A pollster interviewed 500 university seniors who owned credit cards. She reported that 240 had Goldcard, 290 had Supercard, and 270 had Thriftcard.
Of those seniors, the report said that:
80 owned only a Goldcard and a Supercard,
70 owned only a Goldcard and a Thriftcard,
60 owned only a Supercard and a Thriftcard, and
50 owned all three cards.
When the report was submitted for publication in the local campus newspaper, the editor refused to publish it, claiming the poll was not accurate. Was the editor right? Why or why not?
The editor was right. The numbers add up to more than 500.
Using the formula ...
n(A or B or C)
= n(A) + n(B) + n(C) - n(A and B) - n(A and C) - n(B and C) + n(A and B and C)
500 = 240 + 290 + 270 - 80 - 70 - 60 + 50
500 = 640
not true, so data not consistent.
You could also make Venn diagrams to show the data distribution.
To determine if the editor was right or not, we need to check if the numbers provided in the report are consistent.
Let's break down the given information into a Venn diagram to help us visualize the relationships between the different sets of credit cards:
Goldcard (G) = 240
Supercard (S) = 290
Thriftcard (T) = 270
Now let's fill in the overlapping regions based on the given information:
Only G and S = 80
Only G and T = 70
Only S and T = 60
All three G, S, and T = 50.
Now let's calculate the number of seniors who own only one card from each set:
Only G = (Only G and S) + (Only G and T) + (G,S,T) = 80 + 70 + 50 = 200
Only S = (Only G and S) + (Only S and T) + (G,S,T) = 80 + 60 + 50 = 190
Only T = (Only G and T) + (Only S and T) + (G,S,T) = 70 + 60 + 50 = 180
Finally, let's determine the number of seniors who own no cards at all:
None = Total seniors - (Only G) - (Only S) - (Only T) - (G,S,T) = 500 - 200 - 190 - 180 - 50 = 500 - 620 = -120
From the calculation above, we can see that we have a negative value for the number of seniors who own no cards at all. This indicates an inconsistency in the given information. It is not possible to have a negative number of seniors who own no cards.
Therefore, the editor was right to reject the report as it contains an error, most likely a miscount or misreporting of the data. The numbers provided do not add up correctly, making the poll inaccurate.
To determine if the editor was right in claiming that the poll was not accurate, we need to analyze the given information and check if it satisfies certain conditions.
First, let's create a Venn diagram to visualize the relationships between the three types of cards—Goldcard, Supercard, and Thriftcard—and their respective owners.
```
Goldcard (G)
/ \
/ \
Supercard (S) Thriftcard (T)
\ /
\ /
\ /
All three (or overlapping area)
```
Based on the information provided, we have the following data:
- 240 seniors had Goldcard (G)
- 290 seniors had Supercard (S)
- 270 seniors had Thriftcard (T)
- 80 seniors had both Goldcard (G) and Supercard (S) only
- 70 seniors had both Goldcard (G) and Thriftcard (T) only
- 60 seniors had both Supercard (S) and Thriftcard (T) only
- 50 seniors had all three cards (G, S, and T)
Now, let's check if the given data satisfies the conditions for a valid survey or poll:
1. Total number of seniors: The given data mentions that the pollster interviewed 500 university seniors who owned credit cards. So, the total number of seniors should be 500.
Let's check if the number of seniors mentioned in the given data matches this condition:
- Number of seniors with Goldcard (G): 240
- Number of seniors with Supercard (S): 290
- Number of seniors with Thriftcard (T): 270
Summing up the numbers for each card, we have:
240 + 290 + 270 = 800
Since the total number (800) exceeds the total number of seniors interviewed (500), there is an inconsistency. The numbers do not match the condition of the total number of seniors.
2. Overlapping relationships: The given data also includes information about the overlapping relationships between the cards. We can check if these relationships are consistent by comparing them:
- Number of seniors with Goldcard (G) and Supercard (S) only: 80
- Number of seniors with Goldcard (G) and Thriftcard (T) only: 70
- Number of seniors with Supercard (S) and Thriftcard (T) only: 60
- Number of seniors with all three cards (G, S, T): 50
We can verify these relationships by summing up the individual overlaps:
80 + 70 + 60 + 50 = 260
If the overlaps are consistent, this total should match the actual overlaps. However, this total (260) does not match the actual number of overlaps stated in the given data.
Based on these inconsistencies, we can conclude that the editor was right to claim that the poll was not accurate. The data provided is internally inconsistent and does not satisfy the conditions of a valid survey.