For the populations of each of 10 regions, the table shows the proportion that use Device P, the proportion that...

OWNING THE DATASET

Let's start by understanding this dataset with the intention of "owning" it completely.

We have a table comparing usage percentages for two devices (P and Q) across 10 different regions. For each region, we can see:

• The percentage of Device P users in that region
• The percentage of Device Q users in that region
• The percentage of users who use both devices in that region (the overlap)

Key insights from our initial scan:

• Values for Device P range from very small (0.01) to significant (0.45)
• Values for Device Q show a similar spread
• The overlap values are always smaller than either individual device percentage (which makes logical sense)
• The data is currently unsorted, which makes patterns difficult to spot immediately

This is perfect for sorting techniques, which will be our primary strategy for efficient analysis.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "The median percentage for Device P is less than the median percentage for Device Q."
What we're looking for:

• Find the median value for Device P
• Find the median value for Device Q
• Compare these two values

In other words: Is the middle value of all Device P percentages smaller than the middle value of all Device Q percentages?

Let's approach this efficiently using sorting:

First, let's sort the Device P column in ascending order. With 10 regions, the median will be the average of the 5th and 6th values.

After sorting Device P (ascending):

• We can see the values arranged from smallest to largest
• The 5th value is 0.10
• The 6th value is 0.11
• Therefore, median of Device P = \(\frac{0.10 + 0.11}{2} = 0.105\)

Now, let's sort the Device Q column in ascending order:

• The 5th value is 0.20
• The 6th value is 0.21
• Therefore, median of Device Q = \(\frac{0.20 + 0.21}{2} = 0.205\)

Comparing the medians:

• Median of Device P = 0.105
• Median of Device Q = 0.205

Since \(0.105 < 0.205\), Statement 1 is True.

Teaching Callout: Notice how sorting instantly revealed the median values? Without sorting, we would have needed to manually identify and arrange all 10 values for each device - a much more time-consuming and error-prone process. Sorting is often your most powerful first move in table analysis questions.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "For a majority of the regions, more than half of the users of Device Q also use Device P."
What we're looking for:

• For each region, check if \((\mathrm{P \text{ and } Q}) > 0.5 \times \mathrm{Q}\)
• Count how many regions satisfy this condition
• Determine if this count is more than half (>5) of all regions

In other words: In more than 5 regions, do more than 50% of Device Q users also use Device P?

Let's use a smart approach that avoids division:

To check if \((\mathrm{P \text{ and } Q}) > 0.5 \times \mathrm{Q}\), we can rewrite this as a direct comparison to avoid division calculations.

Let's sort Device P in descending order to see the regions with highest usage first. This will help us work with the most significant regions initially:

After sorting, we can identify the top 3 regions for Device P:

• Region 1: Device P = 0.45, Device Q = 0.35, Overlap = 0.09
• Region 3: Device P = 0.30, Device Q = 0.23, Overlap = 0.10
• Region 10: Device P = 0.15, Device Q = 0.27, Overlap = 0.08

Let's check if the condition \((\mathrm{P \text{ and } Q}) > 0.5 \times \mathrm{Q}\) is met for these regions:

For Region 1:

• Is \(0.09 > 0.5 \times 0.35\)?
• Is \(0.09 > 0.175\)?
• No, \(0.09 < 0.175\)

For Region 3:

• Is \(0.10 > 0.5 \times 0.23\)?
• Is \(0.10 > 0.115\)?
• No, \(0.10 < 0.115\)

We've already found two regions where the statement fails. Since we need a majority (more than 5 out of 10) to satisfy the condition, finding even more failing regions isn't necessary. We can confidently conclude that Statement 2 is False.

Teaching Callout: Notice how we avoided division by reframing the question as a direct comparison? Also, we didn't need to check all 10 regions once we found enough counterexamples. When a statement requires a "majority" condition, you can stop as soon as you find enough evidence to confirm or refute it.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "For a majority of the regions, more than half of the users of Device P also use Device Q."
What we're looking for:

• For each region, check if \((\mathrm{P \text{ and } Q}) > 0.5 \times \mathrm{P}\)
• Count how many regions satisfy this condition
• Determine if this count is more than half (>5) of all regions

In other words: In more than 5 regions, do more than 50% of Device P users also use Device Q?

We can use the same comparison approach we used for Statement 2, but with a different condition:

We already know the top 3 regions for Device P from our analysis of Statement 2:

• Region 1: Device P = 0.45, Device Q = 0.35, Overlap = 0.09
• Region 3: Device P = 0.30, Device Q = 0.23, Overlap = 0.10
• Region 10: Device P = 0.15, Device Q = 0.27, Overlap = 0.08

Let's check if the condition \((\mathrm{P \text{ and } Q}) > 0.5 \times \mathrm{P}\) is met for these regions:

For Region 1:

• Is \(0.09 > 0.5 \times 0.45\)?
• Is \(0.09 > 0.225\)?
• No, \(0.09 < 0.225\)

For Region 3:

• Is \(0.10 > 0.5 \times 0.30\)?
• Is \(0.10 > 0.15\)?
• No, \(0.10 < 0.15\)

Just as in Statement 2, we've already found two regions where the statement fails. Since we need a majority (more than 5 out of 10) to satisfy the condition, we can confidently conclude that Statement 3 is False.

Teaching Callout: Here's where our work from Statement 2 paid off! By already having sorted the data and identified the top regions, we could quickly apply the new condition without having to reorganize our data. This kind of strategic order of operations saves significant time.

FINAL ANSWER COMPILATION

After analyzing all three statements:

• Statement 1: True (median of Device P is less than median of Device Q)
• Statement 2: False (not a majority of regions have more than half of Device Q users also using Device P)
• Statement 3: False (not a majority of regions have more than half of Device P users also using Device Q)

The answer pattern is True/False/False.

LEARNING SUMMARY

Skills We Used

• Sorting as a First Move: We used sorting to instantly reveal medians and to identify regions with highest usage percentages
• Comparison Instead of Division: We reframed proportion questions to avoid time-consuming division
• Early Termination: We stopped checking once we found enough counterexamples
• Strategic Question Order: We leveraged our work on Statement 2 to solve Statement 3 more efficiently

Strategic Insights

• Always consider sorting first - It's a one-click operation that can reveal patterns immediately
• For "majority" questions, use the 'half test' - Check if value > half of total
• Look for ways to avoid calculations - Visual comparisons are faster than arithmetic
• Plan your question order strategically - Consider which statements might share analysis steps

Common Mistakes We Avoided

• Manually listing and sorting values instead of using the table's sorting feature
• Performing division calculations when simple comparisons would suffice
• Checking all regions even after finding sufficient counterexamples
• Solving statements in a fixed order rather than in the most efficient sequence

Remember, the goal in table analysis questions isn't just to get the right answer—it's to get there with minimal effort. By using these strategies, we can solve even complex-looking problems with remarkable efficiency.