The designers of a video game hired 100 game testers to play the game to determine whether the level of...

OWNING THE DATASET

Let's start by understanding what we're working with. This table shows test completion data across different levels, with columns showing how many testers completed each level within specific time ranges (1-5 minutes, 6-10 minutes, etc., up to 21-25 minutes).

Key insights we want to identify immediately:

• The distribution of completers across time ranges varies significantly by level
• Level 14 stands out with 0 testers in lower time ranges and a high proportion in upper ranges
• Some levels have testers in both extreme ranges (1-5 and 21-25)
• The "Total completed" column gives us the denominator for any proportion calculations

Strategic approach: Instead of analyzing each level methodically, we'll use sorting and visual scanning to identify patterns that instantly answer our statements.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "For at least one level, the range of completion times is at least 24 minutes."
What we're looking for:

• Any level where at least one person completed in the 1-5 range AND at least one completed in the 21-25 range
• This would create a range of at least 24 minutes (from as low as 1 to as high as 25)

In other words: We need to find at least one level with non-zero values in BOTH the "1-5" AND "21-25" columns.

Let's tackle this statement first because it requires the simplest visual check - we just need to scan for levels with values in both extreme columns.

Scanning the table, we can immediately spot Level 3, which has:

• 65 testers in the 1-5 minute range
• 2 testers in the 21-25 minute range

Since we found a level (Level 3) with testers in both the lowest and highest time ranges, the range of completion times could be as large as 24 minutes (from 1 minute to 25 minutes).

Statement 3: COULD BE GREATER THAN 20 FOR ONE OR MORE.

Teaching note: Notice how we didn't need to check every level or calculate exact ranges. The presence of values in both extreme columns immediately tells us the range could be at least 24 minutes. This pattern recognition saves significant time compared to calculating the exact range for each level.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "For at least one level, the median completion time exceeds 20 minutes."
What we're looking for:

• Any level where more than \(50\%\) of completers finished in the 21-25 minute range
• This would make the median exceed 20 minutes

In other words: We need to find a level where the majority of testers are in the highest time bracket.

For the median to exceed 20 minutes, more than half the completers must be in the 21-25 range. Let's sort the table by "21-25" column (descending) and then by "Total completed" (ascending) to quickly identify levels with the highest proportion in this range.

After sorting, we see Level 14 has the highest proportion with 4 out of 9 completers (approximately \(44\%\)) in the 21-25 range. This is still less than \(50\%\).

Since no level has more than half its completers in the 21-25 range, no level can have a median completion time exceeding 20 minutes.

Statement 2: MUST BE 20 OR LESS.

Teaching note: The power of the \(50\%\) threshold rule for medians saves us from calculating the exact median position for each level. By sorting to find the level with the highest proportion in the upper range and confirming it's below \(50\%\), we can quickly determine the answer.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "For at least one level, the mean completion time exceeds 20 minutes."
What we're looking for:

• Any level where the average completion time is greater than 20 minutes
• This requires calculating or estimating the mean for promising candidates

In other words: We need to find just one level where the average of all completion times exceeds 20.

From our earlier sorting, Level 14 stands out as our best candidate since it has:

• 0 testers in the lower time ranges
• 3 testers in the 11-15 range
• 2 testers in the 16-20 range
• 4 testers in the 21-25 range

Let's calculate the maximum possible mean for Level 14:

• 3 testers × 15 minutes (maximum in 11-15 range) = \(3 \times 15 = 45\)
• 2 testers × 20 minutes (maximum in 16-20 range) = \(2 \times 20 = 40\)
• 4 testers × 25 minutes (maximum in 21-25 range) = \(4 \times 25 = 100\)
• Total: \(185 \div 9 \approx 20.56\) minutes

Since we found a level (Level 14) where the mean completion time exceeds 20 minutes, the statement is true.

Statement 1: COULD BE GREATER THAN 20 FOR ONE OR MORE.

Teaching note: We only needed to calculate one level to prove the statement true. By recognizing Level 14's pattern (high concentration in upper ranges), we immediately focused our calculation efforts where they'd be most productive rather than calculating means for multiple levels.

FINAL ANSWER COMPILATION

Reviewing our analysis of each statement:

• Statement 1: COULD BE GREATER THAN 20 FOR ONE OR MORE
• Statement 2: MUST BE 20 OR LESS
• Statement 3: COULD BE GREATER THAN 20 FOR ONE OR MORE

Therefore, our answer is B (Statements 1 and 3 ONLY are true).

LEARNING SUMMARY

Skills We Used:

• Pattern recognition instead of calculation: Spotting levels with testers in both extreme ranges
• Strategic sorting: Arranging data to highlight the most promising candidates
• Threshold-based analysis: Using the \(50\%\) rule for median questions
• Targeted calculation: Only computing what's necessary to prove or disprove a statement

Strategic Insights:

1. Statement Order Matters: We started with Statement 3 because it required the simplest visual check, then moved to Statement 2 which used a straightforward threshold rule, and finally tackled Statement 1 which required more calculation.

2. The Power of Sorting: By sorting the data strategically, we made patterns immediately visible that would have taken much longer to find through manual checking.

3. One Example Is Enough: For "for at least one level" statements, finding just one example that satisfies the condition is sufficient - we don't need to check every level.

4. Extreme Value Focus: For range questions, always look at the extremes first - it's the quickest way to find large ranges.

Common Mistakes We Avoided:

• Calculating means for multiple levels when we only needed one example
• Checking every level for range values when visual scanning was faster
• Computing exact medians when the \(50\%\) threshold rule gave us the answer immediately
• Processing statements in numerical order rather than efficiency order

By applying these techniques, we transformed what could have been a calculation-heavy problem into a streamlined analysis that focuses on pattern recognition and strategic data evaluation.