Algebra II Probability and Statistics

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Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org

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

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.

Slide 6 / 241 Why do we need this? 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.

Slide 7 / 241 Vocabulary and Set Notation 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}

Slide 8 / 241 Create a Venn Diagram to match the information. U 0 7 A 2 B 9 4 1 8 6 10 3 5 A = {0, 2, 3, 7, 9} B = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Slide 8 (Answer) / 241 Create a Venn Diagram to match the information. U 0 1 7 A 8 Teacher Notes 2 B Move the circles and numbers around to 9 mirror the given 10information. 6 3 [This object is a pull tab] 4 5 A = {0, 2, 3, 7, 9} B = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Slide 9 / 241 Data Displays 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 6 5 4 7 6 5 4 3 2 3 1 2 8 7 6 5 4 3 9 8 7 6 5 4 10 11 12 9 10 11 8 9 10 7 8 9 6 7 8 5 6 7 1 2 3 4 5 6

Slide 10 / 241 Data Displays Use a sample space that helps organize the data effectively. For example, would you be able to effectively display a coin toss in a Venn Diagram or on a chart? Decide how to display the following information. 1. Survey results about what subject students like in school. 2. The different ways you can deal two cards from a deck of cards. 3. Results that compare the number of men and women that like chocolate ice cream over vanilla ice cream. 4. A poll on which grocery store people prefer to go to.

Slide 10 (Answer) / 241 Data Displays Use a sample space that helps organize the data effectively. 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 display the following information. Teacher Notes 1. Venn Diagram or chart 2. Chart 3. Venn Diagram or chart There can be different answers. 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. 2. The different ways you can deal two cards from a deck of cards. 3. Results that compare the number of men and women that like chocolate ice cream over vanilla ice cream. 4. A poll on which grocery store people prefer to go to.

Slide 11 / 241 The Universe The Universe (U) is all aspects that should be considered in a situation. The Universe (U) is basically the same as a sample space also used in probability. Name the Universe (U) of the following: 1. Survey at a local college asking women what they are studying. 2. Calculating the probability that you would draw a red 10 out of a deck of cards. 3. Phone survey on who you will vote for in the U. S. Presidential race.

Slide 11 (Answer) / 241 The Universe 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. Teacher Notes 2. The deck of cards 3. People in the United States that not only picked up their phone, but answered the question. Name the Universe (U) of the following: 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. [This object is a pull tab] 3. Phone survey on who you will vote for in the U. S. Presidential race.

Slide 12 / 241 Empty Set 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.

Slide 13 / 241 Example U A B -3-2 6 17 3 5 1 4-12 7-1 C 15 0 1. 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.

Slide 13 (Answer) / 241 Example U A B -3-2 6 17-1 3 Answer 1. 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. 5 [This object is a pull tab] -12 1. 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} *3. A B = {-3, 15} 0 *4. A B C C = {5} *note: the notation and concept of intersection will be dealt with in the next section of the unit. 7

Slide 14 / 241 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} Men Women 5pm cycling 7pm weight lifting 6pm swimming 4pm water aerobics 3pm nutrition 6am aerobics 10am weight lifting 2pm climbing

Slide 14 (Answer) / 241 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} Men 5pm cycling 7pm weight lifting Answer 4pm water aerobics 3pm nutrition Women 6am aerobics D 10am weight lifting 6pm swimming 2pm climbing [This object is a pull tab]

Slide 15 / 241 2 What is the most popular activity, or activities, at the club? *Answer 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 7pm weight lifting 6pm swimming 4pm water aerobics 3pm nutrition Women 6am aerobics 10am weight lifting 2pm climbing

Slide 15 (Answer) / 241 2 What is the most popular activity, or activities, at the club? *Answer 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 C 3 pm nutrition D 5 pm cycling E 10 am weight diagram lifting is about. F 2 pm climbing G 6 pm swimming H 7 pm weight lifting I Not enough information to tell Answer numbers to tell 5pm what cyclingis most popular or a better explanation of what the 7pm weight lifting 6pm swimming [This object is a pull tab] 4pm water aerobics 3pm nutrition Women 6am aerobics 10am weight lifting 2pm climbing

Slide 16 / 241 3 What are the most popular activities for both men and women at the club? 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 Men 5pm cycling 7pm weight lifting 6pm swimming 4pm water aerobics 3pm nutrition Women 6am aerobics 10am weight lifting 2pm climbing

Slide 16 (Answer) / 241 3 What are the most popular activities for both men and women at the club? A 5 pm cycling B 4 pm water aerobics C 6 am aerobics 3 pm nutrition 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 Answer Men 4 pm water aerobics 5pm cycling 7pm weight lifting 6pm swimming [This object is a pull tab] 4pm water aerobics 3pm nutrition Women 6am aerobics 10am weight lifting 2pm climbing

Slide 17 / 241 4 What is the best display for the sample space (or universe) of rolling an odd number on a single number cube? A S = {1, 2, 3, 4, 5, 6} B 1 2 3 4 5 6 D # 1 2 3 4 5 6 Answer C S = {1, 3, 5} E 1 3 5 4 2 6

Slide 18 / 241 5 What does the following set represent? {3, 6, 7} 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 11 1 5 8 6 7 3 4 0 2 C B 12 9 10

Slide 18 (Answer) / 241 5 What does the following set represent? {3, 6, 7} 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 Answer A 11 1 5 C 6 7 3 8 4 0 2 C B 12 [This object is a pull tab] 9 10

Slide 19 / 241 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 11 1 5 8 6 7 3 4 0 2 C B 12 9 10

Slide 19 (Answer) / 241 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 Answer A 11 1 5 8 6 7 True 3 4 0 2 C [This object is a pull tab] B 12 9 10

Slide 20 / 241 Unions Unions (U) of two or more sets creates a set that includes everything in each set. A C Unions (U) are associated with "or." Examples: Shade in the areas! A U 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") 11 1 5 B 8 6 7 3 4 0 2 12 9 10

Slide 21 / 241 Intersections 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 11 1 5 B 8 6 7 3 4 0 2 12 9 10 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".

Slide 22 / 241 Unions and Intersections Unions (U) and Intersections ( ) are often combined. Find: 1. (A U C) B 2. A B C A 11 1 5 8 6 7 3 4 0 2 C 3. (A C) U (B C) 12 9 10 **Shade the diagram as you go to help. B

Slide 22 (Answer) / 241 Unions and Intersections Unions (U) 1. and (A Intersections U C) B = {0, ( ) 2, are 3, 4, often 8} combined. Answer 2. A B C = {3} A Find: 6 7 1. (A U C) 3. B (A C) U (B C) = {2, 3, 4, 6, 7} 1 2. A B COrder of operations applies. 11 5 8 3 4 0 2 C 3. (A C) U (B C) [This object is a pull tab] 12 9 10 **Shade the diagram as you go to help. B

Slide 23 / 241 Complements 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"

Slide 24 / 241 Examples 1. 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.

Slide 24 (Answer) / 241 Examples 1. 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} Answer 3. If U = {the students at your school} and C = {students that like math}, find ~C. [This object is a pull tab]

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

Slide 25 (Answer) / 241 Examples You can also combine Complements with Intersections and Unions. A C 1. (A C) U ~B = {1, 5, 6, 7, 11} 5 Find: 11 1. (A C) U ~B 6 7 Answer 2. (A U B) ~C 3. C B U ~A 2. (A U B) ~C = {0, 1, 5, 8, 9, 10, 11, 1 12} 3. C B U ~A = {2, 4, 9, 10, 12} 4. ~A U ~B = { } 4. ~A U ~B **Shade the diagram as you go to help. [This object is a pull tab] B 8 3 4 0 2 12 9 10

Slide 26 / 241 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} U A 5 10 12 6 3 B 14 1 8 15 4 2 11 7 9 13 C

Slide 26 (Answer) / 241 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} Answer B U A 5 10 12 6 3 B [This object is a pull tab] 14 1 8 15 4 2 11 7 9 13 C

Slide 27 / 241 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} U A 5 10 12 6 3 B 14 1 8 15 4 2 11 7 9 13 C

Slide 27 (Answer) / 241 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} Answer A U A 5 10 12 6 3 B [This object is a pull tab] 14 1 8 15 4 2 11 7 9 13 C

Slide 28 / 241 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} U A 5 10 12 6 3 B 14 1 8 15 4 2 11 7 9 13 C

Slide 28 (Answer) / 241 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} Answer D U A 5 10 12 6 3 B [This object is a pull tab] 14 1 8 15 4 2 11 7 9 13 C

Slide 29 / 241 10 Find ~B U A A 12 B 27 C 45 D 63 A B 12 18 18 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

Slide 29 (Answer) / 241 10 Find ~B U A A 12 B 27 C 45 D 63 A Answer 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). 12 18 18 15 [This object is a pull tab] 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 30 / 241 11 Find ~(A B) A 45 B 30 C 18 D 12 A B 12 18 18 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

Slide 30 (Answer) / 241 11 Find ~(A B) A 45 B 30 C 18 D 12 A Answer A, it indicates the number of students who do not like BOTH math and English. B 12 18 18 15 [This object is a pull tab] 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

Slide 32 / 241 Goals and Objectives 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.

Slide 33 / 241 Why do we need this? Deciding things such as the cost of insurance can get very complicated. These decisions need to be based on many different elements. For example, who should pay more for health care: a person who smokes or a person who does not smoke? What about car insurance: a female driver, age 45, that drives a brand new Camaro or a 17 year old male driving a used Honda Civic?

Slide 34 / 241 Independence and Conditional Probability 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.

Slide 35 / 241 Independence and Conditional Probability You can also relate this to everyday situations: 1. Are you independent of, or dependent on, your parents and guardians right now? 2. True or false: Smoking causes lung cancer. Is this a dependent or an independent event? 3. Is how you do on a test based on how others study?

Slide 35 (Answer) / 241 Independence and Conditional Probability You can also relate this to everyday situations: Discuss the different 1. Are you independent of, or dependent scenarios on, involved. your parents and guardians right now? 2. True or false: Smoking causes lung cancer. Is this a dependent or an independent event? 3. Is how you do on a test based on how others study? Teacher Notes [This object is a pull tab]

Slide 36 / 241 Independence and Conditional Probability Since many of these situations are based on specific circumstances, we can use probability to study them. The 45 year old female driving a Camaro may have a terrible driving record. Therefore, what she pays for insurance will be dependent on her previous driving and she gets an extremely high rate. The probability she will have another accident is high. While not every smoker will get lung cancer, the probability that people get lung cancer if they smoke is very high. Probability allows us to make predictions! And, therefore, choices.

Slide 36 (Answer) / 241 Independence and Conditional Probability Since many of these situations are based on specific circumstances, we can use probability to study them. Discuss the different scenarios involved. The 45 year old female driving a Camaro may have a terrible driving record. Therefore, what she pays for insurance will be dependent on her previous driving and she gets an extremely high rate. The probability she will have another accident is high. Teacher Notes [This object is a pull tab] While not every smoker will get lung cancer, the probability that people get lung cancer if they smoke is very high. Probability allows us to make predictions! And, therefore, choices.

Slide 37 / 241 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 (Answer) / 241 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 Answer Yes. The fact that you chose blue does not have to affect that your friend chose red. [This object is a pull tab]

Slide 38 / 241 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 (Answer) / 241 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 Answer False. Because you chose the only blue car, your friend cannot choose blue. [This object is a pull tab]

Slide 39 / 241 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

Slide 39 (Answer) / 241 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 Answer 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 60. http://www.netdoctor.co.uk/healthy-living/lungcancer.htm#ixzz2gg5jn4br 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. http://www.cancer.gov/cancertopics/tobacco/smoking [This object is a pull tab]

Slide 40 / 241 Review of General Probability 1. Find the probability of drawing a 6 of clubs followed by a 5 of hearts without replacement. 2. Calculate the probability of throwing 3 heads in a row. 3. There are 20 marbles in a bag. 10 are blue, 4 are red and 6 are white. You draw out two marbles, one at a time, and do not replace them. Calculate the probability of drawing a red marble first, and then a white marble.

Slide 40 (Answer) / 241 Review of General Probability 1. Find the probability of drawing a 6 of clubs followed by a 5 of hearts without replacement. 2. Calculate the probability of throwing 3 heads in a row. Answer 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, fraction or percentage. [This object is a pull tab]

Slide 41 / 241 Review of Mutually Exclusive Events and the Addition Law of Probability 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) =.

Slide 42 / 241 Independence and Conditional Probability Drawing a 6 and drawing a red card from a traditional deck of cards are not mutually exclusive events because two of the 6's are red. These are not mutually exclusive and known as overlapping events. Overlapping events A and B satisfy P(A B).

Slide 43 / 241 Addition Law of Probability 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)

Slide 44 / 241 Independence and Conditional Probability 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.

Slide 44 (Answer) / 241 Independence and Conditional Probability 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. Answer [This object is a pull tab]

Slide 45 / 241 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 (Answer) / 241 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% Answer [This object is a pull tab]

Slide 46 / 241 16 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 (Answer) / 241 16 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 Answer A [This object is a pull tab]

Slide 47 / 241 17 Using the Venn Diagram, how many people like to ski or ride snowmobiles? A 15 B 45 C 69 D 89 30 15 44 People that like to ski. People that like to ride snowmobiles.

Slide 47 (Answer) / 241 17 Using the Venn Diagram, how many people like to ski or ride snowmobiles? A 15 B 45 C 69 D 89 Answer 30 D Just add up all three numbers. Or, you could still use 44 15the formula: 45 + 59-15 People that like to ski. People that like to ride snowmobiles. [This object is a pull tab]

Slide 48 / 241 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%

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Slide 49 / 241 19 Events A and B are NOT mutually exclusive. P(A) = 0.3, P(B) = 0.45 and P(A B) is 0.12. Find P(A U B). A 0.18 B 0.33 C 0.63 D 0.75

Slide 49 (Answer) / 241 19 Events A and B are NOT mutually exclusive. P(A) = 0.3, P(B) = 0.45 and P(A B) is 0.12. Find P(A U B). A 0.18 B 0.33 C 0.63 D 0.75 Answer C 0.3 + 0.4-0.12 = 0.63 [This object is a pull tab]

Slide 50 / 241 Conditional Probability 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:

Slide 50 (Answer) / 241 Conditional Probability 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: [This object is a pull tab]

Slide 51 / 241 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? click click click click click click click click

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

Slide 53 / 241 Example A bag contains plastic disks with numbers on them. You must choose two disks, one at a time without replacing them. The probability that the first disk is odd and the second disk even is 0.23. The probability that the first disk is odd is 0.67. What is the probability of drawing an even number on the second draw given that the first disk was odd?

Slide 53 (Answer) / 241 Example Answer 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 0.23. The probability that the first disk is odd is 0.67. What is the probability of drawing an even number on the second draw given that the first disk was odd? [This object is a pull tab]

Slide 54 / 241 Example Using the Venn Diagram, find the probability that a student is taking music given that they are taking math. Students that take music Students that take math 0.25 0.3 0.45

Slide 54 (Answer) / 241 Example Using the Venn Diagram, find the probability that a student is taking music given that they are taking math. Students that take music Answer Students that take math 0.25 0.3 0.45 [This object is a pull tab]

Slide 55 / 241 Example On any given day, the probability that it will rain and be windy is 18%. At the same time, the probability that it will just rain is 68%. Find the probability that it is windy, Given that it is raining.

Slide 55 (Answer) / 241 Example On any given day, the probability that it will rain and be windy is 18%. At the same time, the probability that it will just rain is 68%. Find the probability that it is windy, Given that it is raining. Answer [This object is a pull tab]

Slide 56 / 241 Formula To decide if the events in a conditional probability situation are independent, use the following formula:

Slide 57 / 241 Example with Formula Use the formula to decide if these two events are independent. Students that take music Students that take math 0.25 0.3 0.45

Slide 57 (Answer) / 241 Example with Formula Use the formula to decide if these two events are independent. Answer Students that take music Students that take math [This object is a pull tab] 0.25 0.3 0.45

Slide 58 / 241 Example with Formula On the other hand, think about rolling dice. Use the formula to decide if rolling a 5, and then a 6 is independent.

Slide 58 (Answer) / 241 Example with Formula On the other hand, think about rolling dice. Use the formula to decide if rolling a 5, and then a 6 is independent. Answer [This object is a pull tab]

Slide 59 / 241 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. A 25% B 38% C 65% D 78%

Slide 59 (Answer) / 241 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. A 25% B 38% C 65% D 78% Answer B [This object is a pull tab]

Slide 60 / 241 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 (Answer) / 241 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% Answer D [This object is a pull tab]

Slide 61 / 241 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) C 75%, P(A U B) D 75%, P(A B) Weightlifting Yoga 30% 10% 35% 25% Preferences of activities at a local gym.

Slide 61 (Answer) / 241 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) C 75%, P(A U B) D 75%, P(A B) Answer Weightlifting B Yoga 30% 10% 35% 25% [This object is a pull tab] Preferences of activities at a local gym.

Slide 62 / 241 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% Preferences of activities at a local gym.

Slide 62 (Answer) / 241 23 Calculate the percentage of people that like yoga, given that they enjoy weightlifting. A 10% B 25% C 30% D 33% Answer Weightlifting Yoga B 30% 10% 35% 25% [This object is a pull tab] Preferences of activities at a local gym.

Slide 63 / 241 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% Preferences of activities at a local gym.

Slide 63 (Answer) / 241 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 Answer Yoga B 30% 10% 35% 25% [This object is a pull tab] Preferences of activities at a local gym.

Slide 64 / 241 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? A 10% B 20% C 30% D 67%

Slide 64 (Answer) / 241 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? A 10% B 20% C 30% D 67% Answer D [This object is a pull tab]

Slide 65 / 241 Permutations & Combinations Return to Table of Contents

Slide 66 / 241 Goals and Objectives 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.

Slide 67 / 241 Why do we need this? 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

Slide 68 / 241 Fundamental Counting Principle Fundamental Counting Principle: If event M can occur in m ways & is followed by event N that can occur in n ways, then the event M followed by the event N can occur in M N ways. - Ex: If a number cube is rolled & a coin is tossed, then there are 6 2, or 12 possible outcomes.

Example: Slide 69 / 241 Fundamental Counting Principle A manager assigns different codes to all the tables in a restaurant to make it easier for the wait staff to identify them. Each code consists of a vowel, A, E, I, O or U, followed by 2 digits from 0 through 9. How many codes could the manager assign using this method?

Example: Slide 69 (Answer) / 241 Fundamental Counting Principle A manager assigns different codes to all the tables in a restaurant to make it easier for the wait staff to identify them. Each code consists of a vowel, A, E, I, O or U, followed by 2 digits from 0 through 9. How many codes could the manager assign using this method? Answer 5 x 10 x 10 = 500 codes [This object is a pull tab]

Slide 70 / 241 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 D 2

Slide 70 (Answer) / 241 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 Answer B D 2 [This object is a pull tab]

Slide 71 / 241 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 B 180 C 300 D 900

Slide 71 (Answer) / 241 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 B 180 C 300 D 900 Answer D [This object is a pull tab]

Slide 72 / 241 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 B 604,800 C 8,000,000 D 10,000,000

Slide 72 (Answer) / 241 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 B 604,800 C 8,000,000 D 10,000,000 Answer C [This object is a pull tab]

Slide 73 / 241 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 B 37,015,056 C 32,292,000 D 6,760,000

Slide 73 (Answer) / 241 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 B 37,015,056 C 32,292,000 D 6,760,000 Answer A [This object is a pull tab]

Slide 74 / 241 Permutations Factorial: n! means the product of all counting numbers beginning w/ n & counting backwards to 1. 0! has a value of 1. Example: 4! = 4 x 3 x 2 x 1 = 24 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)!

Slide 75 / 241 Permutations Example: How many ways can you arrange the letters A, B, C and D?

Slide 75 (Answer) / 241 Permutations Example: How many ways can you arrange the letters A, B, C and D? Answer 4 x 3 x 2 x 1 = 4! = 24 ways [This object is a pull tab]

Slide 76 / 241 Permutations 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

Slide 77 / 241 30 In how many ways can the letters in the word "WEIGHT" be arranged? A 60 B 120 C 720 D 46,656

Slide 77 (Answer) / 241 30 In how many ways can the letters in the word "WEIGHT" be arranged? A 60 B 120 C 720 D 46,656 Answer C [This object is a pull tab]

Slide 78 / 241 31 How many different 4 letter arrangements can be formed from the letters in the word "DECAGON" A 210 B 840 C 5,040 D 823,543

Slide 78 (Answer) / 241 31 How many different 4 letter arrangements can be formed from the letters in the word "DECAGON" A 210 B 840 C 5,040 D 823,543 Answer B [This object is a pull tab]

Slide 79 / 241 32 There are 15 players on a basketball team. In how many ways can the coach select the 5 starting players? A 120 B 360,360 C 12,454,041,600 D 1,307,674,368,000

Slide 79 (Answer) / 241 32 There are 15 players on a basketball team. In how many ways can the coach select the 5 starting players? A 120 B 360,360 C 12,454,041,600 Answer B D 1,307,674,368,000 [This object is a pull tab]

Slide 80 / 241 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? A 254,251,200 B 5,527,200 C 125,000 D 117,600

Slide 80 (Answer) / 241 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? A 254,251,200 B 5,527,200 C 125,000 D 117,600 Answer D [This object is a pull tab]

Slide 81 / 241 Combinations 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!

Slide 82 / 241 Combinations 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! (9-4)! 4! = 9! 5! 4! 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 5 x 4 x 3 x 2 x 1 x 4 x 3 x 2 x 1 click 9 x 8 x 7 x 6 4 x 3 x 2 x 1 click = 126 fruit salads

Slide 83 / 241 Permutation vs. Combination In a permutation, the order matters. In a combination, the order does not matter. Way to remember: "P" and "M" are really close in the alphabet (see underlined words above). "C" and "D" are really close in the alphabet (see underlined words above).

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

Slide 84 (Answer) / 241 34 Determine if this question is asking for a permutation or a combination. How many 3 person committees are possible when selected from a pool of 10 people? A Permutation B Combination Answer B [This object is a pull tab]

Slide 85 / 241 35 How many possible 3 person committees are possible when selected from a pool of 10 people? A 45 B 90 C 120 D 720

Slide 85 (Answer) / 241 35 How many possible 3 person committees are possible when selected from a pool of 10 people? A 45 B 90 C 120 D 720 Answer C [This object is a pull tab]

Slide 86 / 241 36 Determine if this question is asking for a permutation or a combination. How many 4 person committees are possible when selected from a pool of 9 people consisting of a president, vice-president, secretary, and treasurer? A Permutation B Combination

Slide 86 (Answer) / 241 36 Determine if this question is asking for a permutation or a combination. How many 4 person committees are possible when selected from a pool of 9 people consisting of a president, vice-president, secretary, and treasurer? A Permutation B Combination Answer A [This object is a pull tab]

Slide 87 / 241 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 B 504 C 756 D 3,024

Slide 87 (Answer) / 241 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 B 504 C 756 D 3,024 Answer D [This object is a pull tab]

Slide 88 / 241 38 Determine if the question below is asking for a permutation or a combination: How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A Permutation B Combination

Slide 88 (Answer) / 241 38 Determine if the question below is asking for a permutation or a combination: How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A Permutation B Combination Answer B [This object is a pull tab]

Slide 89 / 241 39 How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A 311,875,200 B 133,784,560 C 5,197,920 D 2,598,960

Slide 89 (Answer) / 241 39 How many hands of 5 playing cards can be dealt using a standard deck of 52 cards? A 311,875,200 B 133,784,560 C 5,197,920 D 2,598,960 Answer D [This object is a pull tab]

Slide 90 / 241 Probability Involving Permutations & Combinations Some questions will ask you to calculate the probability of an event, or multiple events, that use the counting techniques of permutations & combinations. When this occurs, calculate the number of outcomes of your event(s) and sample space to create your probability fraction.

Slide 91 / 241 Probability Involving Permutations & Combinations Example: Consider all of the 5-digit numbers that can be made with the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that the number is between 20,000 and 30,000. Is this event a permutation or a combination? Explain how you know. Permutation: with numbers, the order matters click How many outcomes are possible in the sample space? 5 x 4 x 3 x 2 x 1 = 5! = 120 outcomes click

Slide 92 / 241 Probability Involving Permutations & Combinations Example: Consider all of the 5-digit numbers that can be made with the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that the number is between 20,000 and 30,000. 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

Slide 93 / 241 Probability Involving Permutations & Combinations Example: When playing a game of poker, each player is dealt 5 cards from a standard deck of 52. A pair is when 2 cards are the same. What is the probability of getting a pair of Kings? 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! (52-5)! 5! = 2,598,960 hands

Example: Slide 94 / 241 Probability Involving Permutations & Combinations When playing a game of poker, each player is dealt 5 cards from a standard deck of 52. A pair is when 2 cards are the same. What is the probability of getting a pair of Kings? How many outcomes are possible in the event? 4C 2 x 48 C 3 = 6 x 17,296 = 103,776 There are 4 kings in a standard deck and you need 2 of them for the pair in your hand. For the remaining 48 cards, you can be dealt any 3 of them. click What is the probability that you are dealt the pair of Kings? 103,776/2,598,960 = 3.99% click

Slide 95 / 241 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 B 1/60 C 1/20 D 1/10

Slide 95 (Answer) / 241 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 B 1/60 C 1/20 D 1/10 Answer C [This object is a pull tab]

Slide 96 / 241 41 If the letters in the word DECAGON are arranged at random, find the probability that the first letter is a G. A 1/7 B 1/42 C 1/840 D 1/5040

Slide 96 (Answer) / 241 41 If the letters in the word DECAGON are arranged at random, find the probability that the first letter is a G. A 1/7 B 1/42 C 1/840 D 1/5040 Answer A [This object is a pull tab]

Slide 97 / 241 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? A 5/8 B 1/2 C 3/8 D 1/4

Slide 97 (Answer) / 241 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? A 5/8 B 1/2 C 3/8 D 1/4 Answer C [This object is a pull tab]

Slide 98 / 241 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? A 0.017% B 0.17% C 1.7% D 17%

Slide 98 (Answer) / 241 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? A 0.017% B 0.17% C 1.7% D 17% Answer B [This object is a pull tab]

Slide 99 / 241 Measures of Central Tendency Return to Table of Contents

Slide 100 / 241 Goals and Objectives 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.

Slide 101 / 241 Why do we need this? 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.

Slide 102 / 241 Review 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.

Slide 103 / 241 Review: Example 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

Slide 103 (Answer) / 241 Review: Example Example: Find the mean, through median answers and and mode have of them the following set of test scores: check their work. Answer Let students try this on their own. Go Mean: 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 Median: 45, 71, 71, 71, 78, 78, 79, 81, 85, 92, 93, 95, 95, 96, 98 Mode: 71 [This object is a pull tab]

Slide 104 / 241 Review 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.

Slide 105 / 241 Review: Example Find the range and identify any outliers of the following test scores: 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96

Slide 105 (Answer) / 241 Review: Example Find the range Range: and identify 98-45 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: Answer Have students attempt on their own. Go over results. [This object is a pull tab]

Slide 106 / 241 Review 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.

Slide 107 / 241 Interquartile Range Remember making box plots in Algebra 1? lowest number Quartile 1 Median of lower half of data Quartile 2 Median of data Quartile 3 Median of upper half of data highest number

Slide 107 (Answer) / 241 Interquartile Range Remember making box plots in Algebra 1? Refer to statistics unit in Algebra 1 if students need more of highest a review. lowest number Teacher Notes Quartile 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]

Slide 108 / 241 Interquartile Range Interquartile Range is the difference between Q2 and Q1, or Q3 - Q1. Find all three quartiles and calculate the interquartile range for the following test scores. 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96

Slide 108 (Answer) / 241 Interquartile Range Interquartile Range is the difference between Q2 and Q1, or Q3 - Q1. Answer 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: 95-71 = 24 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 [This object is a pull tab]

Slide 109 / 241

Slide 110 / 241

Slide 110 (Answer) / 241

Slide 111 / 241 Standard Deviation Find the standard deviation (σ) for this set of test scores. Remember, we already found the mean - 81.87% 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96

Slide 111 (Answer) / 241 Standard Deviation Find the standard deviation 45 81.87 (σ) = for (-36.87) this 2 = set 1359.40 of test scores. Remember, we already 71 found 81.87 the = (-10.87) mean 2 = 118.16-81.87% 78, 79, 81, 45, 71, 71, 95, 81.87 92, = 95, (13.13) 71, 2 = 85, 172.40 93, 78, 98, 96 Answer 78-81.87 = (-3.87) 2 = 14.98 79 81.87 = (-2.87) 2 = 8.24 81 81.87 = (-0.87) 2 = 0.76 71 81.87 = (-10.87) 2 = 118.16 92 81.87 = (10.13) 2 = 102.62 95 81.87 = (13.13) 2 = 172.40 71 81.87 = (-10.87) 2 = 118.16 85 81.87 = (3.13) 2 = 9.80 93 81.87 = (11.13) 2 = 123.88 78-81.87 = (-3.87) 2 = 14.98 98 81.87 = (16.13) 2 = 260.18 96 81.87 = (14.13) [This object = 199.66 is a pull tab]

Slide 112 / 241 Standard Deviation What do you think would happen to the standard deviation if we eliminated the outlier of 45?

Slide 112 (Answer) / 241 Standard Deviation What do you think would happen to the standard deviation if we eliminated the outlier of 45? Answer Notice how the standard deviation went down. [This object is a pull tab]

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

Slide 113 (Answer) / 241 Standard Deviation Find the standard deviation of the following set of numbers: 6.7, 7.1, 6.5, 7.2, 6.23, 6.9 Answer Have students make a prediction of how big or small the standard deviation will be. 6.7 6.77 = (-0.7) 2 = 0.0049 7.1 6.77 = (0.33) 2 = 0.1089 6.5 6.77 = (-0.27) 2 = 0.729 7.2 6.77 = (0.43) 2 = 0.1849 6.23 6.77 = (-0.54) 2 = 0.2916 6.9 6.77 = (0.13) 2 = 0.0169 [This object is a pull tab]

Slide 114 / 241 Standard Deviation Discuss the standard deviations of both sets that we just calculated. How do each reflect the spread of the data? 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 6.7, 7.1, 6.5, 7.2, 6.23, 6.9

Slide 114 (Answer) / 241 Standard Deviation Discuss the standard deviations of both sets that we just calculated. How do each reflect the spread of the data? Answer 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96 The smaller 6.7, the 7.1, standard 6.5, 7.2, 6.23, 6.9 deviation, the closer the data is to the mean. [This object is a pull tab]

Slide 115 / 241 44 Find the Interquartile Range of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48

Slide 115 (Answer) / 241 44 Find the Interquartile Range of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48 Answer 22, 25, 26, 36, 36, 36, 37, 48, 49, 50 Q1 Q2 Q3 IQR = 48-26 = 22 [This object is a pull tab]

Slide 116 / 241 45 Find the Standard Deviation of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48

Slide 116 (Answer) / 241 45 Find the Standard Deviation of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48 Answer [This object is a pull tab]

Slide 117 / 241 46 Find the Interquartile Range for the following set of data: 1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6

Slide 117 (Answer) / 241 46 Find the Interquartile Range for the following set of data: 1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6 Answer 1.3, 1.6, 2.3, 2.4, 2.6, 3.4, 4.6, 5.7 1.95 Q1 2.5 Q2 4 Q3 IQR = 4-1.95 = 2.05 [This object is a pull tab]

Slide 118 / 241 47 Find the Standard Deviation of the following set of data: 1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6

Slide 118 (Answer) / 241 47 Find the Standard Deviation of the following set of data: 1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6 Answer [This object is a pull tab]

Slide 119 / 241 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 (Answer) / 241 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 Answer The spread would be small, meaning that the data values are fairly close together. [This object is a pull tab]

Slide 120 / 241 Calculators When sets of information are very large, calculators can be very helpful. We will reference operations on a TI-84 for this exercise. Please refer to the manuals of other calculators for alternative directions.

Slide 121 / 241 Calculators Input the following sets into your calculator: L 1 : 1.3, 1.5, 1.7, 1.9, 0.9, 1.3, 1.4, 0.8, 2.1, 1.5, 1.6, 1.7, 1.4, 1.9, 1.3 L 2 : 1.2, 5.7, 0.1, 7.9, 2.0, 0.2, 9.8, 2.1, 4.6, 9.2, 1.1, 4.6, 7.2, 6.4, 9.1 To find L 1 and L 2, go to STAT and then 1: Edit.

Slide 122 / 241 Calculators 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 ).

Slide 122 (Answer) / 241 Calculators Now, calculate the Standard Deviation of each set. 1. Go to calculation screen Students ( 2nd should, Quit get ). 0.354 for L 1 2. Push 2nd, Stat and. 3.450 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 [This object is a pull tab]

Slide 123 / 241 Calculators L 1 : 1.3, 1.5, 1.7, 1.9, 0.9, 1.3, 1.4, 0.8, 2.1, 1.5, 1.6, 1.7, 1.4, 1.9, 1.3 L 2 : 1.2, 5.7, 0.1, 7.9, 2.0, 0.2, 9.8, 2.1, 4.6, 9.2, 1.1, 4.6, 7.2, 6.4, 9.1 For L 1, the standard deviation is 0.354. With L 2, the standard deviation is 3.450. Why is there such a large difference between the two numbers? What does it say about the data?

Slide 123 (Answer) / 241 Calculators 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 0.354. [This object With is a pull Ltab] 2, the standard deviation is 3.450. Why is there such a large difference between the two numbers? What does it say about the data?

Slide 124 / 241 Standard Deviation Standard Deviation (σ) is a number that represents how different the data is from the mean. The smaller the standard deviation, the closer the data is to the mean. The higher the standard deviation, the further the data is from the mean.

Slide 124 (Answer) / 241 Standard Deviation Standard Deviation (σ) is a number that represents how different the data is from the mean. The smaller the standard deviation, the closer the data is to the 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 [This object is a pull tab]

Slide 125 / 241 49 Using a calculator, find the standard deviation of the following set of data. 13.4, 14.6, 17.2, 21.3, 14.5, 17.8, 13.3, 22, 16.7, 17.3, 17.6, 15, 21, 16.8, 19.2, 15.3

Slide 125 (Answer) / 241 49 Using a calculator, find the standard deviation of the following set of data. 13.4, 14.6, 17.2, 21.3, 14.5, 17.8, 13.3, 22, 16.7, 17.3, 17.6, 15, 21, 16.8, 19.2, 15.3 σ = 2.72 Answer [This object is a pull tab]

Slide 126 / 241 50 Find the standard deviation of the following set of data. 12, 7, 14, 16, 25, 17, 8, 20, 20, 30, 12, 14, 17, 17, 25, 6, 6, 22, 25, 26, 24, 16, 17, 19

Slide 126 (Answer) / 241 50 Find the standard deviation of the following set of data. 12, 7, 14, 16, 25, 17, 8, 20, 20, 30, 12, 14, 17, 17, 25, 6, 6, 22, 25, 26, 24, 16, 17, 19 Answer σ = 6.69 [This object is a pull tab]

Slide 127 / 241 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

Slide 127 (Answer) / 241 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 Answer B [This object is a pull tab]

Slide 128 / 241 Standard Deviation and Normal Distribution Return to Table of Contents

Slide 129 / 241 Goals and Objectives Students will be able to calculate the standard deviation of a data set and analyze a a normal distribution.

Slide 130 / 241 Why do we need this? In short, the standard deviation of data represents how close the data is to its mean. It is used to report such things as results from political polls and data from medical experiments. We need to understand how these numbers are calculated to make informed decisions.

Slide 131 / 241 Graphs Check out the following graphs. What do they have in common? Calories in French Fries Diastolic Blood Pressure http://www.trialsjournal.com/content/6/1/5 Intervals of Peaks of Heartbeats http://www.statcrunch.com/grabimage.php?image_id=427473 http://www.swharden.com/blog/2009-08-14-diy-ecg-machine-on-the-cheap/

Slide 131 (Answer) / 241 Graphs Check out the following graphs. What do they have in common? Calories in French Fries Diastolic Blood Pressure http://www.trialsjournal.com/content/6/1/5 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. http://www.statcrunch.com/grabimage.php?image_id=427473 http://www.swharden.com/blog/2009-08-14-diy-ecg-machine-on-the-cheap/ [This object is a pull tab]

Slide 132 / 241 Normal Distribution Many different aspects of life, when measured and graphed, fit this type of distribution. Imagine a what the graph of height for humans, weight for bears or size of homes would look like. Most of the data would be around the same number (the mean), yet there would be some that would be larger or smaller. Finally, you would have the extremes that would be rare. This is called a Normal Distribution.

Slide 133 / 241 Normal Distribution Normal Distributions are very useful when analyzing data. It allows you to calculate the probability that an event will happen as well as a percentile ranking of scores. Consider the following examples...

Slide 134 / 241 Normal Distribution A tennis ball manufacturer measures the height their tennis balls bounce after dropping them from 5 feet off of the ground. The balls will not bounce the same height each time, but should be very close. A graph of this, after many trials, would begin to resemble a normal distribution. From here, you can calculate a mean height of the ball and use that to test other tennis balls from the factory to make sure that the quality is consistent. The blue shaded area would represent the range of acceptable heights.

Slide 135 / 241 A particular engineering school at a university prides itself on producing high quality engineers. Each class coming through has to take an introductory physics class. The professor uses a normal distribution to calculate grades such that only the top 5% of students get As. This ensures the course is challenging and that the best are the ones that continue on. *note: graph does not represent top 5% with As. Normal Distribution http://www.nohsteachers.info/pcaso/ap_statistics/midtermexamreview.htm

Slide 136 / 241 Normal Distribution Using the mean and standard deviation takes into account different spreads of the graph. In fact, knowing the standard deviation of a study can tell you how reliable the study is. Small standard deviations indicate that the mean is a good representation of the information. Large standard deviations tell you that the data was actually very spread out and the mean may not be reliable. Small σ = small spread Large σ = large spread

Slide 137 / 241 Normal Curve Normal Curves are created using the mean (μ or x) of the data and the standard deviation (σ). mean

Slide 137 (Answer) / 241 Normal Curve 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 [This object is a pull tab]

Slide 138 / 241 Normal Distribution Much of the time, instead of having more complicated numbers for μ ± σ, we write 0, ±1 or ±2 representing the number of standard deviations from the mean as shown below. These numbers are also known as z-scores. mean -3-2 -1 0 1 2 3

Slide 138 (Answer) / 241 Normal Distribution Much of the time, instead of having more complicated numbers for μ ± σ, we write 0, ±1 or ±2 representing the number of standard deviations from the mean as shown below. These numbers are also known as z-scores. Teacher Notes mean Take the time to point out the different parts of the graph. [This object is a pull tab] -3-2 -1 0 1 2 3

Slide 139 / 241 Normal Distribution In normal distributions, the area under the curve is what is used to calculate percentages or probabilities. These numbers follow what is called the Empirical Rule and is the same for each distribution. 68% of all data will fall within 1 standard deviation of the mean. 95% of all data falls within 2 standard deviations of the mean. 99.7% of all data falls within 3 standard deviations of the mean. The graph on the next page is an excellent illustration of this.

Slide 140 / 241 Normal Distribution Mean http://rchsbowman.wordpress.com/2009/11/29/statistics-notes-%e2%80%93-properties-of-normal-distribution-2/

Slide 140 (Answer) / 241 Normal Distribution 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. [This object is a pull tab] http://rchsbowman.wordpress.com/2009/11/29/statistics-notes-%e2%80%93-properties-of-normal-distribution-2/

Slide 141 / 241 Normal Distribution Each graph can be used differently even though there is a uniformity about their calculations. http://27gen.files.wordpress.com/2011/09/2011-bell-curve.jpg http://www.mhhe.com/socscience/intro/ibank/set4.htm

Slide 142 / 241 Chart for Examples Use this chart to answer the following questions. http://www.regentsprep.org/regents/math/algtrig/ats2/normallesson.htm

Slide 142 (Answer) / 241 Chart for Examples 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. [This object is a pull tab] http://www.regentsprep.org/regents/math/algtrig/ats2/normallesson.htm

Slide 143 / 241 Examples a) John usually scores an average of 82% on his math tests with a standard deviation of 5%. What is the probability that John will get an between an 82% and an 87% on his next test? b) At 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.

Slide 143 (Answer) / 241 Examples a) John usually scores an average of 82% 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 82% and an 87% on his next test? Answer a) 87% - 82% = 5% which is 1σ. The a 34.1% chance of getting between an 82% and an 87% on his next test. b) 95.4% tells you that there is a spread of 2 standard deviations from b) At Big Mama's Gym, there is a special 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 [This object is a pull tab] pounds. Find the average weight and the standard deviation of the data.

Slide 144 / 241 Example c) A machine at Superfoods Food Factory puts a mean of 44 oz of mayonnaise in their bottles. The machine has a standard of deviation of 0.5 ounces. While filling 1000 bottles with mayonnaise, about how many times will the machine fill a bottle with 45 or more ounces?

Slide 144 (Answer) / 241 Example 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 1000. It will fill approximately 23 bottles with 45 or more ounces. Answer [This object is a pull tab]

Slide 145 / 241 Example d) Scores on the final exam in Mr. Dahlberg's Precalculus classes are normally distributed. He calculates a mean to be 71% with a standard deviation of 7. What is the probability that a student in his classes will get between an 85 and a 92 on the final exam?

Slide 145 (Answer) / 241 Example d) Scores on the final exam in Mr. Dahlberg's Precalculus classes are normally distributed. He calculates a mean to be 71% with a standard deviation of 7. What is the probability that a student in his classes will get between an 85 85 and and a 92 92 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%. Answer [This object is a pull tab]

Slide 146 / 241 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? http://www.regentsprep.org/regents/math/algtrig/ats2/normalprac.htm

Slide 146 (Answer) / 241 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? Answer 15.9% or 16% [This object is a pull tab] http://www.regentsprep.org/regents/math/algtrig/ats2/normalprac.htm

Slide 147 / 241 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 (Answer) / 241 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. Answer 0.841 x 340 = 285.94 286 students [This object is a pull tab]

Slide 148 / 241 54 Which of the following curves represents a mean of 85 and a standard deviation of 6? A B C 67 73 79 85 91 97 103 D 82 83 84 85 86 87 88 70 75 80 85 90 95 100 56 66 76 85 86 96 106

Slide 148 (Answer) / 241 54 Which of the following curves represents a mean of 85 and a standard deviation of 6? A C 67 73 79 85 91 97 103 Answer B D A 82 83 84 85 86 87 88 [This object is a pull tab] 70 75 80 85 90 95 100 56 66 76 85 86 96 106

Slide 149 / 241 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 (Answer) / 241 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? Answer 33 [This object is a pull tab]

Slide 150 / 241 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 (Answer) / 241 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? Answer 18 [This object is a pull tab]

Slide 151 / 241 57 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 (Answer) / 241 57 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? Answer 13.6% [This object is a pull tab]

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. Slide 152 / 241 Z-Score

Slide 153 / 241 Z-Score z-score = A table of z-scores is shown on the next 2 slides. Each score is associated with the amount of area under the normal curve from the score to the left.

Slide 154 / 241 Z-Scores: Negative

Slide 155 / 241 Z-Scores: Positive

Slide 156 / 241 Z-Score 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 =

Slide 157 / 241 Z-Score Example: On a test, your score was 83%. The mean of all of the tests was 79, the data was normally distributed and the standard deviation was 4.25. Find your z-score and then use the table to calculate the percentile. z-score =

Slide 157 (Answer) / 241 Z-Score Example: On a test, your score was 83%. The mean of all of the tests was 79, the data was normally distributed and the standard deviation was 4.25. Find your z-score and then use the table to calculate the percentile. From the table, 0.94 is z-score = associated with 0.8264. That means your score was in the 82nd percentile. Answer [This object is a pull tab]

Slide 158 / 241 Z-Score Your friend took the same test and got a score of 92%. Find your friend's z-score and calculate their percentile.

Slide 158 (Answer) / 241 Z-Score Your friend took the same test and got a score of 92%. Find your friend's z-score and calculate their percentile. Answer From the table, 3.06 is associated with 0.9989. That means your friend's score was in the 99th percentile. [This object is a pull tab]

Slide 159 / 241 58 Find the z-score for a 29 if the mean was 34 and the standard deviation is 2.3.

Slide 159 (Answer) / 241 58 Find the z-score for a 29 if the mean was 34 and the standard deviation is 2.3. Answer -2.17 [This object is a pull tab]

Slide 160 / 241 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 24.2. A -1.86, 1.95% B -1.86, 3.14% C 1.86, 31.4% D 1.86, 97.5%

Slide 160 (Answer) / 241 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 24.2. A -1.86, 1.95% B -1.86, 3.14% C 1.86, 31.4% D 1.86, 97.5% Answer B [This object is a pull tab]

Slide 161 / 241 60 A value has a z-score of 0.82. The mean for the data is 73 and the standard deviation is 2.16. What was the original value?

Slide 161 (Answer) / 241 60 A value has a z-score of 0.82. The mean for the data is 73 and the standard deviation is 2.16. What was the original value? Answer Solving for x = 74.77 [This object is a pull tab]

Slide 162 / 241 61 A student calculated a z-score of -1.25. What percentile does this score fall in?

Slide 162 (Answer) / 241 61 A student calculated a z-score of -1.25. What percentile does this score fall in? Answer The percentage associated with -1.25 is 10.56%. Therefore, the score is in the 10th percentile. [This object is a pull tab]

Slide 163 / 241 62 Find the z-score of 10 if the data set is: 9 11 5 7 10 10 10 11 9 15

Slide 163 (Answer) / 241 62 Find the z-score of 10 if the data set is: 9 11 5 7 10 10 10 11 9 15 Answer z-score = 0.48 [This object is a pull tab]

Slide 164 / 241 Two-Way Frequency Tables Return to Table of Contents

Slide 165 / 241 Goals and Objectives Students will be able to recognize trends with and interpret different association of data in a two-way frequency table.

Slide 166 / 241 Why do we need this? All of us are marketed to on a regular basis. Television, the Internet and magazines are different ways that businesses get us to buy their product or use their service. It is vital to be able to interpret information that is given to us and make smart choices.

Slide 167 / 241 Remember from Algebra 1... 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 1 2 2 6 7 8 9 9 2 1 1 3 4 4 4 5 5 6 8 8 9 3 0 1 3 6 9 4 3 4 8 5 1 4 6 7 2 5 32 40 46 62 72 Ages of Professors at a College *These are all ways to display a collection of data.

Slide 167 (Answer) / 241 Remember from Algebra 1... 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 1 2 2 6 7 8 9 9 2 1 1 3 4 4 4 5 5 6 8 8 9 3 0 1 3 6 9 4 3 4 8 5 1 4 6 7 2 5 Box-and-Whisker Plot Discuss each type of graph. If Leaf students need review, see the statistics unit in Algebra 1. 32 40 46 62 72 Ages of Professors at a College *These are all ways to display a collection of data. [This object is a pull tab]

Slide 168 / 241

Slide 168 (Answer) / 241

Slide 169 / 241 Remember from Algebra 1... Line Plots Scatter Plots

Slide 169 (Answer) / 241 Remember from Algebra 1... Line Plots Teacher Notes Discuss each type of graph. If students need review, see the statistics unit in Algebra 1. Scatter Plots [This object is a pull tab]

Slide 170 / 241 Two-Way Frequency Tables In this section, we are going to study Two-Way Frequency Tables. These displays allow us to study situations that have more than one variable such as how many men and women that exercise regularly. The chart below shows a survey of 100 people.

Slide 171 / 241 Two-Way Frequency Tables 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

Slide 172 / 241 Two-Way Frequency Tables The yellow boxes represent Joint Relative Frequency and the pink boxes represent Marginal Relative Frequency. Joint Relative Frequency is found by dividing the number in that category by the total observations or outcomes. Marginal Relative Frequency is found by totaling the rows and columns.

Slide 172 (Answer) / 241 Two-Way Frequency Tables 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. [This object is a pull tab]

Slide 173 / 241 Two-Way Frequency Tables These relative frequencies directly translate into quantitative statements. Such statements mirror those that are reported in the media. 18% of the men surveyed exercise regularly. 22% of the women surveyed did not exercise regularly. 54% of the people surveyed were women.

Slide 173 (Answer) / 241 Two-Way Frequency Tables 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." [This object is a pull tab] 18% of the men surveyed exercise regularly. 22% of the women surveyed did not exercise regularly. 54% of the people surveyed were women.

Slide 174 / 241 Two-Way Frequency Tables 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.

Slide 174 (Answer) / 241 Two-Way Frequency Tables A teacher asked their class if they had been to an amusement park before or not. Out of 36 students, there were 22 boys and 14 girls. 16 of the boys and 10 of the girls answered that they had been to an amusement park before. Create a relative frequency table from the data collected. Answer [This object is a pull tab]

Slide 175 / 241 Two-Way Frequency Tables Together, write some quantitative statements about the information.

Slide 175 (Answer) / 241 Two-Way Frequency Tables Together, write some quantitative statements about the information. Answer 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 were girls. [This object is a pull tab]

Slide 176 / 241 Two-Way Frequency Tables At a vet clinic over the month of July, the vets saw a total of 150 animals. Out of those animals, 105 were dogs and 45 were cats. 70 of the dogs that were seen needed blood work. 20 of the cats needed blood work. Create a relative frequency table for the information. You will use your table to answer some questions.

Slide 176 (Answer) / 241 Two-Way Frequency Tables At a vet clinic over the month of July, the vets saw a total of 150 animals. Out of those animals, 105 were dogs and 45 were cats. 70 of the dogs that were seen needed blood work. 20 of the cats needed blood work. Create a relative frequency table for the information. You will use your table to answer some questions. Answer [This object is a pull tab]

Slide 177 / 241 63 From the relative frequency table you created, find the joint relative frequency for the dogs that did not need blood work.

Slide 177 (Answer) / 241 63 From the relative frequency table you created, find the joint relative frequency for the dogs that did not need blood work. Answer 0.23 or 23% [This object is a pull tab]

Slide 178 / 241 64 What is the marginal relative frequency of cats that came to the clinic?

Slide 178 (Answer) / 241 64 What is the marginal relative frequency of cats that came to the clinic? Answer 0.3 [This object is a pull tab]

Slide 179 / 241 65 What is the percentage of dogs that came in that needed blood work?

Slide 179 (Answer) / 241 65 What is the percentage of dogs that came in that needed blood work? Answer 47% [This object is a pull tab]

Slide 180 / 241 66 Find the marginal relative frequency for the number of animals which came in and needed blood work?

Slide 180 (Answer) / 241 66 Find the marginal relative frequency for the number of animals which came in and needed blood work? Answer 0.72 [This object is a pull tab]

Slide 181 / 241 Two-Way Frequency Tables From these frequencies, you can also find a useful comparison called Conditional Relative Frequency which is directly correlated to Conditional Probability. To find Conditional Relative Frequency, divide the joint relative frequency by the appropriate marginal relative frequency. 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. 0.17 0.30 57%

Slide 182 / 241 Conditional Relative Frequency and Conditional Probability Conditional Relative Frequency and Conditional Probability go hand in hand. In fact how statistics are reported usually involves some probability.

Slide 183 / 241 Two-Way Frequency Tables Using the table, find the probability that if a pet was brought into the clinic that needed blood work, it would be a dog.

Slide 183 (Answer) / 241 Two-Way Frequency Tables Using the table, find the probability that if a pet was brought into the clinic 0.47that needed blood work, it would be a dog. 0.60 = 78% Answer This means that there is a 78% chance that your dog would need blood work if you brought it in. [This object is a pull tab]

Slide 184 / 241 Two-Way Frequency Tables Using the table, find the probability that if you brought in a cat, it would NOT need blood work?

Slide 184 (Answer) / 241 Two-Way Frequency Tables Answer Using the table, find the probability that 0.17if you brought in a cat, it would NOT need blood 0.40 work? = 43% [This object is a pull tab]

Slide 185 / 241 67 From the table, find the probability that a girl has gone to an amusement park.

Slide 185 (Answer) / 241 67 From the table, find the probability that a girl has gone to an amusement park. Answer 28% [This object is a pull tab]

Slide 186 / 241 68 Find the conditional probability that out of the girls, the person has been to an amusement park.

Slide 186 (Answer) / 241 68 Find the conditional probability that out of the girls, the person has been to an amusement park. Answer 0.28 0.39 = 72% [This object is a pull tab]

Slide 187 / 241 69 What is the probability that if a person has been to an amusement park, it was a boy?

Slide 187 (Answer) / 241 69 What is the probability that if a person has been to an amusement park, it was a boy? Answer 61% [This object is a pull tab]

Slide 188 / 241 70 Find the probability that out of the people that have not gone to an amusement park, it would be a girl.

Slide 188 (Answer) / 241 70 Find the probability that out of the people that have not gone to an amusement park, it would be a girl. Answer 39% [This object is a pull tab]

Slide 189 / 241 Two-Way Frequency Tables Information summarized like this can easily be analyzed when studying certain situations. At the same vet clinic during July, 42 of the same dogs that came in needed an x-ray. 10 of the cats needed an x-ray. Create a frequency table that dispays this information. Find joint and marginal relative frequencies.

Slide 189 (Answer) / 241 Two-Way Frequency Tables Information summarized like this can easily be analyzed when studying certain situations. Answer 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. [This object is a pull tab]

Slide 190 / 241 Two-Way Frequency Tables Find the probability that: a) if you brought in a dog, it would need an x-ray, b) if you brought in a cat, it would need an x-ray.

Slide 190 (Answer) / 241 Two-Way Frequency Tables 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. 0.70 = 40% Cat and x-ray = 0.07 = 23% 0.30 Answer [This object is a pull tab]

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

Slide 191 (Answer) / 241 Two-Way Frequency Tables Answer X-rays that were dogs = X-rays that were cats = 0.28 0.35 = 80% 0.07 0.35 = 20% Out of all of the animals x-rayed, calculate the percentages that were a) dogs and b) cats. [This object is a pull tab]

Slide 192 / 241 Two-Way Frequency Tables Using the information from both tables, what trends can you find in the data? Use quantitative statements to justify your answers.

Slide 192 (Answer) / 241 Two-Way Frequency Tables Using the information from both tables, what trends can you find in the data? One Use possible quantitative example: statements During to justify your answers. July, dogs need more medical attention. 80% of the x-rays taken and 78% of the blood work was from dogs. Answer [This object is a pull tab]

Slide 193 / 241 Two-Way Frequency Tables At USA High School, 300 seniors went on to a 4-year college or university. A survey collected the following data on whether they chose an in-state or an out-of-state school. Use this information to answer the following questions.

Slide 194 / 241 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 (Answer) / 241 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. Answer 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. A [This object is a pull tab]

Slide 195 / 241 72 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 (Answer) / 241 72 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. Answer 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 [This object is a pull tab] college or university. C

Slide 196 / 241 73 The marginal relative frequency of in-state students is: A 0.33 B 0.78 C 0.45 D 0.22

Slide 196 (Answer) / 241 73 The marginal relative frequency of in-state students is: A 0.33 B 0.78 C 0.45 D 0.22 Answer B [This object is a pull tab]

Slide 197 / 241 74 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 (Answer) / 241 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 Answer [This object is a pull tab]

Slide 198 / 241 Sampling and Experiments Return to Table of Contents

Slide 199 / 241 Goals and Objectives Students will be able to recognize appropriate uses and models for statistics, justify their results using data or experimentation, and calculate a margin of error for sets of information.

Slide 200 / 241 Why do we need this? Everyone needs to learn appropriate ways to interpret statistical analyses. Just because someone comes up with a survey and publicizes their results, does not mean that the survey has validity. In today's society, we need to have educated opinions and to question what we are told in the media.

Slide 201 / 241 Sampling 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.

Slide 202 / 241 Sampling 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.

Slide 203 / 241 Sampling There are several aspects of sampling that deserve attention: 1) randomization and bias, 2) sample size, and 3) margin of error.

Slide 204 / 241 75 Which of the following samples would most accurately represent the way people would vote on lowering the drinking age to 18? A Polling 100 random students at all college campuses. 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.

Slide 204 (Answer) / 241 75 Which of the following samples would most accurately represent the way people would vote on lowering the drinking age to 18? A Polling 100 random students at all college campuses. B Asking 10 mothers 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. [This object is a pull tab] Answer D

Slide 205 / 241 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 (Answer) / 241 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. Answer Depending on the goal of the survey, all answers may be viable. Discuss what company would value each sample. [This object is a pull tab]

Slide 206 / 241 Sampling As in the last examples, different samples may get different results. Knowing the purpose of the sampling is very important. If the sample size is too small, if it is not randomized or if the method of obtaining samples is not well thought out, you will get biased results.

Slide 207 / 241 Sampling - Bias 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.

Slide 208 / 241 Sampling - Sample Size Once a sampling method has been well thought out and proven not to be biased, one must consider sample size. As a rule, small sample sizes will result in a large variation while larger sample sizes result in less variation.

Slide 209 / 241 Sampling - Example 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.

Slide 209 (Answer) / 241 Sampling - Example For example: You are a quality control manager at an ice cream factory. Out of the 4,000 gallons of ice cream they produce in one day, you 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 [This object is a pull tab]

Slide 210 / 241 Margin of Error One way businesses and organizations calculate the answer to "what is sufficient" is to decide on the margin of error that they want to be within. We have all seen margins of error reported in polls. Although, they are usually an add-on at the end. **http://polltracker.talkingpointsmemo.com/polls/5097408bebcabf0e3e00010a

Slide 211 / 241 Margin of Error What this means is that Obama actually had a range of votes from 47.8% to 56.2% and Romney had a range that was from 36.8% to 45.2%. If the numbers were looked at a bit differently, it could be a much closer race and lead to reports such as this: **http://www.washingtonpost.com/blogs/the-fix/wp/2012/10/28/minnesota-poll-shows-romney-within-margin-of-error/

Slide 212 / 241 Margin of Error **The margin of error represents an interval that would contain the true population parameter and usually has a 95% confidence level which is two standard deviations. In its simplest form, we can use the margin of error to calculate a sample size as well as use the sample size to calculate the margin of error. This is generally used for surveys that are going to be conducted in the future. To do this, use the formula: M = margin of error n = sample size

Slide 213 / 241 Margin of Error At the ice cream factory, calculate the margin of error if you used a sample of 4 gallons out of the 4000. What is the margin of error if you used a sample of 400?

Slide 213 (Answer) / 241 Margin of Error 4 gallon sample: 400 gallon sample: Answer At the ice cream factory, calculate the margin of error if you used a sample of 4 gallons out of the 4000. What is the margin of error if you used a sample of 400? [This object is a pull tab]

Slide 214 / 241 Margin of Error 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?

Slide 214 (Answer) / 241 Margin of Error Answer 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? [This object is a pull tab]

Slide 215 / 241 77 What is the margin of error for a sample size of 30?

Slide 215 (Answer) / 241 77 What is the margin of error for a sample size of 30? Answer ±18% [This object is a pull tab]

Slide 216 / 241 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% - 27.2% C 3.2% - 24% D 24% - 27.2%

Slide 216 (Answer) / 241 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% - 27.2% C 3.2% - 24% D 24% - 27.2% Answer B [This object is a pull tab]

Slide 217 / 241 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%

Slide 217 (Answer) / 241 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% Answer 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%. [This object is a pull tab]

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

Slide 218 (Answer) / 241 80 Find the sample size needed to achieve a margin of error of ±1%. Answer 10,000 [This object is a pull tab]

Slide 219 / 241 81 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 (Answer) / 241 81 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)? Answer The interval of confidence is 1.75% to 2.25%. Therefore, that 2.25% of 5,000, which is 112. [This object is a pull tab]

Slide 220 / 241 Margin of Error The margin of error calculated by the formula is a very simplified, general method. It will give you the largest possible margin of error and is a good estimate of the numbers you are looking for, but is not as accurate as it could be. The following formulas are used to calculate margin of error a bit more accurately. We will use only the one above and the second below. p = proportion in a decimal n = sample size σ = standard deviation n = sample size

Slide 221 / 241 Margin of Error 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?

Slide 221 (Answer) / 241 Margin of Error 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 [This object is a pull tab]

Slide 222 / 241 Margin of Error Since the sample size was too small to support the claim, we can use a simulation model to find a margin of error. Then, if the original claim falls inside of the confidence interval, we can support that claim.

Slide 223 / 241 Margin of Error Margins of error can also be used to decide if a results of particular experiment are relevant. For example, if 45% of a sample population voted for Jane Doe with a margin of error of ±3%, you could predict that a second or third survey would return results that are in the confidence interval of 42% to 48%. To generalize, we can make a claim (or hypothesis) about a particular event by taking a survey and computing results. From those results, we can make further claims that can be proven or disproved based on the results falling within the original confidence interval. If the expectation a particular hypothesis does not fall within the interval, the hypothesis could be rejected.

Slide 224 / 241 Sampling and Experiments Lab 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

Slide 224 (Answer) / 241 Sampling and Experiments Lab Simulations are used if studies are not cost effective, tests are not ethical or if the calculating the probability is too difficult. Let's perform a simulation now. Click on the lab link below to get started. 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 [This object is a pull tab]

Slide 225 / 241 Sampling and Experiments Lab Lab: Teacher Slides - Part 1: Flipping a Coin Take out a coin or get one from your teacher. Everyone flip the coin 10 times and record whether you get heads or tails. Write this information on the board in the table below. Name(s) Heads tally Tails tally

Slide 226 / 241 Sampling and Experiments Lab Mini-Lab: Teacher Slides - Part 1: Flipping a Coin How many heads did the class get? How many tails? What is your experimental probability for each? What is the longest streak of heads or tails?

Slide 227 / 241 Sampling and Experiments Lab Mini-Lab: Teacher Slides - Part 2: Simulation of Flipping a Coin using a Graphing Calculator The most efficient way of doing simulations is with a calculator or a computer. Let's "flip a coin" again, but use our calculator. 1. Make the following selections on your calculator. Math 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.

Slide 228 / 241 Sampling and Experiments Lab Mini-Lab: Teacher Slides - Part 2: Simulation of Flipping a Coin using a Graphing Calculator 3. Quit out of your List by pressing 2nd Quit 4. To flip the coin 10 times, use randint(0, 1, 10). To flip it another 10 times hit 2nd Enter. 5. Write down the results that you found for 2nd round of flipping the coins in the space below.

Slide 229 / 241 Calculator Simulations Go back to one of our earlier problems... Someone has made a claim that 45% of the students at USA High School have Smart Phones. A student from that school took a survey during one of her classes and found that, out of an English class of 30 students, 11 students had a Smart Phone, which is approximately 37% of her class. Mathematically prove that this original claim is true. A sample size of 30 is way too small to make a decision, so let's use simulations to develop a mean and a margin of error for this problem. Get out your calculator again.

Slide 230 / 241 Calculator Simulations Since 37% of the students in the class had a Smart Phone, assign the numbers 1 to 37 as students having a Smart Phone. Therefore, 38 to 100 will represent students not having a Smart Phone. In your calculator, do randint(1, 100, 100). Store it in L 1. Stat Calc 1-Var Stats Enter x = mean σx = standard deviation Q 1 = 1st Quartile Med = Median Q 3 = 3rd Quartile Compare everyone's mean!

Slide 231 / 241 Calculator Simulations 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 30.14. There were 100 random integers generated, so n = 100. Calculate the Confidence interval.

Slide 231 (Answer) / 241 Calculator Simulations Our calculator gives us enough information that we can easily calculate a more accurate margin of error. From your data, find the confidence interval. Use: Answer 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 30.14. There [This object were is a pull 100 tab] random integers generated, so n = 100. Calculate the Confidence interval.