Question

A student at a university has been doing a project to investigate entertainment habits of students. They have surveyed 100 random students who were each asked three questions:

1. Have you watched a movie in the last week?
2. Have you listened to music in the last week?
3. Have you read a book in the last week?

However, the student keeps a very messy room and has lost some of the results. They have been able to find the following results:

Of all the students surveyed, 46 had watched a movie, 36 had listened to music and 41 had read a book in the last week. Also:

• 4 students answered yes to all three questions
• 10 students answered yes to questions 1 and 2 only
• 12 students answered yes to questions 1 and 3 only
• 20 students answered yes only to question 1
• 18 students answered no to all three questions

Find the missing information and answer the following questions regarding the group surveyed. Give your answers as whole numbers.

Calculate the number of students that:

a)answered yes to question 3 only =

b)answered yes to exactly two questions =

c)had watched a movie or listened to music but had not read a book =

Answer 1

46 had watched a movie , thus 46 students answered yes to questions 1

36 had listened to music, thus 36 students answered yes to questions 2

41 had read a book in the last week, thus 41 students answered yes to questions 3

With all given information, we will create Venn diagram as below.

Students answered yes to at least one question, n(1 U 2 U 3) = 100 - 18 = 82

Now, n(1 U 2 U 3) = n(1) + n(2) + n(3) - n(1 2) - n(1 ​​​​​​​ 3) - n(2 ​​​​​​​ 3) + n(1 ​​​​​​​ 2 3)

82 = 46 + 36 + 41 - 10 - 12 - n(2 ​​​​​​​ 3) + 4

n(2 ​​​​​​​ 3) = 46 + 36 + 41 - 10 - 12 - 82 + 4 = 23

Students answered yes to questions 2 or 3 only = 23

a) From Venn diagram, Students answered yes to question 3 only = Students answered yes to question 3 - (Students answered yes to question 1 and 3 + Students answered yes to question 2 and 3) + Students answered yes to question 1, 2 and 3

= 41 - (12 + 23) + 4 = 10

b) Students answered yes to exactly two questions = (Students answered yes to question 1 and 2 + Students answered yes to question 1 and 3 + Students answered yes to question 2 and 3) - 2* Students answered yes to question 1, 2 and 3

= (10 + 12 + 23) - 2 * 4= 37

c) had watched a movie or listened to music but had not read a book = Students answered yes to question 1 and 2 but no to question 3 = Students answered yes to at least one question - Students answered yes to question 3 only

= 82 - 10

= 72