Question

A survey of 120 college students was taken to determine the musical styles they liked. Of those, 50 students listened to rock, 49 to classical, and 38 to jazz. Also, 24 students listened to rock and jazz, 33 to rock and classical, and 20 to classical and jazz. Finally, 17 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions.

a. How many listened to only rock music? n(only rock)=0
b. How many listened to classical and jazz, but not rock? n(classical and jazz, not rock) =
c. How many listened to classical or jazz, but not rock? n(classical or jazz, not rock)
d. How many listened to music in exactly one of the musical styles? n(exactly one style) = 1
e. How many listened to music in exactly two of the musical styles? n(exactly two styles) = 0
f. How many did not listen to any of the musical styles? n(none) = 0

Answer 1

Explanation:

A survey of 120 college students was taken to determine the musical styles they liked. Of those, 50 students listened to rock, 49 to classical, and 38 to jazz. Additionally, 24 students listened to rock and jazz, 33 to rock and classical, and 20 to classical and jazz. Finally, 17 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions:

a. How many listened to only rock music?

n(only rock) =

b. How many listened to classical and jazz but not rock?

n(classical and jazz not rock) =

c. How many listened to classical or jazz, but not rock?

n(classical or jazz not rock) =

d. How many listened to music in exactly one of the musical styles?

n(exactly one style) =

e. How many listened to music in exactly two of the musical styles?

n(exactly two styles) =

f. How many did not listen to any of the musical styles?

n(none) =

Answer 2

a. 0 listened to only rock music

b. 20 listened to classical and jazz, but not rock

c. 67 listened to classical or jazz, but not rock

d. 0 listened to music in exactly one of the musical styles

e. 77 listened to music in exactly two of the musical styles

f. 0 did not listen to any of the musical styles

Only Rock: n(Rock only) = 50 - 24 - 33 - 17 = 0

Total Rock listeners: n(Rock) = 50

Classical:

Only Classical: n(Classical only) = 49 - 33 - 20 - 17 = 0

Total Classical listeners: n(Classical) = 49

Jazz:

Only Jazz: n(Jazz only) = 38 - 20 - 24 - 17 = 0

Total Jazz listeners: n(Jazz) = 38

Overlaps:

Rock and Jazz: n(Rock ∩ Jazz) = 24

Rock and Classical: n(Rock ∩ Classical) = 33

Classical and Jazz: n(Classical ∩ Jazz) = 20

All three: n(Rock ∩ Classical ∩ Jazz) = 17

a)

n(only rock) = 0

b)

n(classical and jazz, not rock) = 20

c)

n(classical or jazz, not rock) = n(classical) + n(jazz) - n(classical and jazz)

= 49 + 38 - 20

= 67

d)

n(exactly one style) = n(Rock only) + n(Classical only) + n(Jazz only)

= 0 + 0 + 0

= 0

e)

n(exactly two styles) = n(Rock ∩ Classical) + n(Rock ∩ Jazz) + n(Classical ∩ Jazz)

= 33 + 24 + 20

= 77

f) How many did not listen to any of the musical styles?

n(none) = 0