Respuesta :
Answer:
a)109 of the surveyed households only own a hand gun and do not support the ban on hand guns.
b)328 of the surveyed households do not own a gun and support the ban on hand guns
c )112 of the surveyed households do not own a gun and do not support the ban on handguns.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents those who own a hand gun.
-The set B represents those who own a rifle
-The set C represents those who support the ban of handguns.
-D represents those who do not own a gun and do not support the ban on hand guns.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which [tex]a[/tex] are those that only own a hand gun, [tex]A \cap B[/tex] are those who own both a handgun and a rifle, [tex]A \cap C[/tex] are those who own a hand gun and support the ban on handguns, and [tex]A \cap B \cap C[/tex] are those who own both a hand gun and a rifle, and also support the ban on hand guns.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
In which [tex]b[/tex] are those who only own a rifle, and [tex]B \cap C[/tex] are those who own a rifle and support the ban on handguns.
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
In which [tex]c[/tex] are those who support the ban on hand guns and do not own any handguns.
This diagram has the following subsets:
[tex]a,b,c,D(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There were 1000 households surveyed, so:
[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1000[/tex]
We start finding the values from the intersection of three sets.
Solution:
48 own both a hand gun and a rifle and also support the ban on hand guns.
[tex]A \cap B \cap C = 48[/tex]
70 own a hand gun and support the ban on hand guns:
[tex](A \cap C) + (A \cap B \cap C) = 70[/tex]
[tex]A \cap C = 70 - 48[/tex]
[tex]A \cap C = 22[/tex]
206 own a rifle but no hand gun and do not support the ban on hand guns.
[tex]b = 206[/tex].
136 own both a hand gun and a rifle:
[tex](A \cap B) + (A \cap B \cap C) = 136[/tex]
[tex]A \cap B = 136 - 48[/tex]
[tex]A \cap B = 88[/tex]
493 supported the ban on hand guns.
[tex]C = 493[/tex]
437 own a rifle:
[tex]B = 437[/tex]
267 own a hand gun:
[tex]A = 267[/tex]
a) How many of the surveyed households only own a hand gun and do not support the ban on hand guns?
This is the value of [tex]a[/tex], that we can find from the following equation:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]267 = a + 88 + 22 + 48[/tex]
[tex]a = 267 - 158[/tex]
[tex]a = 109[/tex]
109 of the surveyed households only own a hand gun and do not support the ban on hand guns.
b) How many of the surveyed households do not own a gun and support the ban on hand guns?
This is the value of [tex]c[/tex], that we can find from the following equation:
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]493 = c + 22 + (B \cap C) + 48[/tex]
Here, we also have to find [tex]B \cap C[/tex], that are those who own a rifle and support the ban on hand guns. We can find this from the following equation:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]437 = 206 + (B \cap C) + 88 + 48[/tex]
[tex]B \cap C = 437 - 342[/tex]
[tex]B \cap C = 95[/tex]
So
[tex]493 = c + 22 + (B \cap C) + 48[/tex]
[tex]493 = c + 22 + 95 + 48[/tex]
[tex]c = 493 - 165[/tex]
[tex]c = 328[/tex]
328 of the surveyed households do not own a gun and support the ban on hand guns.
a) How many of the surveyed households do not own a gun and do not support the ban on hand guns?
This is the value of d, that can be found from the following equation:
[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1000[/tex]
[tex]109 + 206 + 328 + D + 88 + 22 + 95 + 40 = 1000[/tex]
[tex]D = 1000 - 888[/tex]
[tex]D = 112[/tex]
112 of the surveyed households do not own a gun and do not support the ban on handguns.
