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In a survey, each respondent was asked five questions (Q1 through Q5). The table shows the percentage of respondents who picked each of the four possible responses to the question (strongly agree, agree, somewhat agree, and disagree). For each question, each respondent to the survey picked exactly one of the four possible responses.
| Response | Q1 | Q2 | Q3 | Q4 | Q5 |
|---|---|---|---|---|---|
| Strongly agree | 15% | 6% | 45% | 15% | 11% |
| Agree | 20% | 18% | 29% | 42% | 13% |
| Somewhat agree | 43% | 24% | 18% | 14% | 55% |
| Disagree | 22% | 52% | 8% | 29% | 21% |
For each of the following statements about these survey results, select Yes if the information provided indicates that the statement is correct. Otherwise, select No.
At least half of the respondents picked "disagree" for at least one question.
At least one respondent picked "disagree" for Q2 and "somewhat agree" for Q5.
More than half of the respondents picked "strongly agree" for at least one question.
Let's start by understanding this table with the intention of "owning the dataset." This survey response table shows how participants responded to five different questions, with percentages for each response type.
The table is organized with:
Key insights from our initial scan:
These percentage patterns will be crucial for efficiently analyzing the statements.
Statement 1 Translation:
Original: "At least half of the respondents picked 'disagree' for at least one question."
What we're looking for:
In other words: Is there at least one question where half or more of the respondents disagreed?
Let's scan the "Disagree" row across all five questions to see if any values are at least 50%:
Q1: Less than 50% (not shown explicitly in our data)
Q2: 52% ↠This exceeds our 50% threshold!
Q3: Less than 50% (not shown explicitly)
Q4: Less than 50% (not shown explicitly)
Q5: Less than 50% (not shown explicitly)
We immediately find that Q2 has 52% of respondents picking "disagree." Since \(52\% > 50\%\), at least half of the respondents picked "disagree" for at least one question (specifically Q2).
Statement 1 is Yes.
Notice how we only needed to find a single instance where the "disagree" percentage was 50% or higher. Once we found 52% for Q2, we could immediately conclude without checking further percentages in detail.
Statement 2 Translation:
Original: "At least one respondent picked 'disagree' for Q2 and 'somewhat agree' for Q5."
What we're looking for:
In other words: Is there any overlap between the "disagree Q2" group and the "somewhat agree Q5" group?
From our previous analysis, we already know:
Here's where we can use a powerful pattern recognition principle: When two groups together exceed 100%, there must be some overlap between them.
Since \(52\% + 55\% = 107\% > 100\%\), there must be some respondents who belong to both groups. This overlap is at least 7% of respondents (\(107\% - 100\% = 7\%\)).
Therefore, at least 7% of respondents both disagreed with Q2 and somewhat agreed with Q5.
Statement 2 is Yes.
Notice how we didn't need to perform complex calculations or see individual respondent data. The simple fact that the percentages exceed 100% guarantees that overlap exists.
Statement 3 Translation:
Original: "More than half of the respondents picked 'strongly agree' for at least one question."
What we're looking for:
In other words: Did more than half of all respondents strongly agree with at least one or more questions?
For this statement, we need to scan the "Strongly agree" row to find the highest percentage:
The key insight: Even if there was zero overlap between the people who strongly agreed with different questions (which would maximize the total percentage), the absolute maximum percentage of respondents who strongly agreed with at least one question would be 45%.
Since \(45\% < 50\%\), it's impossible for more than half of the respondents to have strongly agreed with at least one question.
Statement 3 is No.
Notice how we focused only on the highest percentage (45%) and immediately recognized this as a ceiling. We didn't need to calculate the exact percentage of people who strongly agreed with at least one question - we just needed to know it couldn't exceed 45%.
Let's compile our findings:
The answer pattern is: Yes, Yes, No.
This approach transformed what could have been a calculation-heavy problem into a pattern recognition exercise, allowing us to solve all three statements quickly and accurately.
At least half of the respondents picked "disagree" for at least one question.
At least one respondent picked "disagree" for Q2 and "somewhat agree" for Q5.
More than half of the respondents picked "strongly agree" for at least one question.