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Survey Results Analysis Using Venn Diagrams

This section focuses on utilizing Venn diagrams to visualize and interpret survey results effectively. Examples demonstrate how to identify which regions correspond to sets defined by survey responses. Detailed methods outline how to calculate the number of elements in various regions based on given cardinalities. We examine multiple scenarios, such as determining the number of participants who prefer specific music genres and assessing overlaps among them. Students are tasked with analyzing these surveys, ensuring they document all steps to receive full credit for their assignments.

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Survey Results Analysis Using Venn Diagrams

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  1. Section 1.6 Survey Problems

  2. Objectives • Use Venn diagrams to visualize a survey’s results. • Use survey results to complete Venn diagrams and answer questions about the survey.

  3. Example 1: • Determine which numbered regions make up the indicated set. • A U II IV

  4. Example 2: • Determine which numbered regions make up the indicated set. • A – B U II IV

  5. Example 3: • Determine which numbered regions make up the indicated set. • B’ ∩ C ∩ A’ U II V IV VI VIII

  6. Example 4: • If n(A) = 21, n(B) = 29, and n(U) = 48, find the number of elements in each regions I, III, and IV. U II 7 IV

  7. Example 5: • If n(A) = 23, n(B) = 27, and n(U) = 53, find the number of elements in each regions I, III, and IV. U II 8 IV

  8. Example 6: • The Venn diagram show the cardinality of each region. • How many elements belong to set B? U 1 7 2 3 8

  9. Example 7: • How many elements belong to set A? U 1 7 2 3 8

  10. Example 8: • How many elements belong to set A but not set C? U 1 7 2 3 8

  11. Example 9: • How many elements belong to set B but not set A? U 1 7 2 3 8

  12. Example 10: • How many elements belong to set A or set C? U 1 7 2 3 8

  13. Example 11: • How many elements belong to set A and set C, but not to set B? U 1 7 2 3 8

  14. Example 12: • Considering sets A, B, and C, how many elements belong to exactly one of these sets? U 1 7 2 3 8

  15. Section 1.6 Assignment • Classwork: • TB pg. 55/1 – 18 • Must write problems and show ALL work to receive credit for the given assignment.

  16. Example 13: • The accompanying Venn diagram shows the number of elements in region V. Use the given cardinalities to determine the number of elements in each of the other seven regions. • n(U) = 30, n(A) = 11, n(B) = 8, n(C) = 14, n(A∩B) = 3, n(A∩C) = 5, n(B∩C) = 3 U 2

  17. Example 14: • n(U) = 32, n(A) = 21, n(B) = 15, n(C) = 14, n(A∩B) = 6, n(A∩C) = 7, n(B∩C) = 8 U 2

  18. Example 15: • n(U) = 38, n(A) = 26, n(B) = 21, n(C) = 18, n(A∩B) =17, n(A∩C) = 11, n(B∩C) = 8, n(A∩B∩C) = 7 U

  19. Example 16: • TB pg. 55/23 U

  20. Example 17: • TB pg. 55/25

  21. Example 18: • TB pg. 56/31 U

  22. Example 19: • A survey of 80 college students was taken to determine the musical styles they listened to. Forty-two students listened to rock, 34 listened to classical, and 27 to jazz. Twelve students listened to rock and jazz, 14 to rock and classical, and 10 to classical and jazz. Seven students listened to all three musical styles. • Of those surveyed, • How many listened to only rock music? • How many listened to classical and jazz, but not rock? • How many listened to classical or jazz, but not rock? • How many listened to music in exactly one of the musical styles? • How many listened to music in at least two of the musical styles? • How many did not listen to any of the musical styles?

  23. Example 19: Classical Rock 7 U 7 3 5 6 Jazz

  24. Example 19: • Of those surveyed, • How many listened to only rock music? 23 • How many listened to classical and jazz, but not rock? 3 • How many listened to classical or jazz, but not rock? 32 • How many listened to music in exactly one of the musical styles? 52 • How many listened to music in at least two of the musical styles?22 • How many did not listen to any of the musical styles? 6

  25. Section 1.6 Assignment • Classwork: • TB pg. 55/24 – 34 EVEN • Must write problems and show ALL work to receive credit for the given assignment.

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