Table of Contents Click on a topic to go to that section - - PowerPoint PPT Presentation

7th Grade

Statistics and Probability

2017­02­25 www.njctl.org

Table of Contents

• Introduction to Probability
• Experimental and Theoretical
• Word Problems
• Probability of Compound Events
• Sampling
• Measures of Center
• Measures of Variation
• Mean Absolute Deviation
• Glossary

Click on a topic to go to that section

Teacher Notes

Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end

• f the presentation with the

word defined on it.

Introduction to Probability

Return to Table

• f Contents

One way to express probability is to use a fraction. Number of favorable outcomes Total number of possible outcomes Probability

• f an event

P(event) =

Probability

Example: What is the probability of flipping a nickel and the nickel landing on heads? Step 1: What are the possible outcomes? Step 2: What is the number of favorable outcomes?

Step 3: Put it all together to answer the question. The probability of flipping a nickel and landing on heads is:

Probability

Click Click Click

Answer

Probability can be expressed in many forms. For example, the probability of flipping a head can be expressed as:

• r or or

The probability of randomly selecting a blue marble can be expressed as:

• r or or

Probability

Click

When there is no chance of an event occurring, the probability of the event is zero (0). When it is certain that an event will occur, the probability of the event is one (1). I m p

• s

s i b l e U n l i k e l y Equally Likely Likely Certain The less likely it is for an event to occur, the probability is closer to 0 (i.e. smaller fraction). The more likely it is for an event to occur, the probability is closer to 1 (i.e. larger fraction).

Probability

Without counting, can you determine if the probability of picking a red marble is lesser or greater than ? It is very likely you will pick a red marble, so the probability is greater than (or 50% or 0.5).

What is the probability of picking a red marble? Add the probabilities of both events. What is the sum?

Probability

Click Click Click Click

The sum of all possible outcomes is always equal to 1.

There are three choices of jelly beans ­ grape, cherry and orange. If the probability of getting a grape is and the probability of getting cherry is , what is the probability of getting orange? The probability of getting an orange jelly bean is .

Probability

Click

1 Arthur wrote each letter of his name on a separate card and put the cards in a bag. What is the probability of drawing an A from the bag? A B

C D

A R T H U R

Probability = Number of favorable outcomes Total number of possible outcomes

Click for hint

Answer

B

2 Arthur wrote each letter of his name on a separate card and put the cards in a bag. What is the probability of drawing an R from the bag? A B

C D

A R T H U R A

Probability = Number of favorable outcomes Total number of possible outcomes

Click for hint

Answer

C

Probability = Number of favorable outcomes Total number of possible outcomes

3 Matt's teacher puts 5 red, 10 black, and 5 green markers in a bag. What is the probability of Matt drawing a red marker?

Click for hint

A B C D

Answer

B

4 What is the probability of rolling a 5 on a fair number cube?

Answer

5 What is the probability of rolling a composite number on a fair number cube?

Answer

6 What is the probability of rolling a 7 on a fair number cube?

Answer

7 You have black, blue, and white t­shirts in your

• closet. If the probability of picking a black t­shirt is

and the probability of picking a blue t­shirt is , what is the probability of picking a white t­shirt?

Answer

8 If you enter an online contest 4 times and at the time of drawing its announced there were 100 total entries, what are your chances of winning?

Answer

9 Mary chooses an integer at random from 1 to 6. What is the probability that the integer she chooses is a prime number?

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 2011.

A B C D

Answer

B

10 Each of the hats shown below has colored marbles placed inside. Hat A contains five green marbles and four red marbles. Hat B contains six blue marbles and five red marbles. Hat C contains five green marbles and five blue marbles. If a student were to randomly pick one marble from each of these three hats, determine from which hat the student would most likely pick a green

• marble. Justify your answer.

A Hat B Hat C Hat

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

Answer

Determine the fewest number of marbles, if any, and the color

• f these marbles that could be added to each hat so that the

probability of picking a green marble will be one­half in each

• f the three hats.

Hat A contains five green marbles and four red marbles. Hat B contains six blue marbles and five red marbles. Hat C contains five green marbles and five blue marbles.

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 2011

Answer

Hat A needs 1 red marble. Hat B needs 11 green marbles. Hat C does not need marbles.

Experimental & Theoretical Probability

Return to Table

• f Contents

Click on an object. What is the outcome?

Outcomes

Experimental Probability

Flip the coin 5 times and determine the experimental probability of heads. Probability

• f an event

Heads Tails

number of times the outcome happened number of times experiment was repeated Answer

Example 1 ­ Golf A golf course offers a free game to golfers who make a hole­in­one

• n the last hole. Last week, 24 out of 124 golfers achieved this.

Find the experimental probability that a golfer makes a hole­in­one

• n the last hole.

Out of 31 golfers, you could expect 6 to make a hole­in­one on the last hole. Or there is a 19% chance of a golfer making a hole­in­one on the last hole.

Experimental Probability

P(hole­in­one) = # of successes # of trials

Example 2 ­ Surveys Of the first 40 visitors through the turnstiles at an amusement park, 8 visitors agreed to participate in a survey being conducted by park employees. Find the experimental probability that an amusement park visitor will participate in the survey. You could expect 1 out of every 5 people to participate in the

• survey. Or there is a 20% chance of a visitor participating in

the survey.

Experimental Probability

P(participation) = # of successes # of trials

Click

# on Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six Sally rolled a die 10 times and the results are shown below. Use this information to answer the following questions.

Experimental Probability

11 What is the experimental probability of rolling a 5?

** These are the results after 10 rolls of the die

A B C D

# on Die Picture of Roll Results

1 one 3 twos 1 three 0 fours 4 fives 1 six 1 2 3 4 5 6

Answer

D

12 What is the experimental probability of rolling a 4?

** These are the results after 10 rolls of the die

A B C D

# on Die Picture of Roll Results

1 one 3 twos 1 three 0 fours 4 fives 1 six 1 2 3 4 5 6

Answer

C

13 Based on the experimental probability you found, if you rolled the die 100 times, how many sixes would you expect to get?

** These are the results after 10 rolls of the die

A 6 sizes B 10 sixes C 12 sixes D 60 sixes

# on Die Picture of Roll Results

1 one 3 twos 1 three 0 fours 4 fives 1 six 1 2 3 4 5 6

Answer

B

14 Mike flipped a coin 15 times and it landed on tails 11

• times. What is the experimental probability of landing
• n heads?

Answer

Theoretical Probability

What is the theoretical probability of spinning green? Is this a fair probability? Answer

Probability

• f an event

number of favorable outcomes

total number of possible outcomes

Theoretical Probability

Theoretical Probability

Example 1 ­ Marbles Find the theoretical probability of randomly choosing a white marble from the marbles shown. There is a 2 in 5 chance of picking a white marble or a 40% possibility. P(white) = # of favorable outcomes # of possible outcomes

Theoretical Probability

Example 2 ­ Marbles Suppose you randomly choose a gray marble. Find the probability of this event. There is a 3 in 10 chance of picking a gray marble or a 30% possibility. P(gray) = # of favorable outcomes # of possible outcomes

Click

There is a 1 in 2 chance of getting tails when you flip a coin or a 50% possibility.

Theoretical Probability

Example 3 ­ Coins Find the probability of getting tails when you flip a coin. P(tails) = # of favorable outcomes # of possible outcomes

Click

15 What is the theoretical probability of picking a green marble?

R R G W W Y

Y

B

A B C D

Answer

A

16 What is the theoretical probability of picking a black marble?

R R G W W Y Y

B

A B C D

Answer

D

17 What is the theoretical probability of picking a white marble?

R R G W W Y Y

B

A B C D

Answer

C

18 What is the theoretical probability of not picking a white marble?

R R G W W Y Y

B

A B C D

Answer

A

19 What is the theoretical probability of rolling a three? A B C D

Answer

20 What is the theoretical probability of rolling an odd number? A B C D

Answer

A

21 What is the theoretical probability of rolling a number less than 5? A B C D

Answer

A

22 What is the theoretical probability of not rolling a 2? A B C D

Answer

D

23 Seth tossed a fair coin five times and got five heads. The probability that the next toss will be a tail is A B

C D

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Answer

24 Which inequality represents the probability, x, of any event happening? A B

C D

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

Answer

D

25 The spinner shown is divided into 8 equal sections. The arrow on this spinner is spun once. What is the probability that the arrow will land on a section labeled with a number greater than 3? Enter only your fraction.

From PARCC EOY sample test calculator #1

1 1 2 2 2 3 4 5 Answer

26 Reagan will use a random number generator 1,200

• times. Each result will be a digit form 1 to 6. Which

statement best predicts how many times the digit 5 will appear among the 1,200 results? A It will appear exactly 200 times. B It will appear close to 200 times but probably not exactly 200 times. C It will appear exactly 240 times. D It will appear close to 240 times but probably not exactly 240 times.

From PARCC EOY sample test calculator #17

Answer

Class Activity

• Each student flips a coin 10 times and records the number of heads

and the number of tail outcomes.

• Each student calculates the experimental probability of flipping a tail

and flipping a head.

• Use the experimental probabilities determined by each student to

calculate the entire class's experimental probability for flipping a head and flipping a tail.

What is the theoretical probability for flipping a tail? A head? Compare the experimental probability to the theoretical probability for 10 experiments. Compare the experimental probability to the theoretical probability when the experiments for all of the students are considered?

Class Activity

Sampling

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• f Contents

Your task is to count the number of whales in the ocean or the number of squirrels in a park. How could you do this? What problems might you face? A sample is used to make a prediction about an event or gain information about a population. A whole group is called a POPULATION. A part of a group is called a SAMPLE.

Sampling

A sample is considered Example: To find out how many people in New York feel about mass transit, people at a train station are asked their opinion. Is this situation representative of the general population?

Sampling

random (or unbiased) when every possible sample of the same size has an equal chance of being

• selected. If a sample is biased, then information obtained from it

may not be reliable. Answer

Determine whether the situation would produce a random sample. You want to find out about music preferences of people living in your area. You and your friends survey every tenth person who enters the mall nearest you.

Sampling

Answer

27 Food services at your school wants to increase the number of students who eat hot lunch in the cafeteria. They conduct a survey by asking the first 20 students that enter the cafeteria lunch line to determine the students' preferences for hot lunch. Is this survey reliable? Explain your answer. Yes No

Answer

28 The guidance counselors want to organize a career day. They will survey all students whose ID numbers end in a 7 about their grades and career counseling needs. Would this situation produce a random sample? Explain your answer. Yes No

Answer

Yes

29 The local newspaper wants to run an article about reading habits in your town. They conduct a survey by asking people in the town library about the number of magazines to which they subscribe. Would this produce a random sample? Explain your answer. Yes No

Answer

No

How would you estimate the size of a crowd? What methods would you use? Could you use the same methods to estimate the number of wolves on a mountain?

Sampling

A whole group is called a population. A part of a group is called a SAMPLE. When biologists study a group of wolves, they are choosing a

• sample. The population is all the wolves on the mountain.

Population

Sample

Sampling

Suppose this represents all the wolves on the mountain. One way to estimate the number of wolves on a mountain is to use the capture­recapture method.

Sampling

Wildlife biologists first find some wolves and tag them.

Capture­Recapture Method

Then they release them back onto the mountain.

Capture­Recapture Method

They wait until all the wolves have mixed together.Then they find a second group of wolves and count how many are tagged.

Capture­Recapture Method

Biologists use a proportion to estimate the total number of wolves

• n the mountain:

tagged wolves on mountain tagged wolves in second group

total wolves on mountain total wolves in second group

For accuracy, they will often conduct more than one recapture.

=

There are 36 wolves on the mountain

Capture­Recapture Method

Try This: Biologists are trying to determine how many fish are in the Rancocas Creek. They capture 27 fish, tag them and release them back into the Creek. 3 weeks later, they catch 45 fish. 7 of them are tagged. How many fish are in the creek? There are 174 fish in the river.

Capture­Recapture Method

Click

Try This: 315 out of 600 people surveyed voted for Candidate A. How many votes can Candidate A expect in a town with a population of 1500?

Capture­Recapture Method

Answer

30 Eight hundred sixty out of 4,000 people surveyed watched Dancing with the Stars. How many people in the US watched if there are 93.1 million people?

Answer

31 Six out of 150 tires need to be realigned. How many out of 12,000 are going to need to be realigned?

Answer

32 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000.

Answer

120 faulty bulbs

33 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect?

Answer

about 542 votes

34 The chart shows the number of people wearing different types of shoes in Mr. Thomas' English

• class. Suppose that there are 300 students in the
• cafeteria. Predict how many would be wearing

high­top sneakers. Explain your reasoning.

Number of Students Low­top sneakers 12 High­top sneakers 7 Sandals 3 Boots 6 Shoes

Answer

75 students

35 Josephine owns a diner that is open every day for breakfast, lunch, and dinner. She offers a regular menu and a menu with specials for each of the three meals. She wanted to estimate the percentage of her customers that order form the menu with specials. She selected a random sample of 50 customers who had lunch at her diner during a three­month period. She determined that 28% of these people ordered for the menu with specials. Which statement about Josephine's sample is true? A The sample is the percentage of customers who order from the menu with specials. B The sample might not be representative of the population because it only included lunch customers. C The sample shows that exactly 28% of Josephine's customers order from the menu with specials. D No generalizations can be made from this sample, because the sample size of 50 is too small.

From PARCC EOY sample test calculator #13

Answer

Multiple Samples

The student council wanted to determine which lunch was the most popular among their students. They conducted surveys on two random samples of 100 students. Make at least two inferences based on the results.

Student Sample Hamburgers Tacos

Pizza Total

#1

12

14 74 100

#2

12

11 77 100

• Most students prefer pizza.
• More people prefer pizza than hamburgers and tacos combined.

Click

The NJ DOT (Department of Transportation) used two random samples to collect information about NJ drivers. The table below shows what type of vehicles were being driven. Make at least two inferences based on the results of the data.

Driver Sample

Cars SUVs Mini Vans

Motorcycles Total #1

37 43 12

8

100

#2

33 46 11

10

100

Multiple Samples

Answer

SUV's are the most popular vehicles among NJ drivers. Cars are the second most popular vehicles among NJ drivers. SUV's are at least twice as popular as mini vans and motorcycles combined.

The student council would like to sell potato chips at the next basketball game to raise money. They surveyed some students to figure out how many packages of each type of potato chip they would need to buy. For home games, the expected attendance is approximately 250 spectators. Use the chart to answer the next three questions.

Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6

36 How many students participated in each survey?

Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6 Answer

25

37 According to the two random samples, which flavor potato chip should the student council purchase the most of?

Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6

A Regular B BBQ C Cheddar

Answer

B

38 Use the first random sample to evaluate the number of packages of cheddar potato chips the student council should purchase.

Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6 Answer

70

Word Problems

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• f Contents

19 shots made 100 shots attempted = 19% Erica loves soccer! The ladies' coach tells Erica that she scored 19% of her attempts on goal last season. This season, the coach predicts the same percentage for Erica. Erica reports she attempted approximately 1,100 shots on goal last season. Her coach suggests they estimate the number of goals using experimental probability. What do you know about percentages to figure out the relationship of goals scored to goals attempted? Experimental Probability = number of times the outcome happened number of times experiment was repeated number of goals number of attempts Erica's Experimental Probability

=

Word Problems

Click Click Click

is very close to so she makes about 20% of her shots on goal. Let's estimate the number of goals Erica scored. 1,100 is very close to 1,000. So we will estimate that Erica has about 1,000 attempts. About what percent would be a good estimate to use? About how many attempts did Erica take? Erica makes 19% of her shots

• n goal.

Erica takes 1,100 shots on goal.

Word Problem

click click

Erica figures she made about 200 of her shots on goal. Erica wants to find 20% of 1,000. Her math looks like this:

Word Problem

Click

What are the actual values that will give you 19%?

Challenge

Remember sometimes it helps to turn a percent into a decimal prior to solving the problem.

Word Problem

Click

Answer

Last year, Lexi planted 12 tulip bulbs, but only 10

• f them bloomed. This year she intends to plant 60 tulip bulbs.

Use experimental probability to predict how many bulbs will bloom. Based on her experience last year, Lexi can expect 50

• ut
• f 60 tulips to bloom.

Solve this proportion by equivalent fractions.

Experimental Probability

Today you attempted 50 free throws and made 32 of them. Use experimental probability to predict how many free throws you will make tomorrow if you attempt 75 free throws. Based on your performance yesterday,you can expect to make 48 free throws out of 75 attempts. Solve this proportion using cross products.

Experimental Probability

Number of attempts Number of goals Experimental Probability 100 1000 500 2000 30 600 150 1600 Now, its your turn. Calculate the experimental probability for the number of goals.

Experimental Probability

39 Tom was at bat 50 times and hit the ball 10 times. What is the experimental probability for hitting the ball?

Answer

40 Tom was at bat 50 times and hit the ball 10 times. Estimate the number of balls Tom hit if he was at bat 250 times.

Answer

50

41 What is the theoretical probability of randomly selecting a jack from a deck of cards?

Answer

42 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the experimental probability for rolling a 3?

Answer

43 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the theoretical probability for rolling a 3?

Answer

44 Some books are laid on a desk. Two are English, three are mathematics, one is French, and four are social

• studies. Theresa selects an English book and Isabelle

then selects a social studies book. Both girls take their selections to the library to read. If Truman then selects a book at random, what is the probability that he selects an English book?

Answer

45 What is the probability of drawing a king or an ace from a standard deck of cards? A B C D

Answer

C

46 What is the probability of drawing a five or a diamond from a standard deck of cards? A B C D

Answer

A

Lindsey would like to know the number of people at a movie theater that will buy a movie ticket and popcorn. Based on past data, the probability that a person who is selected at random from those that buy movie tickets and also buy popcorn is 0.6. Lindsey designs a simulation to estimate the probability that exactly two in a group of three people selected randomly at a movie theater will buy both a movie ticket and popcorn. For the simulation Lindsey used a number generator that generates random numbers.

• Any number from 1 through 6

represents a person who buys a movie ticket and popcorn.

• Any number from 7 through 9 or 0

represents a person who buys

• nly a movie ticket.
• Use info for next two questions.

From PARCC EOY sample test calculator #3

266 342 847 672 567 268 252 465 429 573 100 818 139 730 910 494 922 155 585 426 593 903 556 981 966 491 186 865 044 147

Movie Theater

47 Part A In the simulation, one result was "100". What does this result simulate? A No one in a group of three randomly­chosen people who buy movie tickets also buys popcorn. B Exactly one person in a group of three randomly­chosen people who buy movie tickets also buys popcorn. C Exactly two people in a group of three randomly­chosen people who buy movie tickets also buy popcorn. D All three people in a group of three randomly­chosen people who buy movie tickets also buy popcorn.

Answer

48 Part B Use the results of the simulation to estimate the probability that exactly two of three people selected at random from those who buy movie tickets will also buy popcorn.

Answer

Probability of Compound Events

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• f Contents

Probability of Compound Events

For the probability of compound events, first ­ decide if the two events are independent or dependent. When the outcome of one event does not affect the outcome of another event, the two events are independent. Use formula: Probability (A and B) = Probability (A) Probability (B)

Independent Example

Select a card from a deck of cards, replace it in the deck, shuffle the deck, and select a second card. What is the probability that you will pick a 6 and then a king? P (6 and a king) = P(6) P(king)

When the outcome of one event affects the outcome of another event, the two events are dependent. Use formula: Probability (A & B) = Probability(A) Probability(B given A)

Dependent Events

Select a card from a deck of cards, do not replace it in the deck, shuffle the deck, and select a second card. What is the probability that you will pick a 6 and then a king?

Dependent Example

P (6 and a king) = P(6) P(king given a six has been selected) Notice your denominator when down by 1. Why?

Independent Dependent Try to name some other independent and dependent events.

Independent & Dependent Examples

49 The names of 6 boys and 10 girls from your class are put in a hat. What is the probability that the first two names chosen will both be boys? (w/o replacement)

Answer

50 A lottery machine generates numbers randomly. Two numbers between 1 and 9 are generated. What is the probability that both numbers are 5?

Answer

51 The TV repair person is in a room with 20 broken TVs. Two sets have broken wires and 5 sets have a faulty computer chip. What is the probability that the first TV repaired has both problems?

Answer

52 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement)

Answer

53 A spinner containing 5 colors: red, blue, yellow, white and green is spun and a die, numbered 1 through 6, is

• rolled. What is the probability of spinning green and

rolling a two?

Answer

54 A drawer contains 5 brown socks, 6 black socks, and 9 navy blue socks. The power is out. What is the probability that Sam chooses two socks that are both black?

Answer

55 At a school fair, the spinner represented in the accompanying diagram is spun twice. What is the probability that it will land in section G the first time and then in section B the second time? A B C

R G B

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

D

Answer

56 A student council has seven officers, of which five are girls and two are boys. If two officers are chosen at random to attend a meeting with the principal, what is the probability that the first officer chosen is a girl and the second is a boy? A B C

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

D

Answer

57 The probability that it will snow on Sunday is . The probability that it will snow on both Sunday and Monday is . What is the probability that it will snow on Monday, if it snowed on Sunday? A B

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

C D

Answer

Measures of Center

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• f Contents

Sometimes we can make general statements about a set of data as shown in this first question.

Generalizations

58 Alexis chose a random sample of 10 jars of almonds from each of two different brands, X and Y. Each jar in the sample was the same

• size. She counted the number of almonds in each jar. Her results

are shown in the plots.

A The number of almonds in jars from Brand X tends to be greater and more consistent than those from Brand Y. B The number of almonds in jars from Brand X tends to be greater and less consistent than those from Brand Y. C The number of almonds in jars from Brand X tends to be fewer and more consistent than those from Brand Y. D The number of almonds in jars from Brand X tends to be fewer and less consistent than those from Brand Y.

From PARCC EOY sample test calculator #7

Brand X Brand Y

40 45 50 55 60 65 70 40 45 50 55 60 65 70

Number of Almonds Number of Almonds

Answer

Other times we will make statements about the data based on measure of center and variation that we can calculate. This will be the topics for the rest of this chapter.

Generalizations

Measures of Center ­ Vocabulary Review

Median ­ The middle data value when the values are written in numerical order Mean (Average) ­ The sum of the data values divided by the number of items Mode ­ The data value that occurs the most often

Measures of Center

Joey wanted to convince his mom to give him some money for a snack from the concession stand. Below are the prices of the different snacks. $1.75, $0.75, $1.25, $0.75, $2.50, $2.00

What is the mean of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Add up all of the numbers. 1.75 + 0.75 + 1.25 + 0.75 + 2.50 + 2.00 = 9.00 Step 2: Divide the sum by the number of items listed. 9.00 / 6 = 1.50 The mean cost of concession stand snacks is $1.50.

Mean Example

Find the mini mean What is the median of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Order the numbers from least to greatest. 0.75, 0.75, 1.25, 1.75, 2.00, 2.50 Step 2: Find the middle value. 0.75, 0.75, 1.25, 1.75, 2.00, 2.50 The median cost of concession stand snacks is $1.50.

Median Example

What do you do when you have two numbers left? (click)

What is the mode of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Look for the number that appears most often. 1.75, 0.75, 1.25, 0.75, 2.50, 2.00 The mode cost of concession stand snacks is $0.75.

Mode Example

Mean $1.50 Median $1.50 Mode $0.75 How can Joey use this information to ask his mom for money? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00

Measures of Center

Time Spent Texting Daily by 7th Grade Students (in minutes)

60 15 30 45 75 90 105 120

Girls

60 15 30 45 75 90 105 120

Boys

Measures of Center

Use the dot plots to compare the 2 samples.

60 15 30 45 75 90 105 120

Girls

Find the mean, median, and mode for the sample of girls.

Measures of Center

Answer Mean 88.5 min or 1 hr 28 min and 30 sec *Remember to find out how many seconds 0.5 is equal to multiply the decimal by how many seconds are in a minute. 0.5 * 60 seconds = 30 seconds Median 90 minutes or 1 hour 30 minutes Mode 60 minutes or 1 hour

60 15 30 45 75 90 105 120

Boys

Measures of Center

Find the mean, median, and mode for the sample of boys. Answer

Measures of Center

Now compare the two measures of center. Make a statement about the average time spent texting daily by 7th grade students.

Girls Boys Mean Median Mode 88.5 90 60 40.5 30 30 and 60

Answer

59 What is the mean of the stem­and­leaf plot?

Stem Leaf

1 1 1 2 2 0 0 3 5 5 4 8

Key: 1 | 1 = 11

Answer

25

60 What is the median of the stem­and­leaf plot?

Stem Leaf

1 1 1 2 2 0 0 3 5 5 4 8

Key: 1 | 1 = 11

Answer

20

61 What is the mode of the stem­and­leaf plot?

Stem Leaf

1 1 1 2 2 0 0 3 5 5 4 8

Key: 1 | 1 = 11

Answer

11 and 20

62 What is the mean of the stem­and­leaf plot?

Stem Leaf

1 8 9 3 7 7 9

Key: 1 | 8 = 1.8

Answer

3

63 What is the median of the stem­and­leaf plot?

Stem Leaf

1 8 9 3 7 7 9

Key: 1 | 8 = 1.8

Answer

3.7

64 What is the mode of the stem­and­leaf plot?

Stem Leaf

1 8 9 3 7 7 9

Key: 1 | 8 = 1.8

Answer

3.7

Miss M's Math Class Scores 90 50 60 70 80 100

1st Period Scores

Measures of Center

Use the dot plots to find the measures of center. Answer

Miss M's Math Class Scores

90 50 60 70 80 100

8th Period Scores

Measures of Center

Use the dot plots to find the measures of center. Answer

Measures of Center

Write a statement comparing the averages of Miss M's 1st period class scores to her 8th period class scores.

Measures of Variation

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• f Contents

Measures of Variation ­ Vocabulary Review

Quartiles ­ are the values that divide the data in four equal parts. Lower (1st) Quartile (Q1) ­ The median of the lower half of the data. Upper (3rd) Quartile (Q3) ­ The median of the upper half of the data. Interquartile Range ­ The difference of the upper quartile and the lower quartile. (Q3 ­ Q1) Range ­ The difference between the greatest data value and the least data value.

1, 5, 8, 3, 2, 5, 2, 8, 9, 5 1, 2, 2, 3, 5, 5, 5, 8, 8, 9 To find the interquartile range of the data set, we first have to find the quartiles. Step 1: Order the numbers from least to greatest.

Interquartile Range

1, 2, 2, 3, 5, 5, 5, 8, 8, 9

Step 2: Find the median. 5 median *Note:

• If the median falls in between two data values, all of the values are

still used to calculate the upper and lower quartiles.

• If the median falls exactly on one of the two data values, than that

values is NOT used to calculate the upper and lower quartiles.

Interquartile Range

1, 2, 2, 3, 5, 5, 5, 8, 8, 9

5

median

8

Upper

2

Lower Quartile Quartile

Step 3: Find the upper and lower quartiles. Find the mean of each half of the data set.

Interquartile Range

Upper Quartile Lower Quartile Step 4: Subtract the lower quartile from the upper quartile. Interquartile Range = ­ 8 ­ 2 = 6

Interquartile Range

1, 2, 2, 3, 5, 5, 5, 8, 8, 9

To find the range, subtract the least value from the greatest value. Greatest Value Least Value = Range ­ 9

­ 1 = 8

Sample Range

1, 2, 2, 3, 5, 5, 5, 8, 8, 9

5

Median

8

Upper

2

Lower Quartile Quartile

1

Least Value

9

Greatest Value

Box­and­Whisker Plot

These 5 values are used to create a box­and­whisker

• plot. To do this, plot all 5 values on the number line

and then connect them to look like a box with whiskers on both sides. click just above the number line to reveal 1 2 3 4 5 6 7 8 9 10

65 What is the median of the data set?

1 2 3 4 5 6 7 8 9 10 Answer

4

66 What is the interquartile range using the given information? Least Value = 3 Lower Quartile = 6 Median = 7 Upper Quartile = 10 Greatest Value = 11

Answer

4

67 What is the range for the following data set? 3, 5, 10, 4, 2, 2, 1

Answer

9

68 What is the interquartile range for the following data set? 3, 5, 10, 4, 2, 2, 1

Answer

3

Mean Absolute Deviation

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• f Contents

Mean absolute deviation ­ the average distance between each data value and the mean.

Mean Absolute Deviation ­ Vocabulary

Mean Absolute Deviation

Find the mean absolute deviation of the following data. Quiz Scores 65, 75, 90, 90, 100 Step 1: Find the mean.

Score Deviation from mean Absolute deviation from mean

65 75 90 90 100

Step 2: Find the absolute deviation. To do this you need to subtract the mean and each data point. Then take the absolute value of each difference.

Mean Absolute Deviation

Mean Absolute Deviation

Step 3: Find the mean absolute deviation (MAD). To do this find the mean using the absolute deviation numbers. The MAD is 11.2 points. Absolute deviation from mean Answer

This means that the average test score is 11.2 points from the mean of 84.

Girls' Team Boys' Team

Comparing Two Data Sets

The number of goals scored by the players on the boys' and girls' LAX teams are displayed below. Compare the variability of the mean goals scored for both teams.

Girls' Team

Boys' Team

Comparing Two Data Sets

Step 1: Find the mean for each team. Answer

Goals Mean Deviation Absolute Mean Dev.

Girls' Team Boys' Team

Comparing Two Data Sets

Step 2: Find the absolute deviations.

Goals Mean Deviation Absolute Mean Dev.

Girls' Team Boys' Team

Comparing Two Data Sets

Step 3: Find the mean absolute deviations. Answer

Comparing Two Data Sets

Comparison Statements 1.25 = 1.25 The variability is equal for both the boys and girls LAX teams. On average, the boy players scored 1 more goal than the girl players. (How do you know this?) Answer

Pages per Chapter in Hunger Games

10 15 20 25 30

x

x x x x x x x x x x x x x x x x x x x x x

x

x x x

x

10 15 20 25 30

x x x

x x

x x x x x x x x x x x x x x x x x

Pages per Chapter in Twilight Use the following data to answer the next seven questions.

Comparing Two Data Sets

69 What is the mean number of pages per chapter in the Hunger Games?

Answer

13.63 pages

70 What is the mean number of pages per chapter in Twilight?

Answer

20.41 pages

71 What is the difference of the means?

Answer

6.78

72 What is the mean absolute deviation of the data set for Hunger Games? (Hint: Round mean to the nearest ones.)

Answer

1.78

73 What is the mean absolute deviation of the data set for Twilight? (Hint: Round mean to the nearest ones.)

Answer

4.32

74 Which book has more variability in the number of pages per chapter? A Hunger Games B Twilight

Answer

B

75 On average, there are ______ pages per chapter in the Hunger Games than in Twilight. A more B less

Answer

B

Glossary

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• f Contents

Teacher Notes

Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end

• f the presentation with the

word defined on it.

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Biased Sample

When every possible sample of the same size does not have an equal chance of being selected.

Asking only flight attendants if they believe flying is safe.

Asking everyone in Hershey Park if they like chocolate.

Asking everyone at ComicCon if they like comic books.

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Capture­Recapture Method

A method of sampling that is used to try and estimate the entire

• population. A sample of animals are caught, tagged, and then

released into the wild. Later a second sample of animals are caught to compute using a ratio the amount of tagged animals to the population as a whole.

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Compound Event

A combination of two or more simple events.

The prob. of flipping heads AND rolling 4 on a die. The prob. of selecting a Jack OR a 3 card.

The prob. of selecting a Jack AND a 3 card.

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Dependent Event

When the outcome of one affects the

• utcome of another event.

Probability (A & B)

= Prob(A)

*Prob(B given A)

The prob. of selecting a Jack AND a 3 card.

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Equally Likely

When all the outcomes have the same chance of occurring.

Sides on a Coin

A Fair Die A Fair Spinner

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Experimental Probability

The ratio of the number of times an event occurs to the total number of times that the activity is performed.

number of times the outcome happened number of times experiment was repeated

Probability

• f an event

Last week, 24 out

• f 124 golfers hit a

hole­in­one on the last hole. Find the experimental probability that a golfer makes this shot.

P(hole­in­one) = # of successes = # of trials 24 124 = 6 31

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Fair

An experiment with equally likely

• utcomes.

Tossing a Coin Rolling a Fair Die Spinning a Fair Spinner

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Independent Event

When the outcome of one event does not affect the outcome of another event.

Probability (A and B)

=

Prob(A)Prob(B)

The prob. of flipping heads AND rolling 4 on a die.

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Interquartile Range

The difference of the upper quartile and the lower quartile.

25% 25% 25% 25%

Q1 Q2 Q3 1,3,3,4,5,6,6,7,8,8 Q1 Q2 Q3

1 2 3 4 5 6 7 8

Q1 Q2 Q3 = Q3 ­ Q1 = Q3 ­ Q1 = 4

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Lower (1st) Quartile

The median of the lower half of data.

25% 25%

25% 25%

Q1 Q2 Q3 1,3,3,4,5,6,6,7,8,8 Q1 Q2 Q3

1 2 3 4 5 6 7 8

Q1

Q2 Q3

Median

}

Median

}

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Mean

Average The sum of the data values divided by the number of items.

1, 2, 3, 4, 5

Set of Data:

1+2+3+4+5 =15 15/5 = 3

The mean is 3.

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Mean Absolute Deviation

The average distance between each data value and the mean.

Subtract the mean from each data point Find the mean of the differences

2, 2, 3, 4, 4

15 5=3

3­2=1 4­3=1 3­3=0

1+1+0+1+1 =4 5=.8

3­2=1 4­3=1

• 1. Find the mean

2. 3.

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Median

The middle data value when the values are written in numerical order.

1, 2, 3, 4, 5

Median

1, 2, 3, 4

Median is 2.5

1+2+3+4 = 10 10/4 = 2.5

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Mode

The data value that occurs the most often. 2, 4, 6, 3, 4

The mode is 4.

2, 4, 6, 2, 4

The mode is 4 and 2.

2, 4, 6, 3, 8

There is no mode.

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Population

A whole group.

Population Sample

• All m&ms in a

bag

• All types of dogs in a

dog park

• All students

wearing glasses in a classroom

NOT just people in a place

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What is the probability

• f flipping a nickel and

the nickel landing on heads?

1 favorable 2 possible

Probability

The ratio of the number of favorable

• utcomes to the total number of

possible outcomes.

Number of favorable outcomes Total number of possible outcomes

Probability

• f an event

P(event)

=

1 or 50% 2 1:2 or 0.5 Many Forms!

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Quartiles

The values that divide the data in four equal parts.

25% 25% 25% 25%

Q1 Q2

Q3

1,3,3,4,5,6,6,7,8,8 Q1 Q2

Q3

1 2 3 4 5 6 7 8

Q1

Q2 Q3

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Random Sample

Unbiased ­ Every possible sample of the same size has an equal chance of being selected

Asking everyone in a classroom if they believe flying is safe. Asking everyone in a classroom if they like chocolate. Asking everyone in a classroom if they like comic books.

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Range

The difference between the greatest data value and the least data value. 2, 4, 7, 12 12 ­ 2 = 10

The range is 10.

5, 9, 10, 40 40 ­ 5 = 35

The range is 35.

1, 5, 9, 18 18 ­ 1 = 17

The range is 17.

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Sample

A part of a group.

Population Sample

random

• r

unbiased

• only red m&ms

in a bag

• only poodles in a dog

park

• only girls wearing

glasses in a classroom

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Theoretical Probability

The ratio of the number of equally likely

• utcomes in an event to the total number of

possible outcomes.

number of favorable outcomes total number of possible outcomes

Probability

• f an event

Find the probability of getting tails when you flip a coin.

P(tails) =

# of favorable

• utcomes

# of possible

• utcomes

1 2

=

Back to Instruction

Upper Quartile

The median of the upper half of data.

25% 25% 25% 25%

Q1 Q2 Q3

1,3,3,4,5,6,6,7,8,8

Q1 Q2 Q3

1 2 3 4 5 6 7 8

Q1

Q2 Q3

Median

}

Median

}

Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull­tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull­tab.

Standards for Mathematical Practices