Sets Independence and Conditional Probability Measures of Central - - PDF document

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Algebra II

Statistics and Probability

www.njctl.org 2013-04-16

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

click on the topic to go to that section

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Sets

Return to Table of Contents

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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.

Sets

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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.

Sets

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Sets

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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Sets

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

Teacher

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Sets

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: Chart for rolling 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

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Sets

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.

Teacher

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Sets

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.

Teacher

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Sets

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.

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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.

Sets

U

A B

4

• 12

7 3

• 3

1 17 5

• 2

15

• 1

6

C

Teacher

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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}

Sets

Men Women

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

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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

Sets

Men Women

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

Sets

Teacher

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

Sets

Teacher

1 2 3 4 5 6

# 1 2 3 4 5 6

1 2 3 4 5 6

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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

Sets

A B C

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

Teacher

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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

Sets

A B C

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

Teacher

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Sets

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")

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Sets

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")

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Sets

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)

Teacher **Shade the diagram as you go to help.

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Teacher

Sets

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}

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Sets

More 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.

Teacher

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Sets

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. Teacher

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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

Sets

Teacher

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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

Sets

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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

Sets

Teacher

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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

Sets

Teacher

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11 Find ~(A B) A 45 B 30 C 18 D 12

A B 12 18 18 15

Sets

Teacher

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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Independence and Conditional Probability

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.

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Independence and Conditional Probability

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?

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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:

Teacher

Independence and Conditional Probability

• 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?

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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.

Teacher

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.

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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

Independence and Conditional Probability

Teacher

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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

Independence and Conditional Probability

Teacher

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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

Independence and Conditional Probability

Teacher

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Independence and Conditional Probability

Review of general probability

• 1. Find the probability
• f drawing a 5 of

hearts after you have drawn, and not replaced, a 6 of clubs.

• 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.

Teacher

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Independence and Conditional Probability

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 die and drawing a card out of a deck are mutually exclusive. Mutually exclusive events A and B satisfy P(A ∩ B) = # .

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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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Independence and Conditional 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)

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Independence and Conditional Probability

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 die or a 3 on a red die.

Find the probability that you draw a face card or a red card.

Teacher

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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
• ut a white or a yellow?

A 20% B 35% C 43% D 57%

Teacher

Independence and Conditional Probability

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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

Teacher

Independence and Conditional Probability

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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

Teacher

Independence and Conditional Probability

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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%

Teacher

Independence and Conditional Probability

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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 ∩

Teacher

Independence and Conditional Probability

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

Teacher

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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.

Independence and Conditional Probability

Examples:

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?

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Independence and Conditional Probability

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).

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Example:

Independence and Conditional Probability

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?

Teacher

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Example:

Independence and Conditional Probability

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.

Teacher

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Independence and Conditional Probability

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.

Teacher

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Independence and Conditional Probability

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

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Independence and Conditional Probability

Example:

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

Teacher

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

Independence and Conditional Probability

Teacher

Example:

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20 In Colorado, the probability that a person owns skiis is 65% and the probability that they own skiis and a snowboard is 25%. Find the probability that a person owns a snowboard given that they already own skiis. A 25% B 38% C 65% D 78%

Independence and Conditional Probability

Teacher

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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%

Independence and Conditional Probability

Teacher

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22 Given the Venn Diagram, what is the probability that a person enjoys both weightlifing 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)

∩ ∩

Independence and Conditional Probability

Teacher 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%

Independence and Conditional Probability

Teacher 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%

Independence and Conditional Probability

Teacher 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%

Independence and Conditional Probability

Teacher

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Measures of Central Tendency

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Measures of Central Tendency

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.

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Measures of Central Tendency

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.

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Measures of Central Tendency

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.

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Measures of Central Tendency

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

Teacher

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Measures of Central Tendency

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.

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78, 79, 81, 45, 71, 71, 95, 92, 95, 71, 85, 93, 78, 98, 96

Measures of Central Tendency

Find the range and identify any outliers of the following test scores:

Teacher

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Measures of Central Tendency

In addition to these familiar terms, we are going to introduce two important measures of spread:

Interquartile Range and Standard Deviation.

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Measures of Central Tendency

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.

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Measures of Central Tendency

Interquartile Range 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 Teacher

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Measures of Central Tendency

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

Teacher

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Measures of Central Tendency

Standard Deviation looks a bit more complicated, but do not be afraid, it is actually fairly straightforward. This formula is itemized on the next slide

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Measures of Central Tendency

Steps for finding Standard Deviation:

• 1. Find the mean for the set of numbers (xm).
• 2. Subtract the mean from each number to find the difference (xn - xm).
• 3. Square each difference (xn - xm)2.
• 4. Add up the squared numbers.
• 5. Divide by the number of data points (n).
• 6. Finally, take the square root of that number. This is the

standard deviation (σ).

Teacher

**The beauty of standard deviation is that it is in the same units as the original set of data.

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Measures of Central Tendency

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%

Teacher

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Measures of Central Tendency

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

Teacher

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Measures of Central Tendency

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

Teacher

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Discuss the standard deviations of both sets that we just

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

Measures of Central Tendency

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

Teacher

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26 Find the Interquartile Range of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48

Measures of Central Tendency

Teacher

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27 Find the Standard Deviation of the following set of numbers: 36, 37, 50, 22, 25, 26, 36, 36, 49, 48

Measures of Central Tendency

Teacher

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28 Find the Interquartile Range for the following set of data:

Measures of Central Tendency

1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6

Teacher

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29 Find the Standard Deviation of the following set of data:

Measures of Central Tendency

1.3, 4.6, 2.3, 5.7, 2.4, 1.6, 3.4, 2.6

Teacher

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30 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

Measures of Central Tendency

Teacher

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Measures of Central Tendency

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.

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Measures of Central Tendency

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.

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Measures of Central Tendency

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).

Teacher

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Measures of Central Tendency

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

Teacher

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Measures of Central Tendency

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.

Teacher

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31 Using a calculator, find the standard deviation of the following set

• f 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

Measures of Central Tendency

Teacher

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32 Find the standard deviation of the following set of data.

Measures of Central Tendency

12, 7, 14, 16, 25, 17, 8, 20, 20, 30, 12, 14, 17, 17, 25, 6, 6, 22, 25, 26, 24, 16, 17, 19

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33 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

Measures of Central Tendency

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Standard Deviation and Normal Distribution

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Goals and Objectives

Students will be able to calculate the standard deviation of a data set and analyze a a normal distribution.

Standard Deviation and Normal Distribution

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Standard Deviation and Normal Distribution

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.

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Standard Deviation and Normal Distribution

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 Teacher Intervals of Peaks of Heartbeats

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

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Standard Deviation and 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

• 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.

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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...

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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

• f acceptable heights.

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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.

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Standard Deviation and Normal Distribution

Teacher

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

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Standard Deviation and Normal Distribution

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

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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.

Standard Deviation and Normal Distribution

Teacher

mean

1 2 3

• 1
• 2
• 3

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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.

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Teacher

Standard Deviation and Normal Distribution

Mean

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

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Each graph can be used differently even though there is a uniformity about their calculations.

Standard Deviation and Normal Distribution

http://27gen.files.wordpress.com/2011/09/2011-bell-curve.jpg http://www.mhhe.com/socscience/intro/ibank/set4.htm

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Standard Deviation and Normal Distribution

Use this chart to answer the following questions.

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

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Standard Deviation and Normal Distribution

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?

Examples:

Teacher

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.

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Standard Deviation and Normal Distribution

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?

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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?

Standard Deviation and Normal Distribution

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34 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?

Standard Deviation and Normal Distribution

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

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35 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.

Standard Deviation and Normal Distribution

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85 91 73 67 97 103 79

36 Which of the following curves represents a mean of 85 and a standard deviation of 6? A B C D

Standard Deviation and Normal Distribution

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

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37 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?

Standard Deviation and Normal Distribution

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38 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?

Standard Deviation and Normal Distribution

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39 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?

Standard Deviation and Normal Distribution

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Standard Deviation and Normal Distribution

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.

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Standard Deviation and Normal Distribution

z-score = A table of z-scores is below. Each score is associated with the amount of area under the normal curve from the score to the left.

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Standard Deviation and Normal Distribution

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 =

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Standard Deviation and Normal Distribution

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.

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Standard Deviation and Normal Distribution

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

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40 Find the z-score for a 29 if the mean was 34 and the standard deviation is 2.3.

Standard Deviation and Normal Distribution

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41 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%

Standard Deviation and Normal Distribution

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42 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?

Standard Deviation and Normal Distribution

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43 A student calculated a z-score of -1.25. What percentile does this score fall in?

Standard Deviation and Normal Distribution

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44 Find the z-score of 10 is the data set is:

Standard Deviation and Normal Distribution

Teacher

9 11 5 7 10 10 10 11 9 15

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Two-Way Frequency Tables

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Goals and Objectives

Students will be able to recognize trends with and interpret different association of data in a two-way frequency table.

Two-way Frequency Tables

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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.

Two-way Frequency Tables

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Two-way Frequency Tables

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.

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Two-way Frequency Tables

Bar Graphs As well as ... Pie Graphs

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Two-way Frequency Tables

Line Plots And...

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Scatter Plots

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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

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Two-way Frequency Tables

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

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Two-way Frequency Tables

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.

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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.

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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.

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Two-way Frequency Tables

Together, write some quantitative statements about the information.

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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.

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45 From the relative frequency table you created, find the joint relative frequency for the dogs that did not need bloodwork.

Two-way Frequency Tables

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46 What is the marginal relative frequency of cats that came to the clinic?

Teacher

Two-way Frequency Tables

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47 What is the percentage of dogs that came in that needed blood work?

Teacher

Two-way Frequency Tables

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48 Find the marginal relative frequency for the number of animals which came in and needed blood work?

Teacher

Two-way Frequency Tables

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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. 0.17 0.30

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

57%

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Teacher

Two-way Frequency Tables

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

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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.

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Two-way Frequency Tables

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

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49 From the table, find the probability that a girl has gone to an amusement park.

Teacher

Two-way Frequency Tables

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50 Find the conditional probability that out of the girls, the person has been to an amusement park.

Teacher

Two-way Frequency Tables

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51 What is the probability that if a person has been to an amusement park, it was a boy?

Teacher

Two-way Frequency Tables

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52 Find the probability that out of the people that have not gone to an amusement park, it would be a girl.

Teacher

Two-way Frequency Tables

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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.

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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.

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Two-way Frequency Tables

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

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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.

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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

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

questions.

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53 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.

Two-way Frequency Tables

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54 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.

Two-way Frequency Tables

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55 The marginal relative frequency of in-state students is: A 0.33 B 0.78 C 0.45 D 0.22

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Two-way Frequency Tables

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56 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

Teacher

Two-way Frequency Tables

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Sampling and Experiments

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Sampling and Experiments

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.

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Sampling and Experiments

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.

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Sampling and Experiments

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.

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Sampling and Experiments

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.

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Sampling and Experiments

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

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57 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.

Sampling and Experiments

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58 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.

Sampling and Experiments

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Sampling and Experiments

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.

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Sampling and Experiments

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.

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Sampling and Experiments

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.

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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.

Teacher

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Sampling and Experiments

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

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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/ Sampling and Experiments

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**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

Sampling and Experiments

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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?

Teacher

Sampling and Experiments

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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:

Sampling and Experiments

Teacher

Is your answer reasonable in this situation?

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

Sampling and Experiments

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60 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%

Sampling and Experiments

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61 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%

Sampling and Experiments

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62 Find the sample size needed to acheive a margin of error of 1%.

Sampling and Experiments

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63 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)?

Sampling and Experiments

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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.

Sampling and Experiments

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

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Sampling and Experiments

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?

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Sampling and Experiments

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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.

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

Sampling and Experiments

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Sampling and Experiments

Simulations are used if studies are not cost effective, tests are not ethical or if the calculating the probability is too difficult. Let's perfom a simulation now. Take out a coin or get one from your

• teacher. Everyone flip the coin 10 times and record whether you get heads
• r tails. Write this infomation on the board.

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?

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Sampling and Experiments

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 If you want to store it in your calculator, To flip the coin 10 times, use randint(0, 1, 10). To flip it another 10 times hit 2nd Enter.

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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 claim that this

• riginal claim be true.

Sampling and Experiments

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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Sampling and Experiments

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:

Sampling and Experiments

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.

Teacher

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Sampling and Experiments

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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Sampling and Experiments

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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Sampling and Experiments

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.

Teacher

Steps are on the next page...

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Sampling and Experiments

• 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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Sampling and Experiments

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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Sampling and Experiments

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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Sampling and Experiments

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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64 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

Sampling and Experiments

Teacher

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65 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?

Sampling and Experiments

Teacher

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66 40% 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? ***www.education.com/study-help/article/ap-statistics-practice-

exam-2

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.

Sampling and Experiments

Teacher