The table shows the population growth rate (in percent per year), the total population, and the total land area of...

OWNING THE DATASET

Let's start by understanding what we're working with. We have a table showing data for \(11\) provinces with three key metrics:

• Population (number of residents)
• Population Growth Rate (annual percentage)
• Land Area (in square miles)

The table has a manageable number of provinces, which means sorting will be our most powerful tool for quickly analyzing relationships between these metrics.

Key insight: When dealing with statements about medians in a table with \(11\) entries, we're looking for the \(6\mathrm{th}\) value in any sorted list. This will help us quickly identify median values without manual calculations.

ANALYZING STATEMENT 2

Let's strategically start with Statement 2 rather than Statement 1, as it will provide crucial information about medians that we can use for other statements.

Statement 2 Translation:
Original: "The province that has the median population growth rate also has the median land area."
What we're looking for:

• Find the province with the median population growth rate
• Check if this same province also has the median land area

In other words: Is the middle-ranked province for growth rate also the middle-ranked province for land area?

Let's sort the data to quickly identify these median values:

1. First, let's sort by Population Growth Rate: When sorted, the \(6\mathrm{th}\) value (the median) is \(1.12\), which belongs to Mio.
2. Now, let's sort by Land Area: When sorted, the \(6\mathrm{th}\) value (the median) is \(1,964\) square miles, which also belongs to Mio.

Since the same province (Mio) has both the median population growth rate and the median land area, Statement 2 is Yes.

Teaching point: Notice how sorting immediately revealed the median values without requiring us to calculate them manually. By starting with Statement 2, we've also identified key median values that will help us evaluate other statements more efficiently.

ANALYZING STATEMENT 1

Now let's examine Statement 1 with our newfound knowledge of the median values.

Statement 1 Translation:
Original: "The three provinces with the smallest land areas all have populations below the median population."
What we're looking for:

• Identify the three provinces with the smallest land areas
• Check if all three have populations below the median population

In other words: Do the three smallest provinces (by area) all have fewer-than-middle residents?

We've already sorted by Land Area when analyzing Statement 2, so we can immediately identify the three smallest provinces: Agae (\(144\) sq mi), Jan (\(378\) sq mi), and Pitan (\(796\) sq mi).

Now, let's sort by Population to see where these three provinces fall:

• When sorted by Population, we can see that Pitan has the \(6\mathrm{th}\) highest population (\(184,405\)), which is exactly the median.
• Since Pitan's population equals the median (not below it), we've found a counterexample.

Statement 1 is No.

Teaching point: We didn't need to check all three provinces - finding just one counterexample is enough to disprove the statement. This is a powerful approach when evaluating "all" statements - look for any exception and you can stop.

ANALYZING STATEMENT 3

Finally, let's tackle Statement 3 with our efficient approach.

Statement 3 Translation:
Original: "No province has values above the median for all three metrics (population, population growth rate, and land area)."
What we're looking for:

• Check if any province exceeds all three median values
• Just one counterexample would make this statement false

In other words: Is there any province that's above average in all three categories?

We already know the median values from our previous work:

• Median Population: \(184,405\) (from Statement 1)
• Median Population Growth Rate: \(1.12\) (from Statement 2)
• Median Land Area: \(1,964\) sq mi (from Statement 2)

Rather than checking every province, let's be strategic. Provinces with extremely high values in one category are good candidates to check first. Looking at larger provinces like Ida:

Checking Ida:

• Population Growth Rate: \(1.38\) (greater than median \(1.12\)) ✓
• Population: \(1,173,108\) (greater than median \(184,405\)) ✓
• Land Area: \(3,287\) sq mi (greater than median \(1,964\)) ✓

We found that Ida exceeds all three median values, which is a direct counterexample to Statement 3.

Statement 3 is No.

Teaching point: When testing statements about "no instance exists," finding just one counterexample is enough. By strategically targeting likely candidates (like larger provinces) rather than checking all \(11\) provinces, we saved significant time.

FINAL ANSWER COMPILATION

Reviewing our findings:

• Statement 1: No (Pitan has exactly the median population, not below it)
• Statement 2: Yes (Mio has both the median growth rate and land area)
• Statement 3: No (Ida exceeds all three median values)

Therefore, our answer is B) Statement 2 ONLY.

LEARNING SUMMARY

Skills We Used:

• Strategic Statement Order: We tackled Statement 2 first because it established key median values we could use elsewhere.
• Sorting for Visual Pattern Recognition: Sorting transformed complex data into easily scannable lists where medians and outliers became immediately visible.
• Early Termination: For both Statement 1 and Statement 3, we stopped as soon as we found a counterexample.

Strategic Insights:

• The "Sort-and-Stop" Approach: Sort by the relevant column, use visual scanning to find patterns, and stop immediately when you find confirming/disconfirming evidence.
• Statement Order Matters: When multiple statements involve the same metrics (like medians), solve the most information-rich statement first.
• Target Likely Counterexamples: For "no instance exists" statements, first check entries with extreme values in related categories.

Common Mistakes We Avoided:

• We didn't pre-sort all columns before beginning analysis - we sorted strategically when needed.
• We didn't check all provinces against all medians (\(33\) comparisons!) - we targeted our approach.
• We didn't verify Statement 1 by checking all three small provinces - we stopped at the first counterexample.

Remember that in table analysis questions, sorting is often your most powerful first step, allowing you to instantly see relationships that would take much longer to find through calculation or manual checking.