In a survey of 400 individual investors regarding subscriptions to the New York Times (NYT), Wall Street Journal (WSJ), and USA Today (UST), the following data were obtained.
120 subscribe to the NYT.
146 subscribe to the WSJ.
68 subscribe to the UST.
36 subscribe to the NYT and WSJ.
27 subscribe to the WSJ and UST.
23 subscribe to the NYT and UST.
132 do not subscribe to any of these newspapers.
(a) How many of the individual investors surveyed subscribe to all three newspapers?
Incorrect: Your answer is incorrect.
investors
(b) How many subscribe to only one of these newspapers?
investors
If you made a Venn diagram, and remember how they work, then
(a) (120+146+68)-(36+27+23)+x = 400-132
(b) now use the Venn diagram to add up the three parts that don't intersect any others.
To answer these questions, we can use a method called the Inclusion-Exclusion Principle. This principle helps us count the number of elements in different sets.
(a) How many of the individual investors surveyed subscribe to all three newspapers?
To find the number of investors who subscribe to all three newspapers, we need to find the overlapping region of the subscriptions. We can start by subtracting the total number of investors who do not subscribe to any of these newspapers from the total number of investors surveyed.
So, the number of investors surveyed who subscribe to at least one of these newspapers is:
Total surveyed investors - Investors not subscribing to any newspaper
Total surveyed investors = 400
Investors not subscribing to any newspaper = 132
400 - 132 = 268
Now, we need to subtract the number of investors who subscribe to only one or two newspapers from this value.
To do that, we need to consider the overlapping regions. We have the following information:
- 120 subscribe to the NYT
- 146 subscribe to the WSJ
- 68 subscribe to the UST
- 36 subscribe to both NYT and WSJ
- 27 subscribe to both WSJ and UST
- 23 subscribe to both NYT and UST
To find the number of investors who subscribe to all three newspapers, we need to subtract the total number of investors who subscribe to exactly one or two newspapers from the total number who subscribe to at least one newspaper.
So, the number of investors who subscribe to all three newspapers is:
268 - (120 + 146 + 68 - 36 - 27 - 23)
Now, you can calculate the answer.
(b) How many subscribe to only one of these newspapers?
To find the number of investors who subscribe to only one of these newspapers, we need to sum the number of investors who subscribe to only one newspaper for each newspaper individually.
So, we need to add:
- The number of investors who subscribe only to the NYT (120 - 36 - 23)
- The number of investors who subscribe only to the WSJ (146 - 36 - 27)
- The number of investors who subscribe only to the UST (68 - 27 - 23)
Now, you can calculate the answer.