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

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Algebra II

Probability and Statistics

2016-01-15 www.njctl.org

Slide 3 / 241 Table of Contents

Sets Independence and Conditional Probability Measures of Central Tendency Two-Way Frequency Tables Standard Deviation and Normal Distribution Sampling and Experiments

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Permutations & Combinations

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Sets

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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}

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A = {0, 2, 3, 7, 9} B = {1, 3, 7, 10} U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Create a Venn Diagram to match the information.

U A B 1 2 3 4 5 6 7 8 9 10

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Venn Diagrams are one example of a sample space that helps us

• rganize 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

T

H T

H T H T

H T H T H T

1 2 3 4 5 6 6 5 4 3 2 1 2 3 4 5 6 3 4 4 5 5 5 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 9 9 9 9 10 10 10 11 11 12

Data Displays Slide 10 / 241

Use a sample space that helps organize the data effectively. For example, would you be able to effectively display a coin toss in a Venn Diagram or on a chart? Decide how to display the following information.

• 1. Survey results about what subject students like in school.
• 2. The different ways you can deal two cards from a deck of cards.
• 3. Results that compare the number of men and women that like

chocolate ice cream over vanilla ice cream.

• 4. A poll on which grocery store people prefer to go to.

Data Displays Slide 11 / 241

The Universe (U) is all aspects that should be considered in a

• situation. The Universe (U) is basically the same as a sample

space also used in probability. Name the Universe (U) of the following:

• 1. Survey at a local college asking women what they are

studying.

• 2. Calculating the probability that you would draw a red

10 out of a deck of cards.

• 3. Phone survey on who you will vote for in the U. S.

Presidential race.

The Universe Slide 12 / 241

The Empty Set (∅ ) is the equivalent of zero when referring to sets. For example, if you asked people at a college their age, the number

• f 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.

Empty Set

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• 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.

U A B 4

• 12

7 3

• 3

1 17 5

• 2

15

• 1

6 C

Example 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

7pm weight lifting 6am aerobics 4pm water aerobics 10am weight lifting 5pm cycling 3pm nutrition 6pm swimming 2pm climbing

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2 What is the most popular activity, or activities, at the club?

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

Men Women

7pm weight lifting 6am aerobics 4pm water aerobics 10am weight lifting 5pm cycling 3pm nutrition 6pm swimming 2pm climbing *Answer as many letters as necessary.

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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 Women

7pm weight lifting 6am aerobics 4pm water aerobics 10am weight lifting 5pm cycling 3pm nutrition 6pm swimming 2pm climbing

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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 C S = {1, 3, 5} D E

1 2 3 4 5 6

# 1 2 3 4 5 6

1 2 3 4 5 6

Answer

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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 B C

3 6 7 4 8 9 2 5 10 11 12 1

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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 B C

3 6 7 4 8 9 2 5 10 11 12 1

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A B C

3 6 7 4 8 9 2 5 10 11 12 1

Unions (U) of two or more sets creates a set that includes everything in each set. 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")

Unions Slide 21 / 241

A B C

3 6 7 4 8 9 2 5 10 11 12 1

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")

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".

Intersections Slide 22 / 241

A B C

3 6 7 4 8 9 2 5 10 11 12 1

Unions (U) and Intersections (∩) are often combined. Find:

• 1. (A U C) ∩ B
• 2. A ∩ B ∩ C
• 3. (A ∩ C) U (B ∩ C)

**Shade the diagram as you go to help.

Unions and Intersections Slide 23 / 241

One last aspect of sets for this unit are Complements. Complements of a set are all elements of the Universe that are NOT in the set. There are several ways to denote a complement: ~A, Ac, A' and not A In this unit, we will use "~A" or "not A" If U = {0, 1, 2, 3, 4, 5, 6} and A = {0, 1, 2, 3}, then the complement of A is {4, 5, 6}

Complements Slide 24 / 241

• 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.

Examples

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You can also combine Complements with Intersections and Unions. A B C

3 6 7 4 8 9 2 5 10 11 12 1

Find:

• 1. (A ∩ C) U ~B
• 2. (A U B) ∩ ~C
• 3. C ∩ B U ~A
• 4. ~A U ~B

**Shade the diagram as you go to help.

Examples 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}

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 U A B C

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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}

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 U A B C

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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}

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 U A B C

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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

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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

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Independence and Conditional Probability

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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.

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You can also relate this to everyday situations:

• 1. Are you independent of, or dependent on, your parents and

guardians right now?

• 2. True or false: Smoking causes lung cancer. Is this a

dependent or an independent event?

• 3. Is how you do on a test based on how others study?

Independence and Conditional Probability Slide 36 / 241

Since many of these situations are based on specific circumstances, we can use probability to study them. The 45 year old female driving a Camaro may have a terrible driving

• record. Therefore, what she pays for insurance will be dependent
• n 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.

Independence and Conditional Probability

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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

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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

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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 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 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) ≠ ∅ .

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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)

Addition Law of Probability Slide 44 / 241

P(A U B) = P(A) + P(B) P(A U B) = P(A) + P(B) - P(A ∩ B) Mutually Exclusive Overlapping Find the probability that you roll a 6 on a green number cube or a 3 on a red number cube. Find the probability that you draw a face card or a red card.

Independence and Conditional Probability 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%

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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

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17 Using the Venn Diagram, how many people like to ski or

ride snowmobiles? A 15 B 45 C 69 D 89

People that like to ski. People that like to ride snowmobiles.

30 15 44

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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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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

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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). These events are only independent if: To calculate conditional probability, use:

Conditional Probability 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 Dependent Two cards are drawn one at a time, and are replaced. What is the probability of drawing two Aces? 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

Conditional Probability Slide 52 / 241

In Venn Diagrams, obviously P(A ∩ B) is the intersection of A and B. Use numbers from the diagram for calculations. A B 20% 10% 50% P(A ∩ B) P(B) = 60% P(A) = 30%

**Add probabilities in all of A to get P(A) and all of B to get P(B).

Venn Diagrams Slide 53 / 241

A bag contains plastic disks with numbers on them. You must choose two disks, one at a time without replacing them. The probability that the first disk is odd and the second disk even is 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?

Example Slide 54 / 241

Students that take music Students that take math

0.25 0.3 0.45

Using the Venn Diagram, find the probability that a student is taking music given that they are taking math.

Example

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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.

Example Slide 56 / 241

To decide if the events in a conditional probability situation are independent, use the following formula:

Formula Slide 57 / 241

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

Example with Formula Slide 58 / 241

On the other hand, think about rolling dice. Use the formula to decide if rolling a 5, and then a 6 is independent.

Example with Formula Slide 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

• wns a snowboard given that they already own skis.

A 25% B 38% C 65% D 78%

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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%

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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 Preferences of activities at a local gym. 30% 10% 35% 25%

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23 Calculate the percentage of people that like yoga, given

that they enjoy weightlifting. A 10% B 25% C 30% D 33%

Weightlifting Yoga Preferences of activities at a local gym. 30% 10% 35% 25%

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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 Preferences of activities at a local gym. 30% 10% 35% 25%

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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%

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Permutations & Combinations

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Slide 66 / 241

Students will be able to calculate the number of possible outcomes using the fundamental counting principle, permutation formula and combination formula. Also, students will be able to calculate the probability of an event occurring when the permutation and combination formulas are involved.

Goals and Objectives

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

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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 ,

• r 12 possible outcomes.

Fundamental Counting Principle Slide 69 / 241

Example: A manager assigns different codes to all the tables in a restaurant to make it easier for the wait staff to identify them. Each code consists

• f a vowel, A, E, I, O or U, followed by 2 digits from 0 through 9.

How many codes could the manager assign using this method?

Fundamental Counting Principle Slide 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

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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

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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

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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

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Factorial: n! means the product of all counting numbers beginning w/ n & counting backwards to 1. 0! has a value of 1. Permutation: an arrangement or listing of objects when no repetition is allowed and order matters. Example: 4! = 4 x 3 x 2 x 1 = 24 Formula for finding the number of permutations of n objects taken r at a time

nPr = n!

(n - r)!

Permutations

Example: ABC and ACB are different permutations of the letters A, B, and C

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Example: How many ways can you arrange the letters A, B, C and D?

Permutations 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

12P3 = 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 12 x 11 x 10 = 1,320 ways

click click click click

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30 In how many ways can the letters in the word

"WEIGHT" be arranged? A 60 B 120 C 720 D 46,656

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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

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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

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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

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Combination: an arrangement or listing of objects when no repetition is allowed and order does not matter.

Combinations

Formula for finding the number of combinations of n objects taken r at a time

nCr = n!

(n - r)! r! 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

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

9C4 = 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 = 126 fruit salads

click click click

9 x 8 x 7 x 6 4 x 3 x 2 x 1

click

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).

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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

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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

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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

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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 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

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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 90 / 241 Probability Involving Permutations & Combinations

Some questions will ask you to calculate the probability of an event,

• r multiple events, that use the counting techniques of permutations

& combinations. When this occurs, calculate the number of

• utcomes 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 How many outcomes are possible in the sample space? 5 x 4 x 3 x 2 x 1 = 5! = 120 outcomes

click 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. What is the probability that the number is between 20,000 & 30,000? 24/120 = 1/5 = 20%

click 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 How many hands of cards can be dealt (the sample space)?

52C5 = click

52! (52 - 5)! 5! = 2,598,960 hands

click

Slide 94 / 241 Probability Involving Permutations & Combinations

How many outcomes are possible in the event?

4C2 x 48C3 = 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. What is the probability that you are dealt the pair of Kings? 103,776/2,598,960 = 3.99%

click click

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?

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40 A committee of 3 students is to be chosen from a group

• f 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

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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

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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

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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%

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Measures of Central Tendency

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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

• f 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

Example: Find the mean, median and mode of the following set of test scores: 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96

Review: Example Slide 104 / 241

Range: The difference between the highest and the lowest numbers in the set of data. Outliers: Numbers that are significantly larger or smaller than the rest of the numbers.

Review Slide 105 / 241

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

Review: Example Slide 106 / 241

The Spread of a set of data is used to describe the variability of the

• information. This looks at how different the numbers are.

Interquartile Range is the difference of the value of quartile 3 and quartile 1. *We will review quartiles in the next slide. Standard Deviation is a measure of how close all of the data is to the mean.

Review Slide 107 / 241

Remember making box plots in Algebra 1?

Quartile 2 Median of data Quartile 3 Median of upper half of data Quartile 1 Median of lower half of data

lowest number highest number

Interquartile Range Slide 108 / 241

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

Interquartile Range

Slide 109 / 241 Slide 110 / 241 Slide 111 / 241

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

Standard Deviation Slide 112 / 241

What do you think would happen to the standard deviation if we eliminated the outlier of 45?

Standard Deviation Slide 113 / 241

Find the standard deviation of the following set of numbers: 6.7, 7.1, 6.5, 7.2, 6.23, 6.9

Standard Deviation Slide 114 / 241

Discuss the standard deviations of both sets that we just

• calculated. How do each reflect the spread of the data?

6.7, 7.1, 6.5, 7.2, 6.23, 6.9 78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96

Standard Deviation

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 116 / 241

45 Find the Standard Deviation of the following set of

numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48

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 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 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 120 / 241

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

Calculators

Slide 121 / 241

Input the following sets into your calculator: L1: 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 L2: 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 L2, go to STAT

and then 1: Edit.

Calculators Slide 122 / 241

Now, calculate the Standard Deviation of each set.

• 1. Go to calculation screen ( 2nd , Quit ).
• 2. Push 2nd , Stat .
• 3. Go to Math.
• 4. Find 7: stdDev().
• 5. Type stdDev(L1) and then stdDev(L2).

Calculators Slide 123 / 241

For L1, the standard deviation is 0.354. With L2, the standard deviation is 3.450. Why is there such a large difference between the two numbers? What does it say about the data? L1: 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 L2: 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

Calculators Slide 124 / 241

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

Standard Deviation 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 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 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 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

Check out the following graphs. What do they have in common?

http://www.trialsjournal.com/content/6/1/5 http://www.statcrunch.com/grabimage.php?image_id=427473

Calories in French Fries Diastolic Blood Pressure Intervals of Peaks of Heartbeats

http://www.swharden.com/blog/2009-08-14-diy-ecg-machine-on-the-cheap/

Graphs Slide 132 / 241

Many different aspects of life, when measured and graphed, fit this type of distribution. Imagine a what the graph of height for humans, weight for bears or size of homes would look like. Most

• f the data would be around the same number (the mean), yet

there would be some that would be larger or smaller. Finally, you would have the extremes that would be rare. This is called a Normal Distribution.

Normal Distribution

Slide 133 / 241

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

Normal Distribution Slide 134 / 241

A tennis ball manufacturer measures the height their tennis balls bounce after dropping them from 5 feet off of the

• ground. The balls will not bounce the same height each

time, but should be very close. A graph of this, after many trials, would begin to resemble a normal distribution. From here, you can calculate a mean height of the ball and use that to test other tennis balls from the factory to make sure that the quality is consistent. The blue shaded area would represent the range of acceptable heights.

Normal Distribution Slide 135 / 241

http://www.nohsteachers.info/PCaso/AP_Statistics/MidtermExamReview.htm

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

Normal Distribution Slide 136 / 241

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

Normal Distribution Slide 137 / 241

Normal Curves are created using the mean (μ or x) of the data and the standard deviation (σ). mean

Normal Curve Slide 138 / 241

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

1 2 3

• 1
• 2
• 3

Normal Distribution

Slide 139 / 241

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

Normal Distribution Slide 140 / 241

Mean

http://rchsbowman.wordpress.com/2009/11/29/statistics-notes-%E2%80%93-properties-of-normal-distribution-2/

Normal Distribution Slide 141 / 241

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

Normal Distribution Slide 142 / 241

Use this chart to answer the following questions.

http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm

Chart for Examples Slide 143 / 241

a) John usually scores an average of 82% on his math tests with a standard deviation of 5%. What is the probability that John will get an between an 82% and an 87% on his next test? b) At Big Mama's Gym, there is a special weight loss program that is a big hit. And, it works! At the start

• f the program 95.4% of all members, centered

about the mean, weighed between 180 and 260

• pounds. Find the average weight and the standard

deviation of the data.

Examples Slide 144 / 241

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

• unces?

Example

Slide 145 / 241

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

Example Slide 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 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

• f 0.6 years. How many students are younger than 16

years? Express your answer to the nearest student.

Slide 148 / 241

85 91 73 67 97 103 79

54 Which of the following curves represents a mean of 85

and a standard deviation of 6? A B C D

85 86 84 83 82 87 88 85 90 95 100 70 80 75 85 56 66 76 106 96 86

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 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 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 152 / 241

The graph we have been using to the right helps us find values that are multiples of 0.5 away from the

• mean. But what about numbers that

are in between? For those, we use a formula for the z-score and a table

• f values.

Z-Score Slide 153 / 241

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

Z-Score Slide 154 / 241 Z-Scores: Negative Slide 155 / 241 Z-Scores: Positive Slide 156 / 241

Z-scores are what is used to calculate all of the percentile values that are reported for standardized tests. Remember how you are given a result of, say, the 94th percentile? This means that you have done better than 94% of the students who have taken the test. Welcome to a major use of z- scores, normal distribution and standard deviation! z-score =

Z-Score

Slide 157 / 241

z-score = Example: On a test, your score was 83%. The mean of all

• f 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 158 / 241

Your friend took the same test and got a score of 92%. Find your friend's z-score and calculate their percentile.

Z-Score Slide 159 / 241

58 Find the z-score for a 29 if the mean was 34 and the

standard deviation is 2.3.

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 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 162 / 241

61 A student calculated a z-score of -1.25. What percentile

does this score fall in?

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 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...

Stem-and-Leaf Plot Box-and-Whisker Plot Frequency Table

llll llll llll l llll llll lll llll l llll lll lll

Ages of people at the gym 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 168 / 241

Slide 169 / 241

Line Plots Scatter Plots

Remember from Algebra 1... Slide 170 / 241

In this section, we are going to study Two-Way Frequency Tables. These displays allow us to study situations that have more than

• ne variable such as how many men and women that exercise
• regularly. The chart below shows a survey of 100 people.

Two-Way Frequency Tables Slide 171 / 241

Two-Way Frequency Tables connect the collection

• f 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

Two-Way Frequency Tables Slide 172 / 241

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. The yellow boxes represent Joint Relative Frequency and the pink boxes represent Marginal Relative Frequency.

Two-Way Frequency Tables Slide 173 / 241

These relative frequencies directly translate into quantitative

• statements. Such statements mirror those that are reported in the

media. · 18% of the men surveyed exercise regularly. · 22% of the women surveyed did not exercise regularly. · 54% of the people surveyed were women.

Two-Way Frequency Tables Slide 174 / 241

A teacher asked their class if they had been to an amusement park before or not. Out of 36 students, there were 22 boys and 14 girls. 16 of the boys and 10 of the girls answered that they had been to an amusement park before. Create a relative frequency table from the data collected.

Two-Way Frequency Tables

Slide 175 / 241

Together, write some quantitative statements about the information.

Two-Way Frequency Tables Slide 176 / 241

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

Two-Way Frequency Tables 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 178 / 241

64 What is the marginal relative frequency of cats that came

to the clinic?

Slide 179 / 241

65 What is the percentage of dogs that came in that needed

blood work?

Slide 180 / 241

66 Find the marginal relative frequency for the number of

animals which came in and needed blood work?

Slide 181 / 241

From these frequencies, you can also find a useful comparison called Conditional Relative Frequency which is directly correlated to Conditional Probability. To find Conditional Relative Frequency, divide the joint relative frequency by the appropriate marginal relative frequency. For example, use the table to find the probability that if a cat was brought in to the clinic, it would not need blood work. 0.17 0.30

Cats that did not need blood work. Cats that came in.

57%

Two-Way Frequency Tables Slide 182 / 241

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

Conditional Relative Frequency and Conditional Probability Slide 183 / 241

Using the table, find the probability that if a pet was brought into the clinic that needed blood work, it would be a dog.

Two-Way Frequency Tables Slide 184 / 241

Using the table, find the probability that if you brought in a cat, it would NOT need blood work?

Two-Way Frequency Tables Slide 185 / 241

67 From the table, find the probability that a girl has gone to

an amusement park.

Slide 186 / 241

68 Find the conditional probability that out of the girls, the

person has been to an amusement park.

Slide 187 / 241

69 What is the probability that if a person has been to an

amusement park, it was a boy?

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 189 / 241

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

Two-Way Frequency Tables Slide 190 / 241

Find the probability that: a) if you brought in a dog, it would need an x-ray, b) if you brought in a cat, it would need an x-ray.

Two-Way Frequency Tables Slide 191 / 241

Out of all of the animals x-rayed, calculate the percentages that were a) dogs and b) cats.

Two-Way Frequency Tables Slide 192 / 241

Using the information from both tables, what trends can you find in the data? Use quantitative statements to justify your answers.

Two-Way Frequency Tables

Slide 193 / 241

At USA High School, 300 seniors went on to a 4-year college or university. A survey collected the following data

• n whether they chose an in-state or an out-of-state
• school. Use this information to answer the following

questions.

Two-Way Frequency Tables 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

• r university are female.

C 73% of females chose an in-state college or university.

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 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 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 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 is a method of getting information about a large population without having to test or ask each element of the population. How many of you have gotten a phone call requesting that you answer survey questions? Such sampling allows the company or agency to get an idea of what people think or, especially, how they will vote.

Sampling Slide 202 / 241

By choosing a certain number of elements to be a sample, you can efficiently gather results and make a quantitative statement about the entire population. This method is used in many different

• situations. Some examples include:

a) quality control in a parts factory or in food production, b) experimentation with different medical treatments, and c) predicting who or what people vote for.

Sampling Slide 203 / 241

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

Sampling 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 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 14. D Question 2000 fishermen at a fishing convention.

Slide 206 / 241

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

Sampling Slide 207 / 241

Bias comes from how a question is asked as well as who is being asked. Surveys or statistics that are biased do not return valid results. For example, if you ask men at an electrician's convention which purse they prefer, would you get valid answers? It is important for questions or surveys to be unbiased. That way, the results mean something.

Sampling - Bias Slide 208 / 241

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

Sampling - Sample Size Slide 209 / 241

For example: You are a quality control manager at an ice cream factory. Out of the 4,000 gallons of ice cream they produce in one day, you choose 4 of those gallons to pull off

• f the line to check for quality.

a) Is this enough? b) If not, decide on a range of values that would be sufficient.

Sampling - Example Slide 210 / 241

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

Margin of Error

Slide 211 / 241

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/

Margin of Error Slide 212 / 241

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

Margin of Error Slide 213 / 241

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

Margin of Error Slide 214 / 241

Likewise, you can use a desired margin of error to find your sample size. Say that you wanted to have a margin of error of 3% when testing quality of the ice cream. Calculate the sample size: Is your answer reasonable in this situation?

Margin of Error Slide 215 / 241

77 What is the margin of error for a sample size of 30?

Slide 216 / 241

78 What is the actual range on a survey that reported 24%

• f 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 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 218 / 241

80 Find the sample size needed to achieve a margin of error

• f ±1%.

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 220 / 241

The margin of error calculated by the formula is a very simplified, general method. It will give you the largest possible margin of error and is a good estimate of the numbers you are looking for, but is not as accurate as it could be. The following formulas are used to calculate margin of error a bit more

• accurately. We will use only the one above and the second below.

p = proportion in a decimal n = sample size σ = standard deviation n = sample size

Margin of Error Slide 221 / 241

Margins of error can also be calculated via simulation models for random sampling. Someone has made a claim that 45% of the students at USA High School have Smart Phones. A student from that school took a survey during one of her classes and found that out of an English class of 30 students, 11 students had a Smart Phone, which is approximately 37% of her class. Can this original claim be true?

Margin of Error Slide 222 / 241

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

Margin of Error

Slide 223 / 241

Margins of error can also be used to decide if a results of particular experiment are relevant. For example, if 45% of a sample population voted for Jane Doe with a margin of error of ±3%, you could predict that a second or third survey would return results that are in the confidence interval

• f 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.

Margin of Error Slide 224 / 241

Simulations are used if studies are not cost effective, tests are not ethical or if the calculating the probability is too difficult. Let's perform a simulation now. Click on the lab link below to get started. Lab - Sampling and Experiments

Sampling and Experiments Lab Slide 225 / 241

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

• n the board in the table below.

Name(s) Heads tally Tails tally

Sampling and Experiments Lab Slide 226 / 241

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

Sampling and Experiments Lab Slide 227 / 241

The most efficient way of doing simulations is with a calculator

• r a computer. Let's "flip a coin" again, but use our calculator.

Math PRB randInt(beginning value, ending value, how many times) Sto 2nd L1

• 2. If you want to store it in your calculator,

To view the list that you stored, press Stat Edit To flip the coin 10 times, use randint(0, 1, 10). Enter this into your calculator & press "Enter". Mini-Lab: Teacher Slides - Part 2: Simulation of Flipping a Coin using a Graphing Calculator

• 1. Make the following selections on your calculator.

Enter your list into the table on the Lab WS.

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• 4. To flip the coin 10 times, use randint(0, 1, 10). To flip it another

10 times hit 2nd Enter. 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

• 5. Write down the results that you found for 2nd round of flipping the

coins in the space below.

Sampling and Experiments Lab

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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

• riginal claim is true.

Go back to one of our earlier problems... 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.

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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 L1. Stat Calc 1-Var Stats Enter x = mean σx = standard deviation Q1 = 1st Quartile Med = Median Q3 = 3rd Quartile Compare everyone's mean!

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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

• f 53.5 and a standard deviation of 30.14. There were 100

random integers generated, so n = 100. Calculate the Confidence interval.

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Now, this is just one simulation. Does it match what everyone in the class got? To be more accurate, you repeat your simulation several times and generate a mean of the means and a mean of the standard deviation. Our class just repeated the simulation several times as we all have random numbers that were generated. Let's calculate a class mean and class standard deviation to then find a more accurate interval.

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Finally, does the claim that 45% of student fall within your class's margin of error? If so, it can be verified. If not, the claim may not be valid.

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Try it again... A mom's local MADD (Mother's Against Drunk Driving) group stated that more than 30% of today's students have drank and drove in the last month. A group of students believe that number is

• verblown and way too high.

In school, a teacher surveyed her class and found out that 5% of her students anonymously admitted to drinking and driving in the last month. Create and execute a simulation 10 times to prove or disprove MADD's claim. Steps are on the next page...

Calculator Simulations

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• 1. Decide on parameters and input into calculator.

randint(beginning value, ending value, how many times) Store in L1.

• 2. Find mean and σ. Record.
• 3. Repeat 9 more times.
• 4. Find the average mean and average standard deviation.
• 5. Use to find the margin of error.
• 6. Generate the confidence interval.
• 7. Determine if the amount in question falls in your interval.

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This method is also useful in determining if there is a true difference between claims such as in treatments or products. To do this, use the difference in claims, develop a mean and a margin of error from simulations and then decide if the difference of Zero (0) falls within your confidence interval. What does a difference of 0 signify?

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For example... Two different different dog foods claim to make a dog's teeth and gums healthier. Dog Food A claims that 25% of the dog's tested had better dental health and Dog Food B claimed that 34% of the dogs had better dental health. Currently, Dog Food B is advertising itself as the top brand of food. Create and execute a simulation to prove or disprove the claim.

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This same type of simulation can be used to estimate how long it will take to collect a certain amount of objects. For example, Johnny Jim's, a sandwich shop is giving out 5 different cards every time someone orders a Macho Sandwich. Once you collect all 5 cards, you get a free sandwich. Create and execute a simulation that gives you an interval for how many sandwiches you should expect to buy before getting a free one.

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82 A grocery store claims that Buggy Brand Juice is the best

because it has an 83% satisfaction rate. You decided to test this claim and ran a simulation 40 times to get a mean of 78% with a margin of error of 4.3%. True or False, you just verified their claim. True False

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83 Two new cancer treatments are being tested in a

• laboratory. If there is no true difference in the treatments,

what number will fall in the confidence interval?

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84Forty percent of the staff in a local school district have a master's degree. One of the schools in the district has only 4 teachers out of 15 with a master's degree. You are asked to design a simulation to determine the probability of getting this few teachers with master's degrees in a group this size. Which of the following assignments of the digits 0 through 9 would be appropriate for modeling this situation?

A Assign "0, 1, 2" as having a master's degree and "4, 5, 6, 7, 8, 9" as not having a degree. B Assign "1, 2, 3, 4, 5" as having a master's degree and "0, 6, 7, 8, 9" as not having a degree. C Assign "0, 1" as having a master's degree and "2, 3, 4, 5, 6, 7, 8, 9" as not having a degree. D Assign "0, 1, 2, 3" as having a master's degree and "4, 5, 6, 7, 8, 9" as not having a degree. E Assign "7, 8, 9" as having a master's degree and "0, 1, 2, 3, 4, 5, 6" as not having a degree.

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