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The designers of a video game hired 100 game testers to play the game to determine whether the level of difficulty was appropriate. Each of the 100 testers made at most 25 attempts to complete each of the game's 15 levels (Levels 1–15). When a tester completed a level, he or she then made no further attempts at that level. For each of the levels, the table shows, among the testers who completed the level, the total number of testers whose number of attempts to complete that level fell in various ranges.
| Level | 1 to 5 | 6 to 10 | 11 to 15 | 16 to 20 | 21 to 25 | Total completed |
|---|---|---|---|---|---|---|
| 1 | 26 | 28 | 15 | 2 | 0 | 71 |
| 2 | 91 | 8 | 1 | 0 | 0 | 100 |
| 3 | 65 | 13 | 9 | 2 | 2 | 91 |
| 4 | 73 | 17 | 5 | 3 | 1 | 99 |
| 5 | 33 | 22 | 6 | 7 | 1 | 69 |
| 6 | 95 | 5 | 0 | 0 | 0 | 100 |
| 7 | 62 | 14 | 10 | 8 | 3 | 97 |
| 8 | 82 | 15 | 2 | 0 | 0 | 99 |
| 9 | 13 | 17 | 11 | 5 | 3 | 49 |
| 10 | 2 | 7 | 6 | 3 | 1 | 19 |
| 11 | 51 | 9 | 8 | 8 | 1 | 77 |
| 12 | 9 | 5 | 8 | 2 | 6 | 30 |
| 13 | 20 | 13 | 10 | 7 | 5 | 55 |
| 14 | 0 | 0 | 3 | 2 | 4 | 9 |
| 15 | 8 | 12 | 6 | 8 | 4 | 38 |
For each of the following statistical measures, select Must be 20 or less for every level if, based on the information provided, the measure must be at most 20 for each of the 15 levels. Otherwise, select Could be greater than 20 for one or more.
Mean of the total number of attempts to complete for all testers who completed the level
Median of the total number of attempts to complete for all testers who completed the level
Range of the total number of attempts to complete for all testers who completed the level
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:
Strategic approach: Instead of analyzing each level methodically, we'll use sorting and visual scanning to identify patterns that instantly answer our statements.
Statement 3 Translation:
Original: "For at least one level, the range of completion times is at least 24 minutes."
What we're looking for:
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:
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.
Statement 2 Translation:
Original: "For at least one level, the median completion time exceeds 20 minutes."
What we're looking for:
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.
Statement 1 Translation:
Original: "For at least one level, the mean completion time exceeds 20 minutes."
What we're looking for:
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:
Let's calculate the maximum possible mean for Level 14:
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.
Reviewing our analysis of each statement:
Therefore, our answer is B (Statements 1 and 3 ONLY are true).
Skills We Used:
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:
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.
Mean of the total number of attempts to complete for all testers who completed the level
Median of the total number of attempts to complete for all testers who completed the level
Range of the total number of attempts to complete for all testers who completed the level