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Answer:
1. A Venn diagram is attached. The amount of people that have health insurance and is confident about the economy is 287.
2. The amount of people that have health insurance and is not confident about the economy is 116.
3. The amount of people that have health insurance or is confident about the economy is 437.
Step-by-step explanation:
1) Using sets theory, we know:
U: 500 people surveyed
A: people who is confident in the economy
B: people that have health insurance
A' and B': people who is not confident in the economy and don't have health insurance.
A=321
B=403
A' and B' =63
In the attached picture we can see a Venn diagram that represents the situation. In order to obtain the intersection of both sets A∩B, we calculate:
(321 - x) + (403 - x) + x + 63 = 500
321 - x + 403 - x + x + 63 +x -500 = 500 +x - 500
321 + 403 + 63 -500 = x
x= 287
2) The amount of people that have health insurance and is not confident about the economy is given by the intersection of the complement of set A and set B:
A' ∩ B can be calculated as:
B - (A ∩ B)= 403 - 287= 116
3) The amount of people that have health insurance or is confident about the economy is given by:
(A∪B) = (A) + (B) - (A ∩ B)
(A∪B) = 321 + 403 - 287= 437
The number of people that have health insurance and are confident about the economy is 287.
What is a set?
A set is a mathematical model for a collection of diverse things; it comprises elements or members, which can be any mathematical object: numbers, symbols, points in space, lines, other geometrical structures, variables, or even other sets.
1) If we use the sets theory, then we can write,
- The number of people who participated in the survey, U=500 people
- Number of people who are confident in the economy, A = 321
- Number of people that have health insurance, B = 403
- Number of people who did not check any boxes, A' and B' =63
Now, The number of people that have health insurance and is confident about the economy can be represented by the intersection of both sets A∩B, therefore,
[tex](321 - x) + (403 - x) + x + 63 = 500\\\\321 - x + 403 - x + x + 63 +x -500 = 500 +x - 500\\\\321 + 403 + 63 -500 = x\\\\x= 287[/tex]
Hence, The number of people that have health insurance and are confident about the economy is 287.
2) The number of people that have health insurance and is not confident about the economy can be represented by the intersection of the complement of set A and set B, therefore,
[tex]\begin{aligned}A' \cap B &=B - (A \cap B)\\\\&= 403 - 287\\\\&= 116\end{aligned}[/tex]
Hence, The number of people that have health insurance and are not confident about the economy is 116.
3) The number of people that have health insurance or are confident about the economy can be represented as:
[tex](A\cup B) = (A) + (B) - (A \cap B)\\\\(A\cup B) = 321 + 403 - 287= 437[/tex]
Hence, The number of people that have health insurance or are confident about the economy is 437.
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