LESSON
8.2
Name
Relative Frequency
8.2
Class
Date
Relative Frequency
Essential Question: How can you recognize possible associations and trends between two
categories of categorical data?
Common Core Math Standards
The student is expected to:
Resource
Locker
S-ID.5
Explore
Summarize categorical data for two categories in two-way frequency
tables. Interpret relative frequencies in the context of the data (including
joint, marginal, and conditional relative frequencies). Recognize possible
associations and trends in the data.
To show what portion of a data set each category in a frequency
table makes up, you can convert the data to relative frequencies. The
relative frequency of a category is the frequency of the category
divided by the total of all frequencies.
The frequency table below shows the results of a survey Kenesha
conducted at school. She asked 80 randomly selected students whether they
preferred basketball, football, or soccer.
Mathematical Practices
MP.4 Modeling
Language Objective
Favorite Sport
Frequency
In a two-way relative frequency table, identify a joint relative frequency
and a marginal relative frequency, and explain what they mean.

You can use relative frequencies to interpret
associations and trends between two categories in
two-way frequency tables.
PREVIEW: LESSON
PERFORMANCE TASK
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Rob
Hainer/Shutterstock
ENGAGE
Essential Question: How can you
recognize possible associations and
trends between two categories of
categorical data?
Relative Frequencies
Basketball
Football
Soccer
Total
20
32
28
80
Use the frequencies to make a relative frequency table expressed with decimals.
Favorite Sport
Basketball
Football
Soccer
Relative Frequency
20
_
= 0.25
80
32
_
= 0.4
28
_
= 0.35

80
80
Total
80
_
= 1
80
Rewrite the relative frequency table using percents instead of decimals.
Favorite Sport
Basketball
Football
Relative Frequency
25%
40%
Soccer
Total
35%
100%
Reflect
1.
20 refer to in part A.
Explain what the numerator and denominator of the ratio _
80
20 refers to the number of students who prefer basketball, and 80 refers to the total
number of students surveyed.
2.
What types of numbers can you use to write relative frequencies?
fractions, decimals, and percents
View the Engage section online. Discuss whether
students believe that boys or girls are more likely to
play a musical instrument. Then preview the Lesson
Performance Task.
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Lesson 2
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Lesson 8.2
4/9/14
5:58 PM
7/25/14 7:48 AM
Explain 1
Two-Way Relative Frequency Tables
EXPLORE
Two types of relative frequencies are found in a relative frequency table:
1. A joint relative frequency is found by dividing a frequency that is not in the Total
row or the Total column by the grand total. It tells what portion of the total has
both of the two specified characteristics.
Relative Frequencies
2. A marginal relative frequency is found by dividing a row total or a column total by
the grand total. It tells what portion of the total has a specified characteristic.
Example 1

INTEGRATE TECHNOLOGY
Complete a two-way relative frequency table from the data in a two-way
frequency table. Identify the joint relative frequencies and the marginal
relative frequencies.
Spreadsheet programs can produce tables that
automatically make calculations for frequency tables
and relative frequency tables. Challenge students to
use formulas to program a spreadsheet that computes
row totals, column totals, and relative frequencies for
a two-way frequency table.
For her survey about sports preferences, Kenesha also recorded the gender of each student.
The results are shown in the two-way frequency table for Kenesha’s data.
Preferred Sport
Gender
Basketball
Football
Soccer
Total
Girl
6
12
18
36
Boy
14
20
10
44
Total
20
32
28
80
CONNECT VOCABULARY
Make sure that students can distinguish between
frequency and relative frequency. Frequency is a
measure of the raw data (for example, 20 students
play basketball), while relative frequency compares
that frequency to the total (for example, 25% of all
students play basketball).
To find the relative frequencies, divide each frequency by the grand total.
Preferred Sport
Basketball
Football
Soccer
Total
Girl
6
_
= 0.075
80
12 = 0.15
_
80
18
_
= 0.225
80
36
_
= 0.45
80
Boy
14 = 0.175
_
80
20
_
= 0.25
80
10
_
= 0.125
80
44 = 0.55
_
80
Total
20
_
= 0.25
80
32
_
= 0.4
80
28
_
= 0.35
80
80
_
=1
80
© Houghton Mifflin Harcourt Publishing Company
Gender
The joint relative frequencies tell what percent of all those surveyed are in each category:
• 7.5% are girls who prefer basketball.
• 17.5% are boys who prefer basketball.
• 15% are girls who prefer football.
• 25% are boys who prefer football.
• 22.5% are girls who prefer soccer.
• 12.5% are boys who prefer soccer.
The marginal relative frequencies tell what percent of totals has a given single characteristic:
• 25% prefer basketball.
• 45% are girls.
• 40% prefer football.
• 55% are boys.
EXPLAIN 1
Two-Way Relative Frequency Tables
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Discuss with students the relationship
• 35% prefer soccer.
Module 8
360
between two-way frequency tables and two-way
relative frequency tables. Make sure they understand
that each relative frequency is calculated by dividing
the corresponding value in a frequency table by the
grand total.
Lesson 2
PROFESSIONAL DEVELOPMENT
IN1_MNLESE389755_U4M08L2.indd 360
Math Background
01/04/14 8:56 PM
Relative frequencies are often used when determining conditional probability.
In the notation of probability, P(A) means probability of A. The conditional
probability of A being true given that B is true is written as P(A|B), which is read
“the probability of A given B.” This conditional probability is given by
P(A ∩ B)
P(B)
P(A|B) = _. In other words, it is found by dividing the number of cases
when both A and B are true by the total number of cases when B is true.
Relative Frequency 360
B
QUESTIONING STRATEGIES
Which values in a frequency table are
compared to the total by a joint relative
frequency? Which values are compared to the total by
a marginal relative frequency? Explain, using
examples. A joint relative frequency compares an
individual frequency to the grand total. For
example, in a table showing gender and sport
preference, the percent of the total who are girls
who like soccer is a joint relative frequency. A
marginal relative frequency compares a category
total, either across a row or down a column, to the
grand total. For example, in the same table, the
percent of the total who are boys is a marginal
relative frequency.
Millie performed a survey of students in the lunch line and recorded which type of fruit
each student selected along with the gender of each student. The two-variable frequency
data she collected is shown in the table.
Fruit
Apple
Banana
Orange
Total
Girl
16
10
14
40
Boy
25
13
14
52
Total
41
23
28
92
Fruit
Apple
Banana
Orange
Total
Girl
17.4%
10.9%
15.2%
43.5%
Boy
27.2%
14.1%
15.2%
56.5%
Total
44.6%
25.0%
30.4%
100.0%
The joint relative frequencies:
•
17.4% are girls who selected an apple.
•
27.2% are boys who selected an apple.
•
10.9% are girls who selected a banana.
•
14.1% are boys who selected a banana.
•
15.2% are girls who selected an orange.
•
15.2% are boys who selected an orange.
44.6% selected an apple.
•
43.5% are girls.
selected a banana.
•
56.5% are boys.
© Houghton Mifflin Harcourt Publishing Company
The marginal relative frequencies:
•
•
•
25%
30.4% selected an orange.
Reflect
3.
Discussion Explain how you can use joint and marginal relative frequencies to check your relative
frequency table.
You can confirm that the sum of each row and column of joint relative frequencies equals
the marginal relative frequencies at the end of the same row or column.
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361
Lesson 2
COLLABORATIVE LEARNING
IN1_MNLESE389755_U4M08L2.indd 361
Whole Class Activity
Have students conduct their own surveys using left-right handed and a threecategory variable of their choice. Then have students organize the data in two-way
frequency and relative frequency tables. Finally, have students calculate
conditional relative frequencies to analyze the data and investigate any possible
association between left-right handed and the other variable.
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Lesson 8.2
01/04/14 8:56 PM
Your Turn
EXPLAIN 2
Use the two-way table of data from another student survey to answer the following
questions.
Conditional Relative Frequencies
Like Aerobic Exercise
Like Weight Lifting
Yes
No
Total
Yes
7
14
21
No
12
7
19
Total
19
21
40
CONNECT VOCABULARY
4.
Find the joint relative frequency of students surveyed who like aerobics exercise but dislike weight lifting.
12
= 0.3 or 30%
40
5.
What is the marginal relative frequency of students surveyed who like weight lifting?
21
= 0.525 or 52.5%
40
Focus on the term conditional relative frequency. A
relative frequency compares a frequency to the grand
total of the data. A conditional relative frequency goes
one step further and compares this frequency in a
conditional sense—that is, under the condition that
only a subset of the grand total is considered.
_
_
Explain 2
Conditional Relative Frequencies
A conditional relative frequency describes what portion of a group with a given characteristic also has another
characteristic. A conditional relative frequency is found by dividing a frequency that is not in the Total row or the
Total column by the total for that row or column.
Example 2

AVOID COMMON ERRORS
When finding conditional relative frequencies,
students can avoid errors by remembering that the
quantity that is given will be in the denominator of
the fraction. For example, to find the conditional
relative frequency that a student is a boy, given that
the student prefers basketball, the number of boys
who prefer basketball must be compared to the total
number who prefer basketball, not to the total who
are boys.
Use the joint relative frequencies to calculate the associated conditional
relative frequencies and describe what each one means.
Use the data from Example 1A. Find the conditional relative frequency that a a person in
Kenesha's survey prefers soccer, given that the person is a girl.
Divide the number of girls who prefer soccer by the total number of girls.

© Houghton Mifflin Harcourt Publishing Company
Number of girls who prefer soccer _
___
= 18 = 0.5 = 50%
36
Total number of girls
Half of the girls in the sample prefer soccer.
Use the data from Example 1B. Find the conditional relative frequency that a student in
Millie's survey chose an orange, given that the student is a boy.
14
Number of boys who chose an orange
____
= _ ≈ 0.269 =
Total number of
boys
52
26.9 %
Your Turn
Use the data from Your Turn Exercises 4 and 5 after Example 1.
6.
QUESTIONING STRATEGIES
When determining a conditional relative
frequency, how do you know what number to
use as the numerator of the fraction? The numerator
will be the number of items that satisfy both the
given condition and the characteristic for which you
are finding the conditional relative frequency.
What is the conditional relative frequency that a student likes to lift weights, given that the student does
not like aerobics?
14
= 0.667 or 66.7%
21
_
7.
Find the conditional relative frequency that a student likes to lift weights, given that the student likes aerobics.
7 = 0.368 or 36.8%
_
19
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Lesson 2
DIFFERENTIATE INSTRUCTION
IN1_MNLESE389755_U4M08L2 362
Multiple Representations
6/10/15 8:46 AM
To help students understand how one characteristic can influence a distribution,
use the real-world example of left-handed pitchers in baseball. About 10% of the
population is left-handed, so all other things being equal, one would expect 10%
of baseball pitchers to be left-handed. In fact, more than 25% of pitchers are
left-handed, meaning that there are more than twice as many left-handers as
would be expected. This suggests that there are more left-handed pitchers because
they enjoy some kind of advantage in being left-handed.
Relative Frequency 362
Explain 3
EXPLAIN 3
Finding Possible Associations
You can analyze two-way frequency tables to locate possible associations or patterns in the data.
Example 3
Finding Possible Associations
Analyze the results of the surveys to determine preferences by gender.
Kenesha is interested in the question, “Does gender influence what type of sport students prefer?”
If there is no influence, then the distribution of gender within each sport preference will roughly
equal the distribution of gender within the whole group. Analyze the results of Kenesha’s survey
from Example 1. Determine which sport each gender is more likely to prefer.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 When looking for possible associations, make

Analyze the data about girls that were surveyed.
Step 1: Identify the percent of all students surveyed who are girls.
36 = 0.45 = 45%
_
80
sure students understand what kind of data should be
expected when no association or influence exists.
Remind students to keep this idea in mind as they
work through a problem. The actual conditional
relative frequencies must always be compared to the
conditional relative frequencies that would be
expected if there were no association between the
variables.
Step 2: Determine each conditional relative frequency.
Basketball
Of the 20 students who
prefer basketball, 6 are
girls.
6
= 0.3 = 30%
20
_
Football
Of the 32 students who
prefer football, 12 are
girls.
12 = 0.375 = 37.5%
_
32
Of the 28 students who
prefer soccer, 18 are
girls.
18
_
≈ 0.643 = 64.3%
28
Step 3: Interpret the results by comparing each conditional relative frequency to the percent
of all students surveyed who are girls, 45%.
Basketball
30% < 45%
QUESTIONING STRATEGIES

Football
Soccer
37.5% < 45%
Girls are less likely to
prefer basketball.
© Houghton Mifflin Harcourt Publishing Company
What numbers in a two-way relative
frequency table can be used to determine
whether a possible association exists? Explain.
Divide the joint relative frequency for a given cell by
the marginal relative frequency for that column.
Compare the quotient to the marginal relative
frequency for the row that includes the given cell. If
the two numbers are different, there is a possible
association.
Soccer
Girls are less likely to
prefer football.
64.3% > 45%
Girls are more likely to
prefer soccer.
Analyze the data about boys that were surveyed.
Step 1: Identify the percent of all students surveyed who are boys.
44
_ = 0. 55 = 55 %
80
Step 2: Determine each conditional relative frequency.
Basketball
Of the 20 students who
prefer basketball, 14 are
boys.
14
_ = 0. 7 = 70 %
20
Module 8
Football
Soccer
Of the 32 students who
Of the 28 students who
prefer football, 20 are
boys.
prefer soccer, 10 are
boys.
20
_ = 0.
32
625 = 62.5 %
363
10
_ = 0.
28
357 = 35.7 %
Lesson 2
LANGUAGE SUPPORT
IN1_MNLESE389755_U4M08L2 363
Connect Vocabulary
There are many words in algebra that are borrowed from other languages or have
shared Latin roots across languages. English learners who speak Spanish may
benefit from looking for words that may already be familiar in their primary
language. Many multi-syllabic words that end with –al in English (such as final,
conditional, and visual) are cognates with words in Spanish that end with –al
(final, condicional, and visual).
363
Lesson 8.2
7/25/14 7:52 AM
Step 3: Interpret the results by comparing each conditional relative frequency to the percent of all students
ELABORATE
surveyed who are boys, 55 %.
Basketball
Football
70% > 55%
Boys are more likely to
prefer basketball.
Soccer
62.5% > 55%
Boys are more likely to prefer
football.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Discuss the similarities between conditional
35.7% < 55%
Boys are less likely to prefer
soccer.
Reflect
8.
relative frequencies and conditional probability.
Students should understand that for events A and B,
the probability of A given B is different from the
probability of B given A.
Making Connections How can the statement “6 out of the 20 students who prefer basketball are girls”
be stated as a conditional relative frequency?
Given that a student prefers basketball, 30% are girls.
Your Turn
9.
Analyze the data given in the Your Turn after Example 1 to determine if liking aerobic exercise influences
whether a person also likes weight lifting. Explain.
SUMMARIZE THE LESSON
Those who like aerobics are less likely than those who do not like aerobics to like
weight lifting. Possible answer: 36.8% of respondents who like aerobics like weight
lifting, while 52.5% of all respondents like weight lifting.
t
How can you determine whether one category
has an influence over the distribution of the
data in a two-way frequency table? First, find what
percent of the grand total is in the category of
interest. Then, find the conditional relative
frequencies for the category given each choice of
the second categorical variable. Compare each
conditional relative frequency to the percent of the
entire group made up by the category. If the two
percents are roughly equal, the category has little
or no influence over the distribution. If the percents
are very different, the category does have an
influence.
Elaborate
10. What does it mean to say there is an association between characteristics in a two-way frequency table?
It means that there is a pattern between the two variables so that the conditional relative
frequencies, or conditional probabilities, are somewhat predictable.
11. Essential Question Check-In How can you use two-way frequency data to recognize possible
associations between the two categories of categorical data?
You can compare conditional relative frequencies with different givens to each other or to
other conditions or the entire data set.
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IN1_MNLESE389755_U4M08L2 364
364
© Houghton Mifflin Harcourt Publishing Company
the marginal relative frequencies to see if certain conditions favor a category relative to
Lesson 2
7/25/14 7:56 AM
Relative Frequency 364
Evaluate: Homework and Practice
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Use the table of frequency data for Exercises 1–4.
Class Survey of Favorite Colors
1.
ASSIGNMENT GUIDE
Concepts and Skills
Practice
Explore
Relative Frequencies
Exercises 1–4
Example 1
Two-Way Relative Frequency Tables
Exercises 5–13
Example 2
Conditional Relative Frequencies
Exercises 14–19,
24, 26–27
Example 3
Finding Possible Associations
Exercises 20–23,
25
Favorite Color
Red
Orange
Yellow
Green
Blue
Purple
Total
Frequency
2
5
1
6
8
2
24
Complete the relative frequency table for this data using decimals rounded to the nearest thousandth.
Class Survey of Favorite Colors
2.
Favorite Color
Red
Orange
Yellow
Green
Blue
Purple
Total
Relative
Frequency
0.083
0.208
0.042
0.25
0.333
0.083
1
Complete the relative frequency table for this data using percents rounded to the nearest tenth.
Class Survey of Favorite Colors
3.
Favorite Color
Red
Orange
Yellow
Green
Blue
Purple
Total
Relative
Frequency
8.3%
20.8%
4.2%
25%
33.3%
8.3%
100.0%
What is the relative frequency of having blue
as a favorite color, expressed as a decimal?
4.
Which color is a favorite color with a relative
frequency of 25%?
0.333
VISUAL CUES
AVOID COMMON ERRORS
When finding conditional relative frequencies, some
students may divide a frequency by the grand total
instead of the frequency’s row total or column total.
Suggest that students outline or highlight the row or
column that is “given” to identify the part of the table
that is relevant to the question.
The following frequency data shows the number of states, including the District of Columbia, that
favored each party in the presidential popular vote in 1976 and in 2012.
2012 Election
© Houghton Mifflin Harcourt Publishing Company
To visually distinguish between different parts of a
two-way relative frequency table, students could use
one color to shade the part of the table that represents
joint relative frequencies (the cells at the intersection
of two categories) and a different color to shade the
part of the table that represents marginal relative
frequencies (the row and column totals).
1976 Election
Republican
Total
Democrat
12 = 23.5%
12 = 23.5%
24 = 47.1%
Republican
15 = 29.4%
12 = 23.5%
27 = 52.9%
Total
27 = 52.9%
24 = 47.1%
51 =
100%
Complete the table above with relative frequencies using percents.
6.
What percent switched from Democrat in 1976 to Republican in 2012?
What type of frequency is this?
23.5%; a joint relative frequency
7.
What percent voted Republican in 1976? What type of frequency is this?
52.9%; a marginal relative frequency
Module 8
Exercise
Lesson 8.2
Democrat
5.
IN1_MNLESE389755_U4M08L2 365
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green
Lesson 2
365
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–13
1 Recall of Information
MP.4 Modeling
14–24
2 Skills/Concepts
MP.4 Modeling
25–27
3 Strategic Thinking
MP.3 Logic
7/25/14 7:59 AM
The results of a survey of 45 students and the foreign language they are studying
are shown in the two-way frequency table.
COGNITIVE STRATEGIES
Students may want to use the list below to solve
conditional relative frequency problems such as this
one: Find the conditional relative frequency that a
student is a boy, given that the student prefers soccer.
Students should fill in each item.
Language
8.
Gender
Chinese
French
Spanish
Total
Girl
2
8
15
25
Boy
4
4
12
20
Total
6
12
27
45
Given: prefers soccer
Fill in the table of two-way relative frequencies using decimals, rounded to the
nearest thousandth.
Find relative frequency of: boy
Frequency of: boys who play soccer
Language
9.
Gender
Chinese
French
Spanish
Total
Girl
0.044
0.178
0.333
0.556
Boy
0.089
0.089
0.267
0.444
Total
0.133
0.267
0.6
1
What fraction of the surveyed students are boys
taking Spanish?
4
≈ 0.267
15
Total: all who prefer soccer
Percent:
boys who prefer soccer ÷ all who prefer soccer
10. What fraction of the surveyed students are taking
Chinese?
2
≈ 0.133
15
_
11. Complete the table of two-way relative frequencies using percents.
Knows the Law
Yes
No
Total
45 = 18%
82%
Yes
160 = 64%
No
25 = 10%
20 = 8%
18%
Total
74%
26%
100%
12. What is the relative frequency of drivers who
know and obey the law?
13. What is the relative frequency of drivers who
know the law?
64%
Module 8
IN1_MNLESE389755_U4M08L2 366
Students may select the wrong value to use as the
denominator when calculating relative frequencies.
Remind students that when finding a joint or
marginal relative frequency, the denominator of the
fraction is the grand total from the frequency table.
When finding a conditional relative frequency, the
denominator of the fraction is the quantity that is
given as the condition.
© Houghton Mifflin Harcourt Publishing Company . image credit: ©Leena
Robinson/Shutterstock
In some states, a driver of a vehicle may not use a handheld cell phone
while driving. In one state with this law, 250 randomly selected drivers
were surveyed to determine the association between drivers who know
the law and drivers who obey the law. The results are shown in the
table below.
Obeys the Law
AVOID COMMON ERRORS
_
74%
366
Lesson 2
7/25/14 8:06 AM
Relative Frequency 366
Refer to the election data from Exercises 5–7. Answer using percents rounded to
the nearest tenth.
14. What is the conditional relative frequency of a state’s popular vote being won by the
Democrat in 2012, given that it was won by the Democrat in 1976?
12
_
= 50%
24
15. What is the conditional relative frequency of a state’s popular vote being won by the
Democrat in 1976, given that it was won by the Democrat in 2012?
12
_
≈ 44.4%
27
Refer to the language data from Exercises 8–10. Answer using
decimals rounded to the nearest thousandth.
16. What fraction of girls are studying French?
8
_
= 0.32
17. What fraction of Spanish students are boys?
12
_
= 0.444
25
27
Refer to the cell phone law data from Exercises 11–13. Answer using percents
rounded to the nearest tenth.
18. What percent of drivers obey the law despite not knowing the law?
45
= 69.2%
65
_
19. What is the conditional relative frequency of drivers who obey the law, given that they
know the law?
160
= 86.5%
185
© Houghton Mifflin Harcourt Publishing Company
_
Use the previously described data to determine whether there are associations
between the categories surveyed.
20. Refer to the election data from Exercises 5–7. Is there an association between the
party that won the popular vote in a state in 1976 and in 2012?
The conditional relative frequency that a state was won by the Democrat
12 = 50%. The
in 2012, given that a Democrat won the state in 1976, is _
24
conditional relative frequency that a state was won by the Republican
12 ≈ 44.4%.
in 2012, given that a Republican won the state in 1976, is _
27
Since 44.4% < 50%, the 2012 Republican candidate was slightly less
likely to win a state that had voted for the party’s candidate in 1976 than
the Democratic opponent.
21. Refer to the language data from Exercises 8–10. Can you use gender to predict a
preference for taking Spanish?
Conditional relative frequency of taking Spanish, given the student is a girl:
15
27
= 60% Percent of all students who take Spanish:
= 60% No, both
45
25
genders are equally likely to be taking Spanish.
_
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Lesson 8.2
_
367
Lesson 2
22. Refer to the language data from Exercises 8–10. Is there an association between
gender and a preference for French?
Conditional relative frequency of taking French, given the student is a girl:
8
= 32%
25
Percent of all students who are taking French:
12
= 26.7%
45
Yes, a girl in the survey is more likely to be taking French than a boy.
_
_
23. Refer to the cell phone law data from Exercises 11–13. Most drivers who don’t know
that it is illegal to operate a cell phone while driving obey the law anyway, presumably
out of a general concern for safe driving. Does this mean there is no association
between knowledge of the cell phone law and obeying the cell phone law?
The conditional relative frequency of obeying the law, given knowledge
of the law, is 86.5% (from Exercise 19), and the percent of all drivers
who obey the law is 82%. There is a slight difference, though both
frequencies are greater than 50%, so there is a possible association
between knowledge of the law and obeying it.
24. Multipart Classification Classify each statement as describing a joint, marginal, or
conditional relative frequency.
a. In a study on age and driving safety, 33% of drivers were considered younger
and a high accident risk.
joint
b. In a study on age and driving safety, 45% of older drivers were considered a high
accident risk.
conditional
c.
© Houghton Mifflin Harcourt Publishing Company
In a study on age and driving safety, 67% of drivers were classified as younger.
marginal
d. In a pre-election poll, 67% of the respondents who preferred the incumbent were
men.
conditional
e. In a pre-election poll, 33% of women preferred the challenger.
conditional
f.
In a pre-election poll, 16% of respondents were men who preferred the
challenger.
joint
Module 8
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Lesson 2
6/9/15 11:45 AM
Relative Frequency 368
JOURNAL
H.O.T. Focus on Higher Order Thinking
Have students describe the steps they take when
creating a two-way relative frequency table. Students
should include instructions for determining both
marginal relative frequencies and joint relative
frequencies.
25. Explain the Error In the survey on gender and fruit selection (Example 1B),
Millicent notices that given a preference for oranges, the conditional relative
frequencies of a student being a boy or a girl are the same. She concludes that there is
no association between gender and orange preference. Explain her error.
Millie missed the fact that there are more boys than girls in the survey,
so a 50% conditional relative frequency of a student who likes an orange
being a girl is higher than the 43.5% marginal relative frequency of girls
in the whole survey. This means girls in the survey actually had a higher
preference for oranges than boys.
26. Communicate Mathematical Ideas Can a joint relative frequency be greater
than either of the conditional relative frequencies associated with it? Explain your
reasoning.
No. The two conditional frequencies are calculated by dividing the joint
frequency by the row total or column total, which is less than the grand
total. The joint relative frequency is found by dividing the same joint
frequency by the grand total and must therefore be less.
27. Explain the Error Refer to the cell phone data from Exercises 11–13. Cole found
the conditional relative frequency that a driver surveyed does not know the law, given
that the driver obeys the law, by dividing 45 by 250. Explain Cole's error.
© Houghton Mifflin Harcourt Publishing Company
Cole should have divided 45 by the total number of drivers that obey the
law, 205.
Module 8
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Lesson 8.2
369
Lesson 2
7/25/14 8:48 AM
Lesson Performance Task
QUESTIONING STRATEGIES
Eighty students were surveyed about playing an instrument. The results are shown in the
two-way frequency table.
How can you check your work to be sure you
have calculated the percents correctly in your
two-way relative frequency table? Add the joint
relative frequencies in each row and each column.
The sum should equal the row or column’s marginal
relative frequency. In addition, the sum of the row
totals and the sum of the column totals should
both equal 1.
Play an Instrument
Gender
Yes
No
Total
Female
28
17
45
Male
20
15
35
Total
48
32
80
a. Complete the two-way relative frequency table for the data.
Play an Instrument
Gender
Yes
No
Total
Female
28
_
= 0.35
80
17
_
= 0.2125
45
_
= 0.5625
Male
_
_
35
_
= 0.4375
Total
48
_
= 0.60
15
= 0.1875
80
32
_
= 0.40
80
80
_
=1
80
20
= 0.25
80
80
80
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 To help students approach the question of
80
80
whether there is an association between gender and
playing an instrument, ask what results would be
expected if gender does not influence whether a
person plays an instrument. Students should
recognize that if gender has no influence on playing
an instrument, one would expect the students who
play instruments to have the same percent of males
and females as the total group of students surveyed.
Alternatively, one would expect the percent of each
gender that plays an instrument to be the same as the
percent of the whole group that plays an instrument.
Urge students to keep these ideas in mind as they
analyze the data.
b. What percent of the students surveyed play an instrument?
What percent of the males surveyed do not play an
instrument? Identify what type of frequency each percent is.
60% of the students surveyed play an instrument. This is a marginal
relative frequency.
20
≈ 0.429 ≈ 42.9%
35
42.9% of the males surveyed do not play an instrument. This is a
conditional relative frequency.
_
© Houghton Mifflin Harcourt Publishing Company
c. Is there an association between the sex of a student and
whether the student plays an instrument? Explain.
Percent of students surveyed who are female: 56.25%
Find the conditional relative frequency that a student surveyed is a
female, given that she plays an instrument.
28
≈ 0.583 = 58.3%
48
Females make up 58.3% of students who play an instrument, but they
only make up 56.25% of the students surveyed. So, females are more
likely to play an instrument than males.
_
Module 8
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Lesson 2
EXTENSION ACTIVITY
IN1_MNLESE389755_U4M08L2.indd 370
Have students conduct a survey to determine whether boys or girls at their school
are more likely to play certain kinds of instruments. For example, they might
choose to ask whether students play a wind instrument, a string instrument,
percussion, or piano. Have students make a two-way relative frequency table from
the data. Then have them analyze the data to determine whether there is an
association between gender and the type of instrument played.
7/25/14 6:54 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Relative Frequency 370