Transcript Pre-Algebra

Frequency Tables, Line Plots, and Histograms
Lesson 12-1
Pre-Algebra
Additional Examples
A survey asked 22 students how many hours of TV
they watched daily. The results are shown. Display the data
in a frequency table. Then make a line plot.
1 3 4 3 1 1 2 3 4 1 3
2 2 1 3 2 1 2 3 2 4 3
List the numbers of
hours in order.
Number
1
2
3
4
Use a tally mark
for each result.
Tally
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Count the tally marks and
record the frequency.
Frequency
6
6
7
3
Frequency Tables, Line Plots, and Histograms
Lesson 12-1
Pre-Algebra
Additional Examples
Twenty-one judges were asked how many cases they were trying
on Monday. The frequency table below shows their responses. Display the
data in a line plot. Then find the range.
“How many cases are you trying?”
Number Frequency
0
3
1
5
2
4
3
5
4
4
Frequency Tables, Line Plots, and Histograms
Lesson 12-1
Pre-Algebra
Additional Examples
(continued)
For a line plot, follow these steps 1 , 2 , and 3 .
3 Write a title that describes the data.
Cases Tried by Judges
x
x
x
x
x
x
x
x
0
1
x
x
x
x
x
x
x
x
x
x
x
x
x
2
3
4
2 Mark an x for each
response.
1 Draw a number line with the choices below it.
The greatest value in the data set is 4 and the least value is 0. So the
range is 4 – 0, or 4.
Box-and-Whisker Plots
Lesson 12-2
Additional Examples
Pre-Algebra
The data below represent the wingspans in centimeters of
captured birds. Make a box-and-whisker plot.
61 35 61 22 33 29 40 62 21 49 72 75 28 21 54
Step 1: Arrange the data in order from least to greatest. Find the median.
21 21 22 28 29 33 35 40 49 54 61 61 62 72 75
Step 2: Find the lower quartile and upper quartile, which are the medians of the
lower and upper halves.
21 21 22 28 29 33 35 40 49 54 61 61 62 72 75
lower quartile = 28
upper quartile = 61
Box-and-Whisker Plots
Lesson 12-2
Additional Examples
Pre-Algebra
(continued)
Step 3: Draw a number line.
Mark the least and greatest values, the median, and the quartiles.
Draw a box from the first to the third quartiles.
Mark the median with a vertical segment.
Draw whiskers from the box to the least and greatest values.
Box-and-Whisker Plots
Lesson 12-2
Additional Examples
Pre-Algebra
Use box-and-whisker plots to compare test scores from
two math classes.
Class A: 92, 84, 76, 68, 90, 67, 82, 71, 79, 85, 79
Class B: 78, 93, 81, 98, 69, 95, 74, 87, 81, 75, 83
Draw a number line for both sets of data. Use the range of data
points to choose a scale.
Draw the second box-and-whisker plot below the first one.
Box-and-Whisker Plots
Lesson 12-2
Additional Examples
Pre-Algebra
Describe the data in the box-and-whisker plot.
The lowest score is 55 and the highest is 85.
One fourth of the scores are at or below 66 and one fourth of the scores
are at or above 80.
Half of the scores are at or between 66 and 80 and thus within 10 points
of the median, 76.
Box-and-Whisker Plots
Lesson 12-2
Additional Examples
Pre-Algebra
The plots below compare the percents of students who
were eligible to those who participated in extracurricular activities
in one school from 1992 to 2002. What conclusions can you draw?
About 95% of the students were eligible to participate in extracurricular
activities.
Around 60% of the students did participate.
A little less than two thirds of the eligible students participated in
extracurricular activities.
Using Graphs to Persuade
Lesson 12-3
Additional Examples
Pre-Algebra
Which title would be more appropriate for the graph
below: “Texas Overwhelms California” or “Areas of California and
Texas”? Explain.
Using Graphs to Persuade
Lesson 12-3
Additional Examples
Pre-Algebra
(continued)
Because of the break in the vertical axis, the bar for Texas appears to be
more than six times the height of the bar for California.
Actually, the area of Texas is about 267,000 mi2, which is not even two times
the area of California, which is about 159,000 mi2.
The title “Texas Overwhelms California” could be misleading. “Areas of Texas
and California” better describes the information in the graph.
Using Graphs to Persuade
Lesson 12-3
Additional Examples
Pre-Algebra
Study the graphs below. Which graph gives the impression of
a sharper increase in rainfall from March to April? Explain.
Using Graphs to Persuade
Lesson 12-3
Additional Examples
Pre-Algebra
(continued)
In the second graph, the months are closer together and the rainfall amounts
are farther apart than in the first graph.
Thus the line appears to climb more rapidly from March to April in the
second graph.
Using Graphs to Persuade
Lesson 12-3
Additional Examples
Pre-Algebra
What makes the graph misleading? Explain.
The “cake” on the right has much more than two times the area of the
cake on the left.
Counting Outcomes and Theoretical Probability
Lesson 12-4
Pre-Algebra
Additional Examples
The school cafeteria sells sandwiches for which you can choose
one item from each of the following categories: two breads (wheat or
white), two meats (ham or turkey), and two condiments (mayonnaise or
mustard). Draw a tree diagram to find the number of sandwich choices.
ham
wheat
turkey
ham
white
turkey
mayonnaise
mustard
mayonnaise
mustard
mayonnaise
mustard
mayonnaise
mustard
There are 8 possible sandwich choices.
Each branch of the “tree”
represents one choice—for
example, wheat-hammayonnaise.
Counting Outcomes and Theoretical Probability
Lesson 12-4
Pre-Algebra
Additional Examples
How many two-digit numbers can be formed for which the first
digit is odd and the second digit is even?
first digit,
possible choices
5
second digit,
possible choices
•
5
numbers,
possible choices
=
25
There are 25 possible two-digit numbers in which the first digit is odd and
the second digit is even.
Counting Outcomes and Theoretical Probability
Lesson 12-4
Pre-Algebra
Additional Examples
Use a tree diagram to show the sample space for guessing
right or wrong on two true-false questions. Then find the probability of
guessing correctly on both questions.
right
right
wrong
right
wrong
The tree diagram shows there are four
possible outcomes, one of which is
guessing correctly on both questions.
wrong
number of favorable outcomes
P(event) = number of possible outcomes
=
Use the probability formula.
1
4
The probability of guessing correctly on two true/false questions is 1 .
4
Counting Outcomes and Theoretical Probability
Lesson 12-4
Pre-Algebra
Additional Examples
In some state lotteries, the winning number is made up of five
digits chosen at random. Suppose a player buys 5 tickets with different
numbers. What is the probability that the player has a winning number?
First find the number of possible outcomes. For each digit, there are 10
possible outcomes, 0 through 9.
1st digit
2nd digit
3rd digit
4th digit
5th digit
outcomes outcomes outcomes outcomes outcomes
•
•
•
•
10
10
10
10
10
total
outcomes
= 100,000
Then find the probability when there are five favorable outcomes.
number of favorable outcomes
5
P(winning number) = number of possible outcomes = 100,000
The probability is
5
1
, or
.
100,000
20,000
Independent and Dependent Events
Lesson 12-5
Pre-Algebra
Additional Examples
You roll a number cube once. Then you roll it again. What is the
probability that you get 5 on the first roll and a number less than 4 on the
second roll?
P(5) =
1
6
There is one 5 among 6 numbers on a number cube.
P(less than 4) =
3
6
There are three numbers less than 4 on a number cube.
P(5, then less than 4) = P(5) • P(less than 4)
1
3
= 6 • 6
=
3
1
, or
36
12
The probability of rolling 5 and then a number less than 4 is 1 .
12
Independent and Dependent Events
Lesson 12-5
Pre-Algebra
Additional Examples
Bluebonnets grow wild in the southwestern United States.
Under the best conditions in the wild, each bluebonnet seed has a 20%
probability of growing. Suppose you plant bluebonnet seeds in your
garden and use a fertilizer that increases to 50% the probability that a
seed will grow. If you select two seeds at random, what is the probability
that both will grow in your garden?
P(a seed grows) = 50%, or 0.50
Write the percent as a decimal.
P(two seeds grow) = P(a seed grows) • P(a seed grows)
= 0.50 • 0.50
Substitute.
= 0.25
Multiply.
= 25%
Write 0.25 as a percent.
The probability that two seeds will grow is 25%.
Independent and Dependent Events
Lesson 12-5
Pre-Algebra
Additional Examples
Three girls and two boys volunteer to represent their class at a
school assembly. The teacher selects one name and then another from a
bag containing the five students’ names. What is the probability that both
representatives will be boys?
P(boy) =
2
5
Two of five students are boys.
P(boy after boy) =
1
4
If a boy’s name is drawn, one of the four
remaining students is a boy.
P(boy, then boy) = P(boy) • P(boy after boy)
2
1
= 5 • 4
=
2
1
, or
20
10
Substitute.
Simplify.
The probability that both representatives will be boys is
1
.
10
Permutations and Combinations
Lesson 12-6
Additional Examples
Pre-Algebra
Find the number of permutations possible for the
letters H, O, M, E, and S.
1st letter
2nd letter
3rd letter
4th letter
5th letter
5 choices
4 choices
3 choices
2 choices
1 choice
•
•
•
•
5
4
3
2
1
There are 120 permutations of the letters H, O, M, E, and S.
= 120
Permutations and Combinations
Lesson 12-6
Pre-Algebra
Additional Examples
In how many ways can you line up 3 students chosen from 7
students for a photograph?
7 students
Choose 3.
7P3
= 7 • 6 • 5 = 210
Simplify
You can line up 3 students from 7 in 210 ways.
Permutations and Combinations
Lesson 12-6
Pre-Algebra
Additional Examples
In how many ways can you choose two states from the
table when you write reports about the areas of states?
State
Area (mi2)
Alabama
50,750
Colorado 103,729
Maine
30,865
Oregon
96,003
Texas
261,914
Make an organized list of all the combinations.
Permutations and Combinations
Lesson 12-6
Pre-Algebra
Additional Examples
(continued)
AL, CO AL, ME
CO, ME
AL, OR AL, TX
CO, OR CO, TX
ME, OR ME, TX
OR, TX
Use abbreviations of each
state’s name. First, list all
pairs containing Alabama.
Continue until every pair
of states is listed.
There are ten ways to choose two states from a list of five.
Permutations and Combinations
Lesson 12-6
Pre-Algebra
Additional Examples
How many different pizzas can you make if you can
choose exactly 5 toppings from 9 that are available?
9 toppings
Choose 5.
P
9 5
9C5 = P
5 5
9 • 81 • 7 • 6 2 • 51
= 5 • 4 • 3 • 2 • 1 = 126 Simplify.
1
1
1
1
You can make 126 different pizzas.
Permutations and Combinations
Lesson 12-6
Additional Examples
Pre-Algebra
Tell which type of arrangement—permutations or
combinations—each problem involves. Explain.
a. How many different groups of three vegetables could you choose from
six different vegetables?
Combinations; the order of the vegetables selected does not matter.
b. In how many different orders can you play 4 DVDs?
Permutations; the order in which you play the DVDs matters.
Experimental Probability
Lesson 12-7
Pre-Algebra
Additional Examples
A medical study tests a new medicine on 3,500 people. It
produces side effects for 1,715 people. Find the experimental
probability that the medicine will cause side effects.
number of times an event occurs
P(event) = number of times an experiment is done
1,715
= 3,500 = 0.49
The experimental probability that the medicine will cause side effects
is 0.49, or 49%.
Experimental Probability
Lesson 12-7
Pre-Algebra
Additional Examples
Simulate the correct guessing of answers on a multiple-choice
test where each problem has four answer choices (A, B, C, and D).
Use a 4-section spinner to simulate each guess. Mark the sections as
1, 2, 3, and 4. Let “1” represent a correct choice.
Here are the results of 50 trials.
22431
32134
13431
12224
43121
42213
21243
34424
33434
32412
number of times an event occurs
10
1
P(event) = number of times an experiment is done = 50 = 5
The experimental probability of guessing correctly is
1
.
5
Random Samples and Surveys
Lesson 12-8
Additional Examples
Pre-Algebra
You want to find out how many people in the community
use computers on a daily basis. Tell whether each survey plan
describes a good sample. Explain.
a. Interview every tenth person leaving a computer store.
This is not a good sample. People leaving a computer store are more
likely to own computers.
b. Interview people at random at the shopping center.
This is a good sample. It is selected at random from the population you
want to study.
c. Interview every tenth student who arrives at school on a school bus.
This is not a good sample. This sample will be composed primarily of
students, but the population you are investigating is the whole community.
Random Samples and Surveys
Lesson 12-8
Pre-Algebra
Additional Examples
From 20,000 calculators produced, a manufacturer takes a
random sample of 300 calculators. The sample has 2 defective calculators.
Estimate the total number of defective calculators.
defective sample calculators
defective calculators
=
sample calculators
calculators
2
n
=
300
20,000
2(20,000) = 300n
2(20,000) 300n
= 300
300
133
n
Estimate: About 133 calculators are defective.
Write a proportion.
Substitute.
Write cross products.
Divide each side by 300.
Simplify.
Problem Solving Strategy: Simulate the Problem
Lesson 12-9
Additional Examples
Pre-Algebra
A softball player has an average of getting a base hit 2
times in every 7 times at bat. What is an experimental probability
that she will get a base hit the next time she is at bat?
You can use a spinner to simulate the problem.
Construct a spinner with seven congruent
sections. Make five of the sections blue and
two of them red. The blue sections represent
not getting a base hit and the red sections
represent getting a base hit. Each spin
represents one time at bat.
Problem Solving Strategy: Simulate the Problem
Lesson 12-9
Pre-Algebra
Additional Examples
(continued)
Use the results given in the table below. “B” stands for blue and “R” stands
for red.
B
B
R
B
B
B
R
B
B
B
B
R
B
B
B
B
B
B
R
B
B
B
B
B
R
B
B
B
R
B
B
B
R
B
B
R
B
B
B
B
R
R
B
B
B
R
B
B
B
B
R
B
B
B
B
B
B
B
R
R
B
B
B
B
R
B
B
R
B
B
B
B
B
B
B
B
B
R
B
B
R
B
R
B
B
B
B
B
B
B
B
R
B
B
B
R
B
B
B
R
Problem Solving Strategy: Simulate the Problem
Lesson 12-9
Pre-Algebra
Additional Examples
(continued)
Make a frequency table.
Makes a Base Hit
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Doesn’t Make a Base Hit
|||| |||| |||| |||| |||| |||| |||| ||||
|||| |||| |||| |||| |||| |||| |||| |||
An experimental probability that she gets a base hit the next time she
is at bat is 0.22, or 22%.