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**Aim: How do we use Venn Diagrams to visualize survey results.**

Do Now: Would you be willing to donate blood? Would you be willing to help serve a free breakfast to blood donors?

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Survey Results A = set of blood donors B = set of breakfast servers A B I II III IV

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Survey Results B A and means intersection or means union I II III not means complement IV How many students are willing to donate blood? How many students are willing to help serve? How many students are willing to donate & help serve? How many students are willing to donate or help serve? How many students are willing to donate but not serve? How many students are willing to serve but not donate? How many students are neither will to donate nor serve? How many students are surveyed?

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**Solving Survey Problems**

Use the survey’s description to define sets and draw a Venn diagram. Use the survey’s results to determine the cardinality for each region in the Venn diagram. Start with the intersection of the sets, then innermost region and work outward. Use the completed Venn diagram to answer the problem’s questions.

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Model Problem In a Gallup poll, 2000 US adults were selected at random and asked to agree or disagree with the following statement: ‘Job opportunities for women are not equal to those for men.’ 1190 people agreed with the statement 700 women agreed with the statement If half the people surveyed were women, a How many men agreed with the statement? b. How many men disagreed with the statement?

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**1190 people agreed with the statement**

Model Problem U 1. 1190 people agreed with the statement W(Women) A(Agree) I II III 700 women agreed with the statement 300 700 490 510 IV 2. Find cardinality of innermost region |U| = 2000 |W| = 1000 |A| = 1190 |W A| = 700 Region I = |W| – |W A| = 1000 – 700 = 300 Region III = |A| – |W A| = 1190 – 700 = 490 Region IV = |U| – Regions I, II, and III = – 1490 = 510

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**‘Job opportunities for women are not equal to those for men.’**

Model Problem ‘Job opportunities for women are not equal to those for men.’ a. How many men agreed with the statement? b. How many men disagreed with the statement? U W(Women) A(Agree) I II III 300 700 490 510 IV 3. a of both sexes agree; 700 women agree: – 700 = 490 men agree. b men surveyed; 490 men agree; 510 disagree.

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**Solving Problems with Venn Diagrams**

A survey of 100 randomly selected students gave the following information. 45 students were taking mathematics 41 students were taking English 40 students were taking history 15 were taking math and English 18 were taking math and history 17 were taking English and history 7 were taking all three. Use a Venn Diagram to find: How many are taking only mathematics only English only history not taking any of these three courses

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**17 19 8 16 7 11 10 12 Model Problem U Understand the problem**

Devise a plan Carry out the plan Look back M E 17 19 8 16 H 7 11 10 = 83 |H| = 40 |M| = 41 |M E| = 15 |E| = 41 |E H| = 17 |M H| = 18 |M E H| = 7 12 Fill in the innermost region first Work your way outward by subtraction until # elements in all regions is known

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Model Problem Sixty people were contacted and responded to a movie survey. The following information was obtained. 6 people liked comedies, dramas, and science fiction. b. 13 people liked comedies, and dramas c. 10 liked comedies and science fiction d. 11 people liked dramas and science fiction e. 26 people liked comedies f. 21 people liked dramas g. 25 people liked science fiction. How many of those surveyed liked none of these categories of movies and how many liked only comedies.

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**7 9 3 6 4 5 10 Model Problem C = set of comedy lovers**

D = set of drama lovers S = set of science fiction lovers |C| = 26 & 2. |D| = 21 |U| = 60 |S| = 25 6 people liked comedies, dramas, and science fiction C D S = 6 11 people liked dramas and science fiction. C D 7 10 liked comedies and science fiction. 9 3 6 13 people liked comedies, and dramas 4 5 only C = 26 – 17 = 9 10 only D = 21 – 18 = 3 only S = 25 – 15 = 10 S

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Model Problem 3. How many of those surveyed liked none of these categories of movies and how many liked only comedies. |U| = 60 liked none: 60 – ( ) = 16 C D 7 9 3 6 4 5 10 16 S

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**How many respondents listened to classical music? **

55 20 70 30 Jazz Model Problem In a survey of musical tastes, respondents were asked: Do you listen to classical music? Do you listen to jazz? The survey results are summarized above. Use the diagram to answer the following questions. How many respondents listened to classical music? How many listened to jazz? How many listened to both? How many listened to classical or jazz? How many listened to classical but not jazz? How many listened to jazz but not classical? Ho many listen to neither? How many were surveyed?

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