Question

a group of 60 adults were surveyed on the ways on how they cope with stress. and the following are the results 34 said they tend to go for a vacation 27 said they would go for a smoke break 41 answered they listen to music 13 tend to go for a vacation and smoke 16 said go for a smoke and listen to music 20 go for a cation and listen to music 28 said two of the coping mechanism how many adults said exactly one of the coping mechanisms

Answer 1

Answer:

The number of adults that said exactly one coping mechanism are 25 adults

Step-by-step explanation:

Total number of adult in the group = 60 adults

The number that go on vacation vacation = 34

The number that go for a smoke break = 27

The number that listen to music = 41

The number that go for a vacation and smoke = 13

The number that smoke and listen to music = 16

The number that go for a vacation and listen to music = 20

The number that said two of the coping mechanism = 28

Set theory

We have;

n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(A∩C) - n(B∩C) + n(A∩B∩C)

Given that the following relations;

n(A∪B∪C) = 60

n(A) = 34

n(B) = 27

n(C) = 41

n(A∩B) = 13

n(A∩C) = 20

n(B∩C) = 16

Therefore;

60 = 34 + 27 + 41 -13 -  20 - 16 + n(A∩B∩C)

∴ n(A∩B∩C) = 60 - (34 + 27 + 41 -13 -  20 - 16) = 7

Given that 28 said exactly 2, we have;

Total number of respondents = Those that said exactly 1 + Those that said exactly 2 + Those that said exactly 3

Total number of respondents = 60

Those that said exactly 2 = 28

Those that said exactly 3 = n(A∩B∩C) = 7

Therefore;

60 = Those that said exactly 1 + 28 + 7

Those that said exactly 1 = 60 - (28 + 7) = 25

Alternatively, we have;

Those that said exactly one = n(A) -(n(A∩B) + n(A∩C) - n(A∩B∩C))  + n(B) - (n(B∩C) + n(A∩B) - n(A∩B∩C)) + n(C) - (n(A∩C) + n(B∩C) - n(A∩B∩C))  

= 34 - (13 + 20 - 7) + 27 - (16 + 13 - 7) + 41 - (20 + 16 - 7) = 25

60 - 28 - 7 = 25

The number of adults that said exactly one coping mechanism = 25.