In a survey, each respondent was asked five questions (Q1 through Q5). The table shows the percentage of respondents who...

Owning the Dataset

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:

• Rows: Different response types (Strongly agree, Somewhat agree, Disagree)
• Columns: Five survey questions (Q1 through Q5)
• Values: Percentages of respondents giving each response type for each question

Key insights from our initial scan:

• The percentages in each column add up to 100% (as we'd expect for a complete survey)
• Q2 has the highest "Disagree" percentage at 52% - the only question where "Disagree" is the majority response
• Q3 has the highest "Strongly agree" percentage at 45% - nearly half of respondents
• Q5 has the highest "Somewhat agree" percentage at 55%

These percentage patterns will be crucial for efficiently analyzing the statements.

Analyzing Statement 1

Statement 1 Translation:
Original: "At least half of the respondents picked 'disagree' for at least one question."
What we're looking for:

• Any question where 50% or more of respondents selected "disagree"
• Only need to find one such question to prove the statement true

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.

Analyzing Statement 2

Statement 2 Translation:
Original: "At least one respondent picked 'disagree' for Q2 and 'somewhat agree' for Q5."
What we're looking for:

• People who both disagreed with Q2 AND somewhat agreed with Q5
• Need to determine if this overlap group exists (even if just one person)

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:

• 52% of respondents disagreed with Q2
• 55% of respondents somewhat agreed with Q5

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.

Analyzing Statement 3

Statement 3 Translation:
Original: "More than half of the respondents picked 'strongly agree' for at least one question."
What we're looking for:

• The percentage of people who strongly agreed with at least one of the five questions
• Whether this percentage exceeds 50%

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:

• Q1: Less than 45% (not explicitly shown)
• Q2: Less than 45% (not explicitly shown)
• Q3: 45% ← Highest "Strongly agree" percentage
• Q4: Less than 45% (not explicitly shown)
• Q5: Less than 45% (not explicitly shown)

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%.

Final Answer Compilation

Let's compile our findings:

• Statement 1: Yes (52% disagreed with Q2, which is more than half)
• Statement 2: Yes (The 52% who disagreed with Q2 and the 55% who somewhat agreed with Q5 must have some overlap)
• Statement 3: No (At most 45% of respondents strongly agreed with at least one question)

The answer pattern is: Yes, Yes, No.

Learning Summary

Skills We Used

• Pattern Recognition: Recognizing that percentages exceeding 100% guarantees overlap
• Setting Bounds: Using maximum values as ceilings to quickly rule out possibilities
• Efficient Scanning: Looking for threshold values (\(≥50\%\)) rather than analyzing every data point
• Leveraging Previous Findings: Using the 52% from Statement 1 in our Statement 2 analysis

Strategic Insights

• When evaluating "at least half" statements, we only need to find a single instance \(≥50\%\)
• For overlap questions, if two groups total >100%, overlap must exist
• For "at least one" questions, the highest single percentage gives us a ceiling value
• The ordering of statements often allows us to reuse information we've already found

Common Mistakes We Avoided

• Calculating exact percentages when we only needed to know if a threshold was met
• Analyzing all five questions in detail when a single value could answer our question
• Getting distracted by unnecessary calculations when pattern recognition was sufficient
• Missing the logical shortcut that percentages exceeding 100% guarantees overlap

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.