CorrectionKey=NL-B;CA-B CorrectionKey=NL-C;CA-C LESSON 8 . 2 … · 2016-11-07 · use formulas to program a spreadsheet that computes row totals, column totals, and relative frequencies

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Explore Relative FrequenciesTo 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.

Favorite Sport Basketball Football Soccer Total

Frequency 20 32 28 80

Use the frequencies to make a relative frequency table expressed with decimals.


Relative Frequency 20 _ 80

= 0.25 80 _ 80

=

Rewrite the relative frequency table using percents instead of decimals.


Relative Frequency 25%

Reflect

1. Explain what the numerator and denominator of the ratio 20 _ 80 refer to in part A.

2. What types of numbers can you use to write relative frequencies?

1

40%

32 _ 80

= 0.4 28 _ 80

= 0.35

35% 100%

20 refers to the number of students who prefer basketball, and 80 refers to the total

number of students surveyed.

fractions, decimals, and percents

Module 8 359 Lesson 2

8 . 2 Relative FrequencyEssential Question: How can you recognize possible associations and trends between two

categories of categorical data?

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Common Core Math StandardsThe student is expected to:

S-ID.5

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.

Mathematical Practices

MP.4 Modeling

Language ObjectiveIn a two-way relative frequency table, identify a joint relative frequency and a marginal relative frequency, and explain what they mean.

HARDCOVER PAGES 291300

Turn to these pages to find this lesson in the hardcover student edition.

Relative Frequency

ENGAGE Essential Question: How can you recognize possible associations and trends between two categories of categorical data?You can use relative frequencies to interpret

associations and trends between two categories in

two-way frequency tables.

PREVIEW: LESSON PERFORMANCE TASKView 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.

359

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore Relative Frequencies

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.

Favorite Sport BasketballFootball

SoccerTotal

Frequency20

3228

80

Use the frequencies to make a relative frequency table expressed with decimals.


SoccerTotal

Relative Frequency 20 _

80 = 0.25

80 _

80 =

Rewrite the relative frequency table using percentages instead of decimals.


SoccerTotal

Relative Frequency25%

Reflect

1. Explain what the numerator and denominator of the ratio 20 _ 80

refer to in part A.

2. What types of numbers can you use to write relative frequencies?

S-ID.5 For the full text of these standards, see the table starting on page CA2.

1

40%

32 _

80 = 0.4 2

8 _ 80

= 0.35

35%100%

20 refers to the number of students who prefer basketball, and 80 refers to the total

number of students surveyed.

fractions, decimals, and percentages

Module 8

359

Lesson 2

8 . 2 Relative Frequency

Essential Question: How can you recognize possible associations and trends between two

categories of categorical data?

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359 Lesson 8 . 2

L E S S O N 8 . 2

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Explain 1 Two-Way Relative Frequency TablesTwo 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.

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 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.

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

To find the relative frequencies, divide each frequency by the grand total.

Preferred Sport

Gender Basketball Football Soccer Total

Girl 6 _ 80

= 0.075 12 _ 80

= 0.15 18 _ 80

= 0.225 36 _ 80

= 0.45

Boy 14 _ 80

= 0.175 20 _ 80

= 0.25 10 _ 80

= 0.125 44 _ 80

= 0.55

Total 20 _ 80

= 0.25 32 _ 80

= 0.4 28 _ 80

= 0.35 80 _ 80

= 1

The joint relative frequencies tell what percent of all those surveyed are in each category:

• 7.5% are girls who prefer basketball.

• 15% are girls who prefer football.

• 22.5% are girls who prefer soccer.

• 17.5% are boys who prefer basketball.

• 25% are boys who prefer football.

• 12.5% are boys who prefer soccer.

The marginal relative frequencies tell what percent of totals has a given single characteristic:

• 25% prefer basketball.

• 40% prefer football.

• 35% prefer soccer.

• 45% are girls.

• 55% are boys.


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EXPLORE Relative Frequencies

INTEGRATE TECHNOLOGYSpreadsheet 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.

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).

EXPLAIN 1 Two-Way Relative Frequency Tables

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Discuss with students the relationship 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.PROFESSIONAL DEVELOPMENT

Math BackgroundRelative 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(A ∩ B) _

P(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


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B 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%

Boy 27.2%

Total 44.6%

The joint relative frequencies:

• are girls who selected an apple.

• are girls who selected a banana.

• are girls who selected an orange.

• are boys who selected an apple.

• are boys who selected a banana.

• are boys who selected an orange.

The marginal relative frequencies:

• selected an apple.

• selected a banana.

• selected an orange.

• are girls.

• are boys.

Reflect

3. Discussion Explain how you can use joint and marginal relative frequencies to check your relative frequency table.

10.9% 15.2%

17.4%

44.6% 43.5%

27.2%

10.9%

25% 56.5%

14.1%

15.2%

30.4%

15.2%

43.5%

56.5%

100.0%30.4%

15.2%14.1%

25.0%

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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COLLABORATIVE LEARNING

Whole Class ActivityHave students conduct their own surveys using left-right handed and a three-category 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.

QUESTIONING STRATEGIESWhich 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.

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Your Turn

Use the two-way table of data from another student survey to answer the following questions.

Like Aerobic Exercise

Like Weight Lifting Yes No Total

Yes 7 14 21

No 12 7 19

Total 19 21 40

4. Find the joint relative frequency of students surveyed who like aerobics exercise but dislike weight lifting.

5. What is the marginal relative frequency of students surveyed who like weight lifting?

Explain 2 Conditional Relative FrequenciesA 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 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.

Number of girls who prefer soccer

___ Total number of girls

= 18 _ 36 = 0.5 = 50%

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.

Number of who chose an orange

____ Total number of

= _ ≈ 0.269 = %

Your Turn

Use the data from Your Turn Exercises 4 and 5 after Example 1.

6. What is the conditional relative frequency that a student likes to lift weights, given that the student does not like aerobics?

7. Find the conditional relative frequency that a student likes to lift weights, given that the student likes aerobics.

12 _ 40

= 0.3 or 30%

14 _ 21

= 0.667 or 66.7%

7 _ 19 = 0.368 or 36.8%

21 _ 40

= 0.525 or 52.5%

boys

boys

14

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DIFFERENTIATE INSTRUCTION

Multiple RepresentationsTo 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.

EXPLAIN 2 Conditional Relative Frequencies

CONNECT VOCABULARY 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.

AVOID COMMON ERRORSWhen 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.

QUESTIONING STRATEGIESWhen 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.



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Explain 3 Finding Possible AssociationsYou can analyze two-way frequency tables to locate possible associations or patterns in the data.

Example 3 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.

Analyze the data about girls that were surveyed.

Step 1: Identify the percent of all students surveyed who are girls.

36 _ 80 = 0.45 = 45%

Step 2: Determine each conditional relative frequency.

Basketball Football Soccer

Of the 20 students who prefer basketball, 6 are girls.

6 _ 20

= 0.3 = 30%

Of the 32 students who prefer football, 12 are girls.

12 _ 32

= 0.375 = 37.5%

Of the 28 students who prefer soccer, 18 are girls.

18 _ 28

≈ 0.643 = 64.3%

Step 3: Interpret the results by comparing each conditional relative frequency to the percent of all students surveyed who are girls, 45%.


30% < 45%

Girls are less likely to prefer basketball.

37.5% < 45%

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.

_80 = 0. = %

Step 2: Determine each conditional relative frequency.


Of the 20 students who

prefer basketball, are boys.

_ 20

= 0. = %

Of the students who prefer football, are boys.

_ = 0. = %

Of the students who prefer soccer, are boys.

_ = 0. = %

44 55 55

14

147 70

20 10

20 10

32 28

32 28

625 35762.5 35.7



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LANGUAGE SUPPORT

Connect VocabularyThere 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).

EXPLAIN 3 Finding Possible Associations

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 When looking for possible associations, make 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.

QUESTIONING STRATEGIESWhat 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.

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Step 3: Interpret the results by comparing each conditional relative frequency to the percent of all students

surveyed who are boys, %.


70% > 55%

Boys are more likely to prefer basketball.

62.5% > 55%

Boys are more likely to prefer football.

35.7% < 55%

Boys are less likely to prefer soccer.

Reflect

8. Making Connections How can the statement “6 out of the 20 students who prefer basketball are girls” be stated as a conditional relative frequency?

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.

t Elaborate

10. What does it mean to say there is an association between characteristics in a two-way frequency table?

11. Essential Question Check-In How can you use two-way frequency data to recognize possible associations between the two categories of categorical data?

Given that a student prefers basketball, 30% are girls.

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.

It means that there is a pattern between the two variables so that the conditional relative

frequencies, or conditional probabilities, are somewhat predictable.

You can compare conditional relative frequencies with different givens to each other or to

the marginal relative frequencies to see if certain conditions favor a category relative to

other conditions or the entire data set.

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ELABORATE INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Discuss the similarities between conditional 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.

SUMMARIZE THE LESSONHow 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.



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• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

Use the table of frequency data for Exercises 1–4.

Class Survey of Favorite Colors

Favorite Color Red Orange Yellow Green Blue Purple Total

Frequency 2 5 1 6 8 2 24

1. Complete the relative frequency table for this data using decimals rounded to the nearest thousandth.



Relative Frequency

2. Complete the relative frequency table for this data using percents rounded to the nearest tenth.



Relative Frequency

3. 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%?

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

1976 Election Democrat Republican Total

Democrat 12 = 12 = 24 =

Republican 15 = 12 = 27 =

Total 27 = 24 = 51 =

5. 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?

7. What percent voted Republican in 1976? What type of frequency is this?

0.083 0.208 0.042 0.25 0.333

0.333 green

0.083 1

8.3% 20.8% 4.2% 25% 33.3% 8.3% 100.0%

23.5%

29.4%

52.9%

23.5%

23.5%

47.1%

47.1%

52.9%

100%

52.9%; a marginal relative frequency

23.5%; a joint relative frequency



IN1_MNLESE389755_U4M08L2 365 7/25/14 7:59 AMExercise 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

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreRelative Frequencies

Exercises 1–4

Example 1Two-Way Relative Frequency Tables

Exercises 5–13

Example 2Conditional Relative Frequencies

Exercises 14–19, 24, 26–27

Example 3Finding Possible Associations

Exercises 20–23, 25

VISUAL CUESTo 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).

AVOID COMMON ERRORSWhen 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.

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The results of a survey of 45 students and the foreign language they are studying are shown in the two-way frequency table.

Language

Gender Chinese French Spanish Total

Girl 2 8 15 25

Boy 4 4 12 20

Total 6 12 27 45

8. Fill in the table of two-way relative frequencies using decimals, rounded to the nearest thousandth.

Language

Gender Chinese French Spanish Total

Girl

Boy

Total

9. What fraction of the surveyed students are boys taking Spanish?

10. What fraction of the surveyed students are taking Chinese?

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.

11. Complete the table of two-way relative frequencies using percents.

Knows the Law

Obeys the Law Yes No Total

Yes 160 = 64% 45 = 18%

No 25 = 10% 20 = 8%

Total

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?

0.044

0.089

0.133

0.178

0.089

0.267

0.333 0.556

0.267 0.444

0.6 1

4 _ 15

≈ 0.267 2 _ 15

≈ 0.133

74% 26%

82%

18%

100%

64% 74%




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COGNITIVE STRATEGIESStudents 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.

Given:

Find relative frequency of:

Frequency of:

Total:

Percent: ÷

AVOID COMMON ERRORSStudents 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.

prefers soccer

boy

boys who play soccer

all who prefer soccer

all who prefer soccerboys who prefer soccer



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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?

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?

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? 17. What fraction of Spanish students are boys?

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?

19. What is the conditional relative frequency of drivers who obey the law, given that they know the law?

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?

21. Refer to the language data from Exercises 8–10. Can you use gender to predict a preference for taking Spanish?

12_ 24

= 50%

12_ 27

≈ 44.4%

8 _ 25

= 0.32 12 _ 27

= 0.444

45_ 65

= 69.2%

160_ 185

= 86.5%

The conditional relative frequency that a state was won by the Democrat in 2012, given that a Democrat won the state in 1976, is 12_

24 = 50%. The conditional relative frequency that a state was won by the Republican in 2012, given that a Republican won the state in 1976, is 12_

27 ≈ 44.4%. 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.

Conditional relative frequency of taking Spanish, given the student is a girl: 15_ 25

= 60% Percent of all students who take Spanish: 27_ 45

= 60% No, both genders are equally likely to be taking Spanish.


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22. Refer to the language data from Exercises 8–10. Is there an association between gender and a preference for French?

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?

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.

b. In a study on age and driving safety, 45% of older drivers were considered a high accident risk.

c. In a study on age and driving safety, 67% of drivers were classified as younger.

d. In a pre-election poll, 67% of the respondents who preferred the incumbent were men.

e. In a pre-election poll, 33% of women preferred the challenger.

f. In a pre-election poll, 16% of respondents were men who preferred the challenger.

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.

Conditional relative frequency of taking French, given the student is a girl: 8 _ 25

= 32%Percent of all students who are taking French: 12 _ 45

= 26.7%Yes, a girl in the survey is more likely to be taking French than a boy.

joint

joint

conditional

marginal

conditional

conditional




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H.O.T. Focus on Higher Order Thinking

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.

26. Communicate Mathematical Ideas Can a joint relative frequency be greater than either of the conditional relative frequencies associated with it? Explain your reasoning.

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.

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.

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.

Cole should have divided 45 by the total number of drivers that obey the law, 205.



IN1_MNLESE389755_U4M08L2 369 7/25/14 8:48 AM

JOURNALHave 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.

369 Lesson 8 . 2


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Lesson Performance TaskEighty students were surveyed about playing an instrument. The results are shown in the

two-way frequency table.

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

Male

Total

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.

c. Is there an association between the sex of a student and whether the student plays an instrument? Explain.

28 _ 80

= 0.35 17 _ 80

= 0.2125 45 _ 80

= 0.5625

20 _ 80

= 0.25 15 _ 80

= 0.1875 35 _ 80

= 0.4375

48 _ 80

= 0.60 32 _ 80

= 0.40 80 _ 80

= 1

60% of the students surveyed play an instrument. This is a marginal relative frequency.

20 _ 35

≈ 0.429 ≈ 42.9%

42.9% of the males surveyed do not play an instrument. This is a conditional relative frequency.

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 _ 48

≈ 0.583 = 58.3%

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.




EXTENSION ACTIVITY

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.

QUESTIONING STRATEGIESHow 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.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 To help students approach the question of 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.

Scoring Rubric2 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.



Documents

CorrectionKey=NL-B;CA-B CorrectionKey=NL-C;CA-C LESSON 8 . 2 … · 2016-11-07 · use formulas to program a spreadsheet that computes row totals, column totals, and relative frequencies