Students from School A and School B were asked whether they watch TV or use the Internet after finishing their homework. The Venn diagram and the two-way table show the results from the two surveys. Which statement is true?
| School B | Internet | Not Internet | Total |
|---|---|---|---|
| TV | 30 | 5 | 35 |
| Not TV | 11 | 4 | 15 |
| Total | 41 | 9 | 50 |
A. More students do both activities at A than at B.
B. More students watch TV at B than at A.
C. More students do neither activity at B than at A.
D. More students were surveyed at A than at B.
Answer (2)
Without the Venn diagram for School A, it is impossible to definitively determine which statement is true. The answer cannot be provided without additional information.
Explanation
Analyze School B data Let's analyze the data from School B first. From the table, we can see the following:
Number of students who use the Internet and watch TV: 30
Number of students who watch TV but do not use the Internet: 5
Number of students who use the Internet but do not watch TV: 11
Number of students who do neither: 4
Total number of students surveyed: 50
Represent School A data with variables Now, let's assume we have the data from School A (which is not provided, but we'll represent it with variables):
Let A b o t h be the number of students at School A who do both activities (watch TV and use the Internet).
Let A T V be the number of students at School A who watch TV.
Let A n e i t h er be the number of students at School A who do neither activity.
Let A t o t a l be the total number of students surveyed at School A.
Evaluate each statement We will now evaluate each statement:
'More students do both activities at A than at B.' This means 30"> A b o t h > 30 .
'More students watch TV at B than at A.' The number of students who watch TV at B is 35 (30 + 5). This means A_{TV}"> 35 > A T V .
'More students do neither activity at B than at A.' This means A_{neither}"> 4 > A n e i t h er .
'More students were surveyed at A than at B.' This means 50"> A t o t a l > 50 .
Analyze a hypothetical scenario for School A Without the Venn diagram for School A, we cannot definitively determine which statement is true. However, we can analyze the possible scenarios. Let's consider a hypothetical Venn diagram for School A and see if we can deduce any information.
Let's assume:
A b o t h = 25
A T V o n l y = 10
A I n t er n e t o n l y = 15
A n e i t h er = 5
Then:
A T V = 25 + 10 = 35
A t o t a l = 25 + 10 + 15 + 5 = 55
In this scenario:
30"> A b o t h ( 25 ) > 30 is false.
A_{TV} (35)"> 35 > A T V ( 35 ) is false.
A_{neither} (5)"> 4 > A n e i t h er ( 5 ) is false.
50"> A t o t a l ( 55 ) > 50 is true.
Final Analysis and Conclusion Since we don't have the Venn diagram for School A, we cannot definitively say which statement is true. However, based on the structure of the problem, we can assume that one of the statements can be determined to be true with the missing information. Let's re-examine the options, keeping in mind that we are looking for a statement that must be true given any possible Venn diagram for School A.
If the Venn diagram for School A showed that the total number of students surveyed was, say, 40, then the statement 'More students were surveyed at A than at B' would be false. If the total was 60, then the statement would be true. So, without the Venn diagram, we can't know for sure. The same logic applies to the other statements.
However, if we look at the table for School B, we see that 35 students watch TV. If any Venn diagram for School A has more than 35 students who watch TV, then the statement 'More students watch TV at B than at A' would be false. But, if every possible Venn diagram for School A has fewer than 35 students who watch TV, then the statement would be true. Without the Venn diagram, we can't know for sure.
Let's assume the Venn Diagram for School A shows the following:
Watch TV and use internet: 10
Watch TV only: 0
Use internet only 0
Neither: 0 Total: 10
In this case:
More students do both activities at A than at B: False (10 < 30)
More students watch TV at B than at A: True (35 > 10)
More students do neither activity at B than at A: True (4 > 0)
More students were surveyed at A than at B: False (10 < 50)
Let's assume the Venn Diagram for School A shows the following:
Watch TV and use internet: 40
Watch TV only: 10
Use internet only 10
Neither: 10 Total: 70
In this case:
More students do both activities at A than at B: True (40 > 30)
More students watch TV at B than at A: False (35 < 50)
More students do neither activity at B than at A: False (4 > 10)
More students were surveyed at A than at B: True (70 > 50)
Since we don't have the Venn Diagram for School A, we cannot definitively answer this question.
Examples
This type of problem is useful in market research. Imagine two different customer segments (School A and School B) and you want to understand their preferences for two products (TV and Internet). By comparing the data, you can identify which product is more popular in each segment and tailor your marketing strategies accordingly. For example, if more customers in School A use the Internet and watch TV compared to School B, you might focus on digital advertising for School A and traditional TV ads for School B. The data helps in making informed decisions about resource allocation and marketing channels.
Without the data from School A, we cannot determine which statement is true. All the statements rely on comparisons that require information not provided in the question. Therefore, we need the Venn diagram or survey results for School A to assess the accuracy of the statements.
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