Math

QuestionSurvey 112 students on music preferences: 42 techno, 42 hip hop, 39 rap. Find cardinality for each Venn diagram region.

Studdy Solution

STEP 1

Assumptions1. The total number of students surveyed is112. . The number of students who listen to techno is42.
3. The number of students who listen to hip hop is42.
4. The number of students who listen to rap is39.
5. The number of students who listen to both techno and rap is18.
6. The number of students who listen to both techno and hip hop is24.
7. The number of students who listen to both hip hop and rap is17.
8. The number of students who listen to all three musical styles is10.

STEP 2

First, we need to calculate the number of students who listen to only one type of music. We can do this by subtracting the number of students who listen to two types of music and all three types of music from the total number of students who listen to each type of music.
For technoTechnoonly=Totaltechno(Technoandrap+TechnoandhiphopAllthree)Techno\, only = Total\, techno - (Techno\, and\, rap + Techno\, and\, hip\, hop - All\, three)

STEP 3

Plug in the given values to calculate the number of students who listen to techno only.
Technoonly=42(18+2410)Techno\, only =42 - (18 +24 -10)

STEP 4

Calculate the number of students who listen to techno only.
Technoonly=42(18+2410)=10Techno\, only =42 - (18 +24 -10) =10

STEP 5

Repeat the same process for hip hop and rap.
For hip hopHiphoponly=Totalhiphop(Technoandhiphop+HiphopandrapAllthree)Hip\, hop\, only = Total\, hip\, hop - (Techno\, and\, hip\, hop + Hip\, hop\, and\, rap - All\, three)

STEP 6

Plug in the given values to calculate the number of students who listen to hip hop only.
Hiphoponly=42(24+1710)Hip\, hop\, only =42 - (24 +17 -10)

STEP 7

Calculate the number of students who listen to hip hop only.
Hiphoponly=42(24+1710)=11Hip\, hop\, only =42 - (24 +17 -10) =11

STEP 8

For rapRaponly=Totalrap(Technoandrap+HiphopandrapAllthree)Rap\, only = Total\, rap - (Techno\, and\, rap + Hip\, hop\, and\, rap - All\, three)

STEP 9

Plug in the given values to calculate the number of students who listen to rap only.
Raponly=39(18+17)Rap\, only =39 - (18 +17 -)

STEP 10

Calculate the number of students who listen to rap only.
Raponly=39(18+1710)=14Rap\, only =39 - (18 +17 -10) =14

STEP 11

Now, we need to find the number of students who listen to two types of music but not the third one. We can do this by subtracting the number of students who listen to all three types of music from the total number of students who listen to two types of music.
For techno and rap but not hip hopTechnoandraponly=TechnoandrapAllthreeTechno\, and\, rap\, only = Techno\, and\, rap - All\, three

STEP 12

Plug in the given values to calculate the number of students who listen to techno and rap but not hip hop.
Technoandraponly=1810Techno\, and\, rap\, only =18 -10

STEP 13

Calculate the number of students who listen to techno and rap but not hip hop.
Technoandraponly=1810=8Techno\, and\, rap\, only =18 -10 =8

STEP 14

Repeat the same process for the other two pairs of musical styles.
For techno and hip hop but not rapTechnoandhiphoponly=TechnoandhiphopAllthreeTechno\, and\, hip\, hop\, only = Techno\, and\, hip\, hop - All\, three

STEP 15

Plug in the given values to calculate the number of students who listen to techno and hip hop but not rap.
Technoandhiphoponly=2410Techno\, and\, hip\, hop\, only =24 -10

STEP 16

Calculate the number of students who listen to techno and hip hop but not rap.
Technoandhiphoponly=2410=14Techno\, and\, hip\, hop\, only =24 -10 =14

STEP 17

For hip hop and rap but not technoHiphopandraponly=HiphopandrapAllthreeHip\, hop\, and\, rap\, only = Hip\, hop\, and\, rap - All\, three

STEP 18

Plug in the given values to calculate the number of students who listen to hip hop and rap but not techno.
Hiphopandraponly=1710Hip\, hop\, and\, rap\, only =17 -10

STEP 19

Calculate the number of students who listen to hip hop and rap but not techno.
Hiphopandraponly=1710=7Hip\, hop\, and\, rap\, only =17 -10 =7

STEP 20

Finally, we can construct the Venn diagram with the calculated values. The Venn diagram will have three circles representing techno, hip hop, and rap. The intersection of all three circles will represent the students who listen to all three types of music. The intersections of two circles will represent the students who listen to those two types of music but not the third one. The remaining parts of the circles will represent the students who listen to only one type of music.

STEP 21

The cardinality for each region is as follows1. Techno only10. Hip hop only113. Rap only144. Techno and rap only85. Techno and hip hop only146. Hip hop and rap only77. All three10

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