slide 1 / 188content.njctl.org › courses › math › 7th-grade › ... · slide 11 / 188 1...
TRANSCRIPT
Slide 3 / 188
Table of Contents
· Introduction to Probability· Experimental and Theoretical
· Word Problems· Probability of Compound Events
Click on a topic to go to that section.
· Sampling
· Measures of Center· Measures of Variation· Mean Absolute Deviation· Glossary
Slide 5 / 188
One way to express probability is to use a fraction.
Number of favorable outcomes
Total number of possible outcomes
Probability of an event
=
P(event)
Probability
Slide 6 / 188
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: 1 2
click
click
click
Probability
Slide 7 / 188
Probability can be expressed in many forms. For example, the probability of flipping a head can be expressed as:
1 or 50% or 1:2 or 0.5 2
The probability of randomly selecting a blue marble can be expressed as:
1 or 1:6 or 16.7% or .167 6
Probability
Slide 8 / 188
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).
0 14
12
34
1
Impo
ssibl
eUnli
kely
Equall
y Like
ly
Likely
Certai
n
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
Slide 9 / 188
Without counting, can you determine if the probability of picking a red marble is lesser or greater than 1/2?
It is very likely you will pick a red marble, so the probability is greater than 1/2 (or 50% or 0.5)
Click to Reveal
What is the probability of picking a red marble? 56Click
toReveal
Add the probabilities of both events. What is the sum?
1 + 5 = 16 6
Click to Reveal
Probability
Slide 10 / 188
Note: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 3/10 and the probability of getting cherry is 1/5, what is the probability of getting orange?
3 + 1 + ? = 110 5 ?
5 + ? = 110 ?
The probability of getting an orange jelly bean is 5 . 10
Probability
Slide 11 / 188
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 0
B 1/6C 1/2D 1
A R T H U R
Probability = Number of favorable outcomes Total number of possible outcomesClick for hint
Slide 12 / 188
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 0
B 1/6C 1/3D 1
A R T H U R
Probability = Number of favorable outcomes Total number of possible outcomes
A
Click for hint
Slide 13 / 188
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?
A 0
B 1/4
C 1/10
D 10/20
Probability = Number of favorable outcomes Total number of possible outcomes
Click for hint
Slide 14 / 188
4 What is the probability of rolling a 5 on a fair number cube?
Slide 15 / 188
5 What is the probability of rolling a composite number on a fair number cube?
Slide 16 / 188
6 What is the probability of rolling a 7 on a fair number cube?
Slide 17 / 188
7 You have black, blue, and white t-shirts in your closet. If the probability of picking a black t-shirt is 1/3 and the probability of picking a blue t-shirt is 1/2, what is the probability of picking a white t-shirt?
Slide 18 / 188
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?
Slide 19 / 188
9 Mary chooses an integer at random from 1 to 6. What is the probability that the integer she chooses is a prime number?
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.
Slide 20 / 188
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
Slide 21 / 188
Determine the fewest number of marbles, if any, and the color of these marbles that could be added to each hat so that the probability of picking a green marble will be one-half in each of 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
Slide 23 / 188
Click on an object. What is the outcome?Outcomes
Slide 24 / 188
number of times the outcome happened number of times experiment was repeated
Experimental Probability
Flip the coin 5 times and determine the experimental probability of heads.
Probability of an event
Heads Tails
Ans
wer
Slide 25 / 188
Example 1 - GolfA golf course offers a free game to golfers who make a hole-in-one on the last hole. Last week, 24 out of 124 golfers achieved this. Find the experimental probability that a golfer makes a hole-in-one on the last hole.
Out of 31 golfers, you could expect 6 to make a hole-in-one on the last hole. Or there is a 19% chance of a golfer making a hole-in-one on the last hole.
Experimental Probability
P(hole-in-one) = # of successes # of trials
= 24 124
= 6 31
Slide 26 / 188
Example 2 - SurveysOf 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
= 8 40
= 1 5
Slide 27 / 188
# 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
Slide 28 / 188
11 What is the experimental probability of rolling a 5?
A 1/2
B 5/4
C 4/5
D 2/5
# on Die Picture of Roll Results 1 1 one
2 3 twos
3 1 three
4 0 fours
5 4 fives
6 1 six
These are the results after 10 rolls of the die
Ans
wer
Slide 29 / 188
12 What is the experimental probability of rolling a 4?
A 1/2
B 5/4
C 0
D 4/4
# on Die Picture of Roll Results 1 1 one
2 3 twos
3 1 three
4 0 fours
5 4 fives
6 1 six
These are the results after 10 rolls of the die
Ans
wer
Slide 30 / 188
13 Based on the experimental probability you found, if you rolled the die 100 times, how many sixes would you expect to get?
A 6 sizes
B 10 sixes
C 12 sixes
D 60 sixes
These are the results after 10 rolls of the die
Ans
wer
# on Die Picture of Roll Results 1 1 one
2 3 twos
3 1 three
4 0 fours
5 4 fives
6 1 six
Slide 31 / 188
14 Mike flipped a coin 15 times and it landed on tails 11 times. What is the experimental probability of landing on heads?
Slide 32 / 188
Theoretical Probability
What is the theoretical probability of spinning green?
Is this a fair probability?
Slide 33 / 188
Probability of an event
Theoretical Probability
number of favorable outcomestotal number of possible outcomes
Slide 34 / 188
Theoretical Probability
Example 1 - MarblesFind 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
4 2 10 5
==
Slide 35 / 188
Theoretical ProbabilityExample 2 - MarblesSuppose 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
3 10
=
Slide 36 / 188
There is a 1 in 2 chance of getting tails when you flip a coin or a 50% possibility.
Theoretical ProbabilityExample 3 - CoinsFind the probability of getting tails when you flip a coin.
P(tails) = # of favorable outcomes # of possible outcomes
1 2
=
Slide 37 / 188
15 What is the theoretical probability of picking a green marble?
A 1/8
B 7/8
C 1/7
D 1
R
R
G
W
W
Y
Y
B
Slide 38 / 188
16 What is the theoretical probability of picking a black marble?
A 1/8
B 7/8
C 1/7
D 0
R
R
G
W
W
Y
Y
B
Slide 39 / 188
17 What is the theoretical probability of picking a white marble?
A 1/8
B 7/8
C 1/4
D 1
R
R
G
W
W
Y
Y
B
Slide 40 / 188
18 What is the theoretical probability of not picking a white marble?
A 3/4
B 7/8
C 1/7
D 1
R
R
G
W
W
YY
B
Slide 41 / 188
19 What is the theoretical probability of rolling a three?
A 1/2B 3C 1/6D 1
Slide 42 / 188
20 What is the theoretical probability of rolling an odd number? A 1/2B 3C 1/6D 5/6
Slide 43 / 188
21 What is the theoretical probability of rolling a number less than 5?A 2/3B 4C 1/6D 5/6
Slide 44 / 188
22 What is the theoretical probability of not rolling a 2?
A 2/3B 2C 1/6D 5/6
Slide 45 / 188
23 Seth tossed a fair coin five times and got five heads. The probability that the next toss will be a tail is
A 0
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.
Slide 46 / 188
24 Which inequality represents the probability, x, of any event happening?
A x ≥ 0B 0 < x < 1C x < 1D 0 ≤ x ≤ 1
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
Slide 47 / 188
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
Slide 48 / 188
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
Slide 49 / 188
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.
Slide 50 / 188
Answer the following:
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
Slide 52 / 188
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
Slide 53 / 188
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.
Slide 54 / 188
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
Slide 55 / 188
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.
YesNo
Slide 56 / 188
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.YesNo
Slide 57 / 188
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.
YesNo
Slide 58 / 188
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
Slide 59 / 188
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
Slide 60 / 188
Suppose this represents all the wolves on the mountain.
One way to estimate the number of wolves on a mountain is to use the capture-recapture method.
Sampling
Slide 61 / 188
Wildlife biologists first find some wolves and tag them.
Capture-Recapture Method
Slide 62 / 188
Then they release them back onto the mountain.
Capture-Recapture Method
Slide 63 / 188
They wait until all the wolves have mixed together.Then they find a second group of wolves and count how many are tagged.
Capture-Recapture Method
Slide 64 / 188
Biologists use a proportion to estimate the total number of wolves on 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.
=
8 2 w 9
2w = 72 w = 36
=
There are 36 wolves on the mountain
Capture-Recapture Method
Slide 65 / 188
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?
27 7 f 45
27(45) = 7f1215 = 7f173.57 = f
=
There are 174 fish in the river
Capture-Recapture Method
Slide 66 / 188
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?
Capture-Recapture Method
Slide 67 / 188
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?
Slide 68 / 188
31 Six out of 150 tires need to be realigned. How many out of 12,000 are going to need to be realigned?
Slide 69 / 188
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.
Slide 70 / 188
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?
Slide 71 / 188
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 high-top sneakers. Explain your reasoning.
Number of Students
Low-top sneakers 12
High-top sneakers 7
Sandals 3
Boots 6
Shoes
Ans
wer
Slide 72 / 188
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 three-month 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 popultation 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
Slide 73 / 188
Multiple SamplesThe 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.
Slide 74 / 188
Try This!
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
Ans
wer
Slide 75 / 188
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
Slide 76 / 188
36 How many students participated in each survey?
Student Sample Regular BBQ Cheddar
#1 8 10 7
#2 8 11 6
Ans
wer
Slide 77 / 188
37 According to the two random samples, which flavor potato chip should the student council purchase the most of?
A RegularB BBQC Cheddar
Student Sample Regular BBQ Cheddar
#1 8 10 7
#2 8 11 6
Ans
wer
Slide 78 / 188
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
Ans
wer
Slide 80 / 188
19 shots made 100 shots attempted
= 19%
Example 1 - SoccerErica 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
=Move to Reveal
Move to Reveal
click to reveal
Word Problem
Slide 81 / 188
19 100
20 100
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 on goal.
Erica takes 1,100 shots on goal.
clickclick
Word Problem
Slide 82 / 188
Erica figures she made about 200 of her shots on goal.
Erica wants to find 20% of 1,000. Her math looks like this:
click to reveal
Word Problem
Slide 83 / 188
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.Click for hint
Word Problem
Slide 84 / 188
Example 2 - GardeningLast year, Lexi planted 12 tulip bulbs, but only 10 of 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 out of 60 tulips to bloom.
Solve this proportion by equivalent fractions.
Experimental Probability
10 bloom 12 total
x bloom60 total
=
10 bloom 12 total
50 bloom 60 total
=
Slide 85 / 188
Example 3 - BasketballToday you attempted 50 free throws and made32 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
32 75 = 50 x
2400 = 50x
48 = x
32 made 50 attempts
x made 75 attempts
=
Slide 86 / 188
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
Slide 87 / 188
39 Tom was at bat 50 times and hit the ball 10 times. What is the experimental probability for hitting the ball?
Slide 88 / 188
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.
Slide 89 / 188
41 What is the theoretical probability of randomly selecting a jack from a deck of cards?
Slide 90 / 188
42 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the experimental probability for rolling a 3?
Slide 91 / 188
43 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the theoretical probability for rolling a 3?
Slide 92 / 188
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?
Slide 93 / 188
45 What is the probability of drawing a king or an ace from a standard deck of cards?
A 2/52B 4/52C 2/13D 8/52
Slide 94 / 188
46 What is the probability of drawing a five or a diamond from a standard deck of cards?
A 4/13B 13/52C 2/13D 16/52
Slide 95 / 188
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 only a movie ticket.
· Use info for next two questions.
From PARCC EOY sample test calculator #3
Slide 96 / 188
47 Part A
In the simulation, one result was "100". What does this result simulate?
A No one in a group of three randomly-chosen people who buy movie tickets also buys popcorn.
B Exactly one person in a group of three randomly-chosen people who buy movie tickets also buys popcorn.
C Exactly two people in a group of three randomly-chosen people who buy movie tickets also buy popcorn.
D All three people in a group of three randomly-chosen people who buy movie tickets also buy popcorn.
Slide 97 / 188
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.
Slide 99 / 188
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)
Slide 100 / 188
Independent ExampleSelect 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) 4 4 = 1 52 52 169
Slide 101 / 188
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
Slide 102 / 188
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) 4 4 = 4 52 51 663
Notice your demominator when down by 1. Why?
Slide 103 / 188
Independent Dependent
Try to name some other independent and dependent events.
Independent & Dependent Examples
Slide 104 / 188
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)
Slide 105 / 188
50 A lottery machine generates numbers randomly. Two numbers between 1 and 9 are generated. What is the probability that both numbers are 5?
Slide 106 / 188
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?
Slide 107 / 188
52 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement)
Slide 108 / 188
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?
Slide 109 / 188
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?
Slide 110 / 188
Slide 111 / 188
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
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.
Slide 112 / 188
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 2
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.
Slide 114 / 188
Sometimes we can make general statements about a set of data as shown in this first question.
Generalizations
Slide 115 / 188
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
Slide 116 / 188
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
Slide 117 / 188
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
Slide 118 / 188
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
Slide 119 / 188
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
Slide 120 / 188
Find the mini mean
1.25 + 1.75 = 1.50
2
What do you do when you have two numbers
left? (click)
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
Slide 121 / 188
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
Slide 122 / 188
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
Slide 123 / 188
Time Spent Texting Daily by
7th Grade Students
(in minutes)
600 15 30 45 75 90 105 120
Girls
600 15 30 45 75 90 105 120
Boys
Measures of Center
Use the dot plots to compare the 2 samples.
Slide 124 / 188
600 15 30 45 75 90 105 120
Girls
Find the mean, median, and mode for the sample of girls.
Measures of Center
Slide 125 / 188
600 15 30 45 75 90 105 120
Boys
Measures of Center
Find the mean, median, and mode for the sample of boys.
Slide 126 / 188
Girls BoysMean 88.5 40.5Median 90 30Mode 60 30 and 60
Measures of Center
Now compare the two measures of center.
Make a statement about the average time spent texting daily by 7th grade students.
Ans
wer
Slide 127 / 188
59 What is the mean of the stem-and-leaf plot?
Stem Leaf
1 1 1 22 0 0 3
5 54 8
Key: 1 | 1 = 11
Slide 128 / 188
60 What is the median of the stem-and-leaf plot?
Stem Leaf
1 1 1 22 0 0 3
5 54 8
Key: 1 | 1 = 11
Slide 129 / 188
61 What is the mode of the stem-and-leaf plot?
Stem Leaf
1 1 1 22 0 0 3
5 54 8
Key: 1 | 1 = 11
Slide 130 / 188
62 What is the mean of the stem-and-leaf plot?
Stem Leaf
1 8 93 7 7 9
Key: 1 | 8 = 1.8
Slide 131 / 188
63 What is the median of the stem-and-leaf plot?
Stem Leaf
1 8 93 7 7 9
Key: 1 | 8 = 1.8
Slide 132 / 188
64 What is the mode of the stem-and-leaf plot?
Stem Leaf
1 8 93 7 7 9
Key: 1 | 8 = 1.8
Slide 133 / 188
Miss M's Math Class Scores
9050 60 70 80 100
1st Period Scores
Measures of CenterUse the dot plots to find the measures of center.
Slide 134 / 188
Miss M's Math Class Scores
9050 60 70 80 100
8th Period Scores
Measures of CenterUse the dot plots to find the measures of center.
Slide 135 / 188
Measures of Center
Write a statement comparing the averages of Miss M's 1st period class scores to her 8th period class scores.
Slide 137 / 188
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.
Slide 138 / 188
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
Slide 139 / 188
1, 2, 2, 3, 5, 5, 5, 8, 8, 9
Step 2 : Find the median.
5median
*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
Slide 140 / 188
1, 2, 2, 3, 5, 5, 5, 8, 8, 9
5median
8Upper
2Lower
Quartile Quartile
Step 3 : Find the upper and lower quartiles.
Find the mean of each half of the data set.
Interquartile Range
Slide 141 / 188
Upper LowerQuartile Quartile
Step 4 : Subtract the lower quartile from the upper quartile.
Interquartile= Range-
8 - 2 = 6
Interquartile Range
Slide 142 / 188
1, 2, 2, 3, 5, 5, 5, 8, 8, 9
· To find the range, subtract the least value from the greatest value.
Greatest LeastValue Value = Range-
9 - 1 = 8
Sample Range
Slide 143 / 188
1, 2, 2, 3, 5, 5, 5, 8, 8, 9
5Median
8Upper
2Lower
Quartile Quartile
1Least
Value
9Greatest Value
Box-and-Whisker Plot
These 5 values are used to create a box-and-whisker 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
10 2 3 4 5 6 7 8 9 10
Slide 144 / 188
65 What is the median of the data set?
10 2 3 4 5 6 7 8 9 10
Slide 145 / 188
66 What is the interquartile range using the given information?
Least Value = 3
Lower Quartile = 6
Median = 7
Upper Quartile = 10
Greatest Value = 11
Slide 146 / 188
67 What is the range for the following data set?
3, 5, 10, 4, 2, 2, 1
Slide 147 / 188
68 What is the interquartile range for the following data set?
3, 5, 10, 4, 2, 2, 1
Slide 149 / 188
Mean absolute deviation - the average distance between each data value and the mean.
Mean Absolute Deviation - Vocabulary
Slide 150 / 188
Mean Absolute Deviation
Find the mean absolute deviation of the following data.
Quiz Scores
65, 75, 90, 90, 100
Step 1: Find the mean.
65 + 75 + 90 + 90 + 100 = 420 = 84 5 5
Slide 151 / 188
Score Deviation from mean
Absolute deviation from
mean65 65 - 84 = -19 |-19| = 1975 75 - 84 = -9 |-9| = 990 90 - 84 = 6 |6| = 690 90 - 84 = 6 |6| = 6100 100 - 84 = 16 |16| = 16
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
Slide 152 / 188
Absolute deviation from mean
|-19| = 19|-9| = 9|6| = 6|6| = 6|16| = 16
Mean Absolute DeviationStep 3: Find the mean absolute deviation (MAD). To do this find the mean using the absolute deviation numbers.
19 + 9 + 6 + 6 + 16 5
= 56 = 11.2 5
The MAD is
11.2 points.
Ans
wer
Slide 153 / 188
5 6 4 32 1 7 8 9 10 5 6 4 32 1 7 8 9 10
Girls' Team Boys' Team
Comparing Two Data SetsThe 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.
Slide 154 / 188
5 6 4 32 1 7 8 9 10
Girls' Team
5 6 4 32 1 7 8 9 10
Boys' Team
Comparing Two Data Sets
Step 1: Find the mean for each team.
Slide 155 / 188
Goals Mean Deviation Absolute Mean Dev. Goals Mean Deviation Absolute
Mean Dev.
Girls' Team Boys' Team
Comparing Two Data SetsStep 2: Find the absolute deviations.
Slide 156 / 188
Girls' Team Boys' Team
Comparing Two Data Sets
Step 3: Find the mean absolute deviations.
Slide 157 / 188
Comparing Two Data Sets
Comparison Statements1.25 = 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?)
Slide 158 / 188
Pages per Chapter in Hunger Games
10 15 20 25 30
x
xxxx
xxx
xxxxxxx
xxx
xxxx x
xxx x
10 15 20 25 30
xxxxx
xx
xx
xx
xx x
xx x
xxx
xx
Pages per Chapter in Twilight
Use the following data to answer the next seven questions.
Comparing Two Data Sets
Slide 159 / 188
69 What is the mean number of pages per chapter in the Hunger Games?
Slide 160 / 188
70 What is the mean number of pages per chapter in Twilight?
Slide 161 / 188
71 What is the difference of the means?
Slide 162 / 188
72 What is the mean absolute deviation of the data set for Hunger Games?
(Hint: Round mean to the nearest ones.)
Slide 163 / 188
73 What is the mean absolute deviation of the data set for Twilight?
(Hint: Round mean to the nearest ones.)
Slide 164 / 188
74 Which book has more variability in the number of pages per chapter?
A Hunger GamesB Twilight
Slide 165 / 188
75 On average, there are ______ pages per chapter in the Hunger Games than in Twilight.
A more
B less
Slide 167 / 188
Back to
Instruction
Biased SampleWhen 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.
Slide 168 / 188
Back to
Instruction
Capture-Recapture MethodA 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.
Slide 169 / 188
Back to
Instruction
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.
Slide 170 / 188
Back to
Instruction
Dependent Event
When the outcome of one affects the outcome of another event.
Probability (A & B) = Prob(A) *Prob(B given A)
The prob. of selecting a Jack
AND a 3 card.
Slide 171 / 188
Back to
Instruction
Equally Likely
When all the outcomes have the same chance of occurring.
Sides on a Coin
A Fair Die A Fair Spinner
Slide 172 / 188
Back to
Instruction
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 of an event
Last week, 24 out of 124 golfers hit a hole-in-one on the last hole. Find the
experimental probability that a golfer makes this
shot.
P(hole-in-one) =
# of successes = # of trials
24 124 = 6
31
Slide 173 / 188
Back to
Instruction
Fair
An experiment with equally likely outcomes.
Tossing a Coin
Rolling a Fair Die
Spinning a Fair Spinner
Slide 174 / 188
Back to
Instruction
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.
Slide 175 / 188
Back to
Instruction
Interquartile Range
The difference of the upper quartile and the lower quartile.
25% 25% 25% 25%
Q1 Q2 Q31,3,3,4,5,6,6,7,8,8
Q1 Q2 Q3
1 2 3 4 5 6 7 8
Q1 Q2 Q3
= Q3 - Q1 = Q3 - Q1= 4
Slide 176 / 188
Back to
Instruction
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
}
Slide 177 / 188
Back to
Instruction
MeanAverage
The sum of the data values divided by the number of items.
1, 2, 3, 4, 5Set of Data: 1+2+3+4+5
=15
15/5 = 3The mean
is 3.
Slide 178 / 188
Back to
Instruction
Mean Absolute Deviation
The average distance between each data value and the mean.
Find the mean Subtract
the mean from
each data point
Find the mean of the differences2,2,3,4,4
15 5=3
3-2=1
4-3=13-3=0 1+1+0+1+1
=4 5=.8
3-2=1
4-3=1
1. 2.3.
Slide 179 / 188
Back to
Instruction
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
Slide 180 / 188
Back to
Instruction
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.
Slide 181 / 188
Back to
Instruction
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
Slide 182 / 188
Back to
Instruction
What is the probability of flipping a nickel and the nickel landing on
heads?
1 favorable 2 possible
Probability
The ratio of the number of favorable outcomes to the total number of
possible outcomes.
Number of favorable outcomes
Total number of possible outcomes
Probability of an event
P(event)
= 1 or 50% 2
1:2 or 0.5
Many Forms!
Slide 183 / 188
Back to
Instruction
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
Slide 184 / 188
Back to
Instruction
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.
Slide 185 / 188
Back to
Instruction
Range
The difference between the greatest data value and the least data value.
2, 4, 7, 1212 - 2 = 10The range
is 10.
5, 9, 10, 4040 - 5 = 35The range
is 35.
1, 5, 9, 1818 - 1 = 17The range
is 17.
Slide 186 / 188
Back to
Instruction
Sample
A part of a group.
Population
Sample
random or
unbiased
· only red m&ms in a bag
· only poodles in a dog park
· only girls wearing glasses in a classroom
Slide 187 / 188
Back to
Instruction
Theoretical Probability
The ratio of the number of equally likely outcomes in an event to the total number of
possible outcomes.
number of favorable outcomes total number ofpossible outcomes
Probability of an event
Find the probability of
getting tails when you flip a coin.
P(tails) =
# of favorable outcomes # of possible outcomes
1 2
=
Slide 188 / 188
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}