mutually Exclusive Events and the Addition Law of Probability Slide 41 / 241 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 B satisfy P(A B) =. Independence and Conditional Probability Slide 42 / 241 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 B satisfy P(A B). Addition Law of Probability Slide 43 / 241 Using the Addition Law of Probability: if two events are mutually exclusive, then P(A U B) = P(A) + P(B) if two events are overlapping, then P(A U B) = P(A) + P(B) - P(A B)

23 Independence and Conditional Probability Slide 44 / 241 Mutually Exclusive Overlapping P(A U B) = P(A) + P(B) P(A U B) = P(A) + P(B) - P(A B) 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. Independence and Conditional Probability Slide 44 () / 241 Mutually Exclusive Overlapping P(A U B) = P(A) + P(B) P(A U B) = P(A) + P(B) - P(A B) 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. 15 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% B 35% C 43% D 57% Slide 45 / 241

24 15 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% B 35% C C 43% D 57% Slide 45 () / 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 0.15 B 0.20 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 0.15 B 0.20 C 0.50 D 0.65 A Slide 46 () / 241

25 17 Using the Venn Diagram, how many people like to ski or ride snowmobiles? A 15 B 45 C 69 D Slide 47 / 241 People that like to ski. People that like to ride snowmobiles. 17 Using the Venn Diagram, how many people like to ski or ride snowmobiles? A 15 B 45 C 69 D D Just add up all three numbers. Or, you could still use 44 15the formula: Slide 47 () / 241 People that like to ski. People that like to ride snowmobiles. 18 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% B 40.6% C 53.1% D 65.6% Slide 48 / 241

26 Slide 48 () / Events A and B are NOT mutually exclusive. P(A) = 0.3, P(B) = 0.45 and P(A B) is Find P(A U B). A 0.18 B 0.33 C 0.63 D 0.75 Slide 49 / Events A and B are NOT mutually exclusive. P(A) = 0.3, P(B) = 0.45 and P(A B) is Find P(A U B). A 0.18 B 0.33 C 0.63 D 0.75 C = 0.63 Slide 49 () / 241

27 Conditional Probability Slide 50 / 241 Conditional Probability is the probability of an event (B), given that another (A) has already occurred. The notation for conditional probability is P(B A) or P(B given A). To calculate conditional probability, use: These events are only independent if: Conditional Probability Slide 50 () / 241 Conditional Probability is the probability Be sure students of an event (B), given that another (A) has already understand occurred. to The divide notation by for conditional probability is P(B A) the or P(B probability given A). of the first event. Teacher Notes To calculate conditional probability, use: These events are only independent if: To calculate P(B A), we use what is given, or P(A) P(B), if the events are independent and P(A) P(B 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 51 / 241 click click click click click click click click

28 Venn Diagrams Slide 52 / 241 In Venn Diagrams, obviously P(A B) is the intersection of A and B. Use numbers from the diagram for calculations. P(A) = 30% A 20% 10% 50% B P(B) = 60% P(A B) **Add probabilities in all of A to get P(A) and all of B to get P(B). Example Slide 53 / 241 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? Example Slide 53 () / 241 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?

29 Example Slide 54 / 241 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 Example Slide 54 () / 241 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 Example Slide 55 / 241 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.

30 Example Slide 55 () / 241 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. Formula Slide 56 / 241 To decide if the events in a conditional probability situation are independent, use the following formula: Example with Formula Slide 57 / 241 Use the formula to decide if these two events are independent. Students that take music Students that take math

31 Example with Formula Slide 57 () / 241 Use the formula to decide if these two events are independent. Students that take music Students that take math Example with Formula Slide 58 / 241 On the other hand, think about rolling dice. Use the formula to decide if rolling a 5, and then a 6 is independent. Example with Formula Slide 58 () / 241 On the other hand, think about rolling dice. Use the formula to decide if rolling a 5, and then a 6 is independent.

32 20 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. Slide 59 / 241 A 25% B 38% C 65% D 78% 20 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. Slide 59 () / 241 A 25% B 38% C 65% D 78% B 21 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% B 81.2% C 71.3% D 26.2% Slide 60 / 241

33 21 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% B 81.2% C 71.3% D 26.2% D Slide 60 () / 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 B) B 10%, P(A B) Weightlifting Yoga C 75%, P(A U B) D 75%, P(A B) Slide 61 / % 10% 35% 25% Preferences of activities at a local gym. 22 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 B) B 10%, P(A B) Weightlifting Yoga C 75%, P(A U B) B D 75%, P(A B) Slide 61 () / % 10% 35% 25% Preferences of activities at a local gym.

34 23 Calculate the percentage of people that like yoga, given that they enjoy weightlifting. A 10% B 25% C 30% D 33% Weightlifting Yoga 30% 10% 35% 25% Slide 62 / 241 Preferences of activities at a local gym. 23 Calculate the percentage of people that like yoga, given that they enjoy weightlifting. A 10% B 25% C 30% D 33% Weightlifting Yoga B 30% 10% 35% 25% Slide 62 () / 241 Preferences of activities at a local gym. 24 What percentage of gym members asked about their preferences did not like either weightlifting or yoga? A 10% B 25% C 30% D 65% Weightlifting Yoga 30% 10% 35% 25% Slide 63 / 241 Preferences of activities at a local gym.

35 24 What percentage of gym members asked about their preferences did not like either weightlifting or yoga? A 10% B 25% C 30% D 65% Weightlifting Yoga B 30% 10% 35% 25% Slide 63 () / 241 Preferences of activities at a local gym. 25 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? Slide 64 / 241 A 10% B 20% C 30% D 67% 25 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? Slide 64 () / 241 A 10% B 20% C 30% D 67% D

36 Slide 65 / 241 Permutations & Combinations Return to Table of Contents Goals and Objectives Slide 66 / 241 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. Why do we need this? Slide 67 / 241 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

37 Fundamental Counting Principle Slide 68 / 241 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. Fundamental Counting Principle Slide 69 / 241 Example: 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? Fundamental Counting Principle Slide 69 () / 241 Example: 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? 5 x 10 x 10 = 500 codes

38 26 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 Slide 70 / 241 B 8 C 4 D 2 26 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 B 8 C 4 B Slide 70 () / 241 D 2 27 At a restaurant, there are 10 beverages, 5 salad choices, 6 main courses, and 3 desserts. How many possible meals can be made? A 90 Slide 71 / 241 B 180 C 300 D 900

39 27 At a restaurant, there are 10 beverages, 5 salad choices, 6 main courses, and 3 desserts. How many possible meals can be made? A 90 Slide 71 () / 241 B 180 C 300 D 900 D 28 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 483,840 Slide 72 / 241 B 604,800 C 8,000,000 D 10,000, 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 483,840 Slide 72 () / 241 B 604,800 C 8,000,000 D 10,000,000 C

40 29 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 73 / 241 B 37,015,056 C 32,292,000 D 6,760, 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 73 () / 241 B 37,015,056 C 32,292,000 D 6,760,000 A Permutations Slide 74 / 241 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 Permutation: an arrangement or listing of objects when no repetition is allowed and order matters. Example: ABC and ACB are different permutations of the letters A, B, and C Formula for finding the number of permutations of n objects taken r at a time np r = n! (n - r)!

41 Permutations Slide 75 / 241 Example: How many ways can you arrange the letters A, B, C and D? Permutations Slide 75 () / 241 Example: How many ways can you arrange the letters A, B, C and D? 4 x 3 x 2 x 1 = 4! = 24 ways Permutations Slide 76 / 241 Example: 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 12 x 11 x 10 = 1,320 ways click

42 30 In how many ways can the letters in the word "WEIGHT" be arranged? A 60 Slide 77 / 241 B 120 C 720 D 46, In how many ways can the letters in the word "WEIGHT" be arranged? A 60 Slide 77 () / 241 B 120 C 720 D 46,656 C 31 How many different 4 letter arrangements can be formed from the letters in the word "DECAGON" Slide 78 / 241 A 210 B 840 C 5,040 D 823,543

43 31 How many different 4 letter arrangements can be formed from the letters in the word "DECAGON" Slide 78 () / 241 A 210 B 840 C 5,040 D 823,543 B 32 There are 15 players on a basketball team. In how many ways can the coach select the 5 starting players? Slide 79 / 241 A 120 B 360,360 C 12,454,041,600 D 1,307,674,368, There are 15 players on a basketball team. In how many ways can the coach select the 5 starting players? Slide 79 () / 241 A 120 B 360,360 C 12,454,041,600 D 1,307,674,368,000 B

44 33 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? Slide 80 / 241 A 254,251,200 B 5,527,200 C 125,000 D 117, 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? Slide 80 () / 241 A 254,251,200 B 5,527,200 C 125,000 D 117,600 D Combinations Slide 81 / 241 Combination: an arrangement or listing of objects when no repetition is allowed and order does not matter. Example: ABC and ACB are the same combination of the letters A, B, C, and D Example: ABC and ADB are different combinations of the letters A, B, C, and D Formula for finding the number of combinations of n objects taken r at a time nc r = n! (n - r)! r!

45 Combinations Slide 82 / 241 Example: 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 Permutation vs. Combination Slide 83 / 241 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). 34 Determine if this question is asking for a permutation or a combination. Slide 84 / 241 How many 3 person committees are possible when selected from a pool of 10 people? A Permutation B Combination

46 34 Determine if this question is asking for a permutation or a combination. Slide 84 () / 241 How many 3 person committees are possible when selected from a pool of 10 people? A Permutation B Combination B 35 How many possible 3 person committees are possible when selected from a pool of 10 people? Slide 85 / 241 A 45 B 90 C 120 D How many possible 3 person committees are possible when selected from a pool of 10 people? Slide 85 () / 241 A 45 B 90 C 120 D 720 C

47 36 Determine if this question is asking for a permutation or a combination. Slide 86 / 241 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 B Combination 36 Determine if this question is asking for a permutation or a combination. Slide 86 () / 241 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 B Combination A 37 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 / 241 B 504 C 756 D 3,024

48 37 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 () / 241 B 504 C 756 D 3,024 D 38 Determine if the question below is asking for a permutation or a combination: Slide 88 / 241 How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A Permutation B Combination 38 Determine if the question below is asking for a permutation or a combination: Slide 88 () / 241 How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A Permutation B Combination B

49 39 How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? Slide 89 / 241 A 311,875,200 B 133,784,560 C 5,197,920 D 2,598, How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? Slide 89 () / 241 A 311,875,200 B 133,784,560 C 5,197,920 D 2,598,960 D Probability Involving Permutations & Combinations Slide 90 / 241 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.

50 Probability Involving Permutations & Combinations Example: Slide 91 / 241 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 Probability Involving Permutations & Combinations Example: Slide 92 / 241 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. 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 Probability Involving Permutations & Combinations Example: Slide 93 / 241 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? 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!

51 Probability Involving Permutations & Combinations Example: Slide 94 / 241 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 40 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 Slide 95 / 241 B 1/60 C 1/20 D 1/10 40 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 Slide 95 () / 241 B 1/60 C 1/20 D 1/10 C

52 41 If the letters in the word DECAGON are arranged at random, find the probability that the first letter is a G. Slide 96 / 241 A 1/7 B 1/42 C 1/840 D 1/ If the letters in the word DECAGON are arranged at random, find the probability that the first letter is a G. Slide 96 () / 241 A 1/7 B 1/42 C 1/840 D 1/5040 A 42 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? Slide 97 / 241 A 5/8 B 1/2 C 3/8 D 1/4

53 42 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? Slide 97 () / 241 A 5/8 B 1/2 C 3/8 D 1/4 C 43 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? Slide 98 / 241 A 0.017% B 0.17% C 1.7% D 17% 43 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? Slide 98 () / 241 A 0.017% B 0.17% C 1.7% D 17% B

54 Slide 99 / 241 Measures of Central Tendency Return to Table of Contents Goals and Objectives Slide 100 / 241 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. Why do we need this? Slide 101 / 241 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.

55 Review Slide 102 / 241 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. Mode: The number that appears most frequently in the data set. Review: Example Slide 103 / 241 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 Review: Example Slide 103 () / 241 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 Mode: 71

56 Review Slide 104 / 241 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. Review: Example Slide 105 / 241 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 Review: Example Slide 105 () / 241 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.

57 Review Slide 106 / 241 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. Interquartile Range Slide 107 / 241 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 Interquartile Range Slide 107 () / 241 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 2 Median of data number Quartile 1 Median of lower half of data Quartile 3 Median of upper half of [This dataobject is a pull tab]

58 Interquartile Range Slide 108 / 241 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 Interquartile Range Slide 108 () / 241 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 Slide 109 / 241

59 Slide 110 / 241 Slide 110 () / 241 Standard Deviation Slide 111 / 241 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

60 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 111 () / 241 Standard Deviation Slide 112 / 241 What do you think would happen to the standard deviation if we eliminated the outlier of 45? Standard Deviation Slide 112 () / 241 What do you think would happen to the standard deviation if we eliminated the outlier of 45? Notice how the standard deviation went down.

61 Standard Deviation Slide 113 / 241 Find the standard deviation of the following set of numbers: 6.7, 7.1, 6.5, 7.2, 6.23, 6.9 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 113 () / 241 Standard Deviation Slide 114 / 241 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

62 Standard Deviation Slide 114 () / 241 Discuss the standard deviations of both sets that we just calculated. How do each reflect the spread of the data? 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, Find the Interquartile Range of the following set of numbers: Slide 115 / , 37, 50, 22, 25, 26, 36, 36, 49, Find the Interquartile Range of the following set of numbers: Slide 115 () / , 37, 50, 22, 25, 26, 36, 36, 36, 36, 49, 37, 48 48, 49, 50 Q1 Q2 Q3 IQR = = 22

63 45 Find the Standard Deviation of the following set of numbers: Slide 116 / , 37, 50, 22, 25, 26, 36, 36, 49, Find the Standard Deviation of the following set of numbers: Slide 116 () / , 37, 50, 22, 25, 26, 36, 36, 49, Find the Interquartile Range for the following set of data: Slide 117 / , 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6

64 46 Find the Interquartile Range for the following set of data: Slide 117 () / , 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 = = Find the Standard Deviation of the following set of data: Slide 118 / , 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, Find the Standard Deviation of the following set of data: Slide 118 () / , 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6

65 48 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. B 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. B The spread is large The spread would be small, meaning that the data values are fairly close together. Slide 119 () / 241 Calculators Slide 120 / 241 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.

66 Calculators Slide 121 / 241 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. Calculators Slide 122 / 241 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). Calculators Slide 122 () / 241 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

67 Calculators Slide 123 / 241 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? Calculators Slide 123 () / 241 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? Standard Deviation Slide 124 / 241 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.

68 Standard Deviation Slide 124 () / 241 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 49 Using a calculator, find the standard deviation of the following set of data. Slide 125 / , 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, Using a calculator, find the standard deviation of the following set of data. Slide 125 () / , 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

69 50 Find the standard deviation of the following set of data. Slide 126 / , 7, 14, 16, 25, 17, 8, 20, 20, 30, 12, 14, 17, 17, 25, 6, 6, 22, 25, 26, 24, 16, 17, Find the standard deviation of the following set of data. Slide 126 () / , 7, 14, 16, 25, 17, 8, 20, 20, 30, 12, 14, 17, 17, 25, 6, 6, 22, 25, σ 26, = , 16, 17, 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, 2.4 B 1.2, 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 / 241

70 51 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, 2.4 B 1.2, 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 B Slide 127 () / 241 Slide 128 / 241 Standard Deviation and Normal Distribution Return to Table of Contents Goals and Objectives Slide 129 / 241 Students will be able to calculate the standard deviation of a data set and analyze a a normal distribution.

71 Why do we need this? Slide 130 / 241 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. Graphs Slide 131 / 241 Check out the following graphs. What do they have in common? Diastolic Blood Pressure Calories in French Fries Intervals of Peaks of Heartbeats Graphs Slide 131 () / 241 Check out the following graphs. What do they have in common? Diastolic Blood Pressure Calories in French Fries 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.

72 Normal Distribution Slide 132 / 241 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. Normal Distribution Slide 133 / 241 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... Normal Distribution Slide 134 / 241 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.

73 Normal Distribution 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. Slide 135 / 241 *note: graph does not represent top 5% with As. Normal Distribution Slide 136 / 241 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 Normal Curve Slide 137 / 241 Normal Curves are created using the mean (μ or x) of the data and the standard deviation (σ). mean

74 Normal Curve Slide 137 () / 241 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 Normal Distribution Slide 138 / 241 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 Normal Distribution Slide 138 () / 241 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

75 Normal Distribution Slide 139 / 241 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. 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. Normal Distribution Slide 140 / 241 Mean Normal Distribution Slide 140 () / 241 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.

76 Normal Distribution Slide 141 / 241 Each graph can be used differently even though there is a uniformity about their calculations. Chart for Examples Slide 142 / 241 Use this chart to answer the following questions. Chart for Examples Slide 142 () / 241 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.

77 Examples Slide 143 / 241 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 Big 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. Examples Slide 143 () / 241 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 Big 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. Example Slide 144 / 241 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?

78 Example Slide 144 () / 241 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. Example Slide 145 / 241 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? Example Slide 145 () / 241 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%.

79 52 Battery 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 / Battery 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 146 () / 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 / 241

80 53 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 () / x 340 = students 54 Which of the following curves represents a mean of 85 and a standard deviation of 6? A B Slide 148 / 241 C D Which of the following curves represents a mean of 85 and a standard deviation of 6? A C B D A Slide 148 () /

81 55 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? 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 -3 from the mean? Slide 150 / 241

82 56 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 () / 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% Slide 151 () / 241

83 Z-Score Slide 152 / 241 The graph we have been using to the right helps us find values that are multiples of 0.5 away from the mean. But what about numbers that are in between? For those, we use a formula for the z-score and a table of values. 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. Z-Scores: Negative Slide 154 / 241

84 Z-Scores: Positive Slide 155 / 241 Z-Score Slide 156 / 241 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! z-score = Z-Score Slide 157 / 241 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 =

85 Z-Score Slide 157 () / 241 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. Z-Score Slide 158 / 241 Your friend took the same test and got a score of 92%. Find your friend's z-score and calculate their percentile. Z-Score Slide 158 () / 241 Your friend took the same test and got a score of 92%. Find your friend's z-score and calculate their percentile. From the table, 3.06 is associated with That means your friend's score was in the 99th percentile.

86 58 Find the z-score for a 29 if the mean was 34 and the standard deviation is 2.3. Slide 159 / Find the z-score for a 29 if the mean was 34 and the standard deviation is 2.3. Slide 159 () / 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% B -1.86, 3.14% C 1.86, 31.4% D 1.86, 97.5% Slide 160 / 241

87 59 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% B -1.86, 3.14% C 1.86, 31.4% B D 1.86, 97.5% Slide 160 () / 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 161 () / 241 Solving for x = 74.77

88 61 A student calculated a z-score of What percentile does this score fall in? Slide 162 / A student calculated a z-score of What percentile does this score fall in? Slide 162 () / 241 The percentage associated with is 10.56%. Therefore, the score is in the 10th percentile. 62 Find the z-score of 10 if the data set is: Slide 163 /

89 62 Find the z-score of 10 if the data set is: Slide 163 () / z-score = 0.48 Slide 164 / 241 Two-Way Frequency Tables Return to Table of Contents Goals and Objectives Slide 165 / 241 Students will be able to recognize trends with and interpret different association of data in a two-way frequency table.

90 Why do we need this? Slide 166 / 241 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. Remember from Algebra 1... Slide 167 / 241 Frequency Table Stem-and-Leaf Plot Ages of people at the gym Box-and-Whisker Plot llll llll llll l llll llll lll llll lll llll l lll Stem Leaf Ages of Professors at a College *These are all ways to display a collection of data. Remember from Algebra 1... Slide 167 () / 241 Frequency Table llll llll llll l llll llll lll llll lll llll l lll Stem-and-Leaf Plot Ages of people at the gym Teacher Notes Stem Box-and-Whisker Plot 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.

91 Slide 168 / 241 Slide 168 () / 241 Remember from Algebra 1... Slide 169 / 241 Line Plots Scatter Plots

92 Remember from Algebra 1... Slide 169 () / 241 Line Plots Teacher Notes Discuss each type of graph. If students need review, see the statistics unit in Scatter Algebra Plots1. Two-Way Frequency Tables Slide 170 / 241 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 Slide 171 / 241 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

93 Two-Way Frequency Tables Slide 172 / 241 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. Two-Way Frequency Tables Slide 172 () / 241 The yellow boxes represent Joint Relative Frequency and the pink boxes represent Marginal Relative Frequency. 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. Two-Way Frequency Tables Slide 173 / 241 These relative frequencies directly translate into quantitative statements. Such statements mirror those that are reported in the media. 18% of the men surveyed exercise regularly. 22% of the women surveyed did not exercise regularly. 54% of the people surveyed were women.

94 Two-Way Frequency Tables Slide 173 () / 241 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. Two-Way Frequency Tables Slide 174 / 241 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. Two-Way Frequency Tables Slide 174 () / 241 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.

95 Two-Way Frequency Tables Slide 175 / 241 Together, write some quantitative statements about the information. Two-Way Frequency Tables Slide 175 () / 241 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] Two-Way Frequency Tables Slide 176 / 241 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.

96 Two-Way Frequency Tables Slide 176 () / 241 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. 63 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. Slide 177 () / or 23%

97 64 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? Slide 178 () / What is the percentage of dogs that came in that needed blood work? Slide 179 / 241

98 65 What is the percentage of dogs that came in that needed blood work? Slide 179 () / % 66 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? Slide 180 () /

99 Two-Way Frequency Tables Slide 181 / 241 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. 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 % Conditional Relative Frequency and Conditional Probability Slide 182 / 241 Conditional Relative Frequency and Conditional Probability go hand in hand. In fact how statistics are reported usually involves some probability. Two-Way Frequency Tables Slide 183 / 241 Using the table, find the probability that if a pet was brought into the clinic that needed blood work, it would be a dog.

100 Two-Way Frequency Tables Slide 183 () / 241 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. Two-Way Frequency Tables Slide 184 / 241 Using the table, find the probability that if you brought in a cat, it would NOT need blood work? Two-Way Frequency Tables Slide 184 () / 241 Using the table, find the probability that 0.17if you brought in a cat, it would NOT need blood 0.40 work? = 43%

101 67 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. Slide 185 () / % 68 Find the conditional probability that out of the girls, the person has been to an amusement park. Slide 186 / 241

102 68 Find the conditional probability that out of the girls, the person has been to an amusement park. Slide 186 () / = 72% 69 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? Slide 187 () / %

103 70 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. Slide 188 () / % Two-Way Frequency Tables Slide 189 / 241 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.

104 Two-Way Frequency Tables Slide 189 () / 241 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. Two-Way Frequency Tables Slide 190 / 241 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. Two-Way Frequency Tables Slide 190 () / 241 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

105 Two-Way Frequency Tables Slide 191 / 241 Out of all of the animals x-rayed, calculate the percentages that were a) dogs and b) cats. Two-Way Frequency Tables Slide 191 () / 241 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. 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

106 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 192 () / 241 Two-Way Frequency Tables Slide 193 / 241 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. 71 Based 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. B 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 194 / 241

107 71 Based 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 B 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 194 () / Based 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. B 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 / Based 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 B 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 () / 241

108 73 The marginal relative frequency of in-state students is: A 0.33 B 0.78 C 0.45 D 0.22 Slide 196 / The marginal relative frequency of in-state students is: A 0.33 B 0.78 C 0.45 D 0.22 B Slide 196 () / The joint relative frequency that a female would choose an out-of-state college or university is: A 0.12 B 0.45 C 0.22 D 0.10 Slide 197 / 241

109 74 The joint relative frequency that a female would choose an out-of-state college or university is: A 0.12 B 0.45 D C 0.22 D 0.10 Slide 197 () / 241 Slide 198 / 241 Sampling and Experiments Return to Table of Contents Goals and Objectives Slide 199 / 241 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.

110 Why do we need this? Slide 200 / 241 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. Sampling Slide 201 / 241 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. Sampling Slide 202 / 241 By 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.

111 Sampling Slide 203 / 241 There are several aspects of sampling that deserve attention: 1) randomization and bias, 2) sample size, and 3) margin of error. 75 Which of the following samples would most accurately represent the way people would vote on lowering the drinking age to 18? Slide 204 / 241 A Polling 100 random students at all college campuses. B 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. 75 Which of the following samples would most accurately represent the way people would vote on lowering the drinking age to 18? Slide 204 () / 241 A Polling 100 random students at all college campuses. B 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.

112 76 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. B 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. B 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 205 () / 241 Sampling Slide 206 / 241 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.

113 Sampling - Bias Slide 207 / 241 Bias 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. Sampling - Sample Size Slide 208 / 241 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. Sampling - Example Slide 209 / 241 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.

114 Sampling - Example Slide 209 () / 241 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 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 210 / 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: Slide 211 / 241 **

115 Margin of Error Slide 212 / 241 **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. To do this, use the formula: M = margin of error n = sample size Margin of Error Slide 213 / 241 At the ice cream factory, calculate the margin of error if you used a sample of 4 gallons out of the What is the margin of error if you used a sample of 400? Margin of Error Slide 213 () / 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 What is the margin of error if you used a sample of 400?

116 Margin of Error Slide 214 / 241 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: Is your answer reasonable in this situation? Margin of Error Slide 214 () / 241 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? BUT..is this reasonable? 77 What is the margin of error for a sample size of 30? Slide 215 / 241

117 77 What is the margin of error for a sample size of 30? Slide 215 () / 241 ±18% 78 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% B 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% B 20.8% % C 3.2% - 24% D 24% % B Slide 216 () / 241

118 79 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. Slide 217 / 241 A 5%, 20% to 30% B 15%, 1% to 31% C 20%, 0% to 36% D 25%, 0% to 50% 79 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% B 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 217 () / Find the sample size needed to achieve a margin of error of ±1%. Slide 218 / 241

119 80 Find the sample size needed to achieve a margin of error of ±1%. Slide 218 () / , 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 219 () / 241

120 Margin of Error Slide 220 / 241 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 Margin of Error Slide 221 / 241 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? Margin of Error Slide 221 () / 241 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

121 Margin of Error Slide 222 / 241 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. Margin of Error Slide 223 / 241 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. Sampling and Experiments Lab Slide 224 / 241 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

122 Sampling and Experiments Lab Slide 224 () / 241 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 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 225 / 241 Sampling and Experiments Lab Slide 226 / 241 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?

123 Sampling and Experiments Lab Mini-Lab: Teacher Slides - Part 2: Simulation of Flipping a Coin using a Graphing Calculator Slide 227 / 241 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 PRB 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, Sto 2nd L 1 To view the list that you stored, press Stat Edit Enter your list into the table on the Lab WS. Sampling and Experiments Lab Slide 228 / 241 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. Calculator Simulations Slide 229 / 241 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.

124 Calculator Simulations Slide 230 / 241 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! Calculator Simulations Slide 231 / 241 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. Calculator Simulations Slide 231 () / 241 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.

125 Calculator Simulations Slide 232 / 241 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. Calculator Simulations Slide 232 () / 241 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 Calculator Simulations Slide 233 / 241 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.

126 Calculator Simulations Slide 233 () / 241 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 Calculator Simulations Slide 234 / 241 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... Calculator Simulations Slide 234 () / 241 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...

127 Calculator Simulations Slide 235 / 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. 6. Generate the confidence interval. 7. Determine if the amount in question falls in your interval. Calculator Simulations Slide 236 / 241 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? Calculator Simulations Slide 236 () / 241 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?