The table shows, for each of 10 Irish weather stations and each of 8 calendar months, the average (arithmetic mean)...

Owning the Dataset

Let's take a strategic approach to understanding this table of average snowfall days in Ireland. The data shows the average number of days with snowfall across multiple weather stations over a 40-year period.

Looking at the table, we immediately notice:

• The data represents averages over 40 years, not yearly values
• Values range from small decimals to mid-single digits
• Each station has different patterns across months
• Some stations excel in specific months while others have higher totals

Key insights that will help us solve efficiently:

• Claremorris has the highest January average at 6.5 days
• Clones has the highest total average across all months at 24.1 days
• Cork's February average is 4.1 days
• Valentia's highest monthly average is 1.4 days (in February)

These observations will be crucial for evaluating the statements without unnecessary calculations.

Analyzing Statement 1

Statement 1 Translation:

Original: "The station with the greatest average number of days with snow in January is the same as the station with the greatest average total number of days with snow in a year."

What we're looking for:

• Which station has the highest January average?
• Which station has the highest yearly total average?
• Are these the same station?

In other words: Does the station with most January snow days also have the most total snow days?

Let's approach this efficiently. Instead of checking every station, we can directly identify:

• The station with the highest January average is Claremorris (6.5 days)
• The station with the highest yearly total average is Clones (24.1 days)

Since these are different stations (Claremorris ≠ Clones), the statement is MIGHT NOT BE TRUE.

Teaching note: Notice how we avoided calculating 40-year totals entirely. We only needed to identify which stations had the highest values in each category and check if they matched.

Analyzing Statement 3

Statement 3 Translation:

Original: "The Valentia station had at least 2 days with snow in at least one month."

What we're looking for:

• Did Valentia ever experience 2 or more snow days in any single month?
• Does the 40-year average data tell us anything conclusive about this?

In other words: Can we determine if Valentia had 2+ snow days in at least one month during those 40 years?

This is where understanding averages becomes crucial. Looking at Valentia's data, we see its highest monthly average is 1.4 days (February).

Let's think about what this average means:

• \(1.4 \times 40 = 56\) total days with snow in February over 40 years
• If every February had exactly 1 day with snow, that would be 40 days total
• Since \(56 > 40\), some years must have had more than 1 day with snow

Mathematically, it's impossible to have an average of 1.4 unless some years had 2 or more days. Therefore, Valentia must have had at least one month with 2 or more snow days.

The statement is MUST BE TRUE.

Teaching note: This demonstrates the mathematical necessity principle. When an average exceeds a whole number over many instances, at least some instances must exceed that whole number.

Analyzing Statement 2

Statement 2 Translation:

Original: "Cork had at least 2 days with snow every February."

What we're looking for:

• Did Cork have a minimum of 2 snow days in every single February?
• Can we determine this from the 40-year average?

In other words: Does Cork's February average of 4.1 days guarantee a minimum of 2 days every year?

This requires understanding a fundamental principle about averages: they tell us nothing definitive about minimums unless we have additional constraints.

For Cork's February average of 4.1 days:

• This means \(4.1 \times 40 = 164\) total days with snow over 40 Februaries
• This average could be achieved with some years having less than 2 days
• For example, we could have 38 years with 4 days each (152 days) and 2 years with 6 days each (12 days), totaling 164 days (average = 4.1)
• Or we could have 35 years with 5 days each (175 days) and 5 years with 0 days (0 days), still averaging around 4.1 days

Without additional information about the distribution, we cannot guarantee a minimum of 2 days every February. The statement is MIGHT NOT BE TRUE.

Teaching note: This illustrates the critical "Average Principle" - averages never guarantee minimums without additional constraints.

Final Answer Compilation

After analyzing all three statements:

• Statement 1: MIGHT NOT BE TRUE (different stations have highest January vs. total)
• Statement 2: MIGHT NOT BE TRUE (an average doesn't guarantee a minimum)
• Statement 3: MUST BE TRUE (mathematical necessity - some months must have had 2+ days)

Therefore, the correct answer is: Statement 3 ONLY MUST BE TRUE.

Learning Summary

Skills We Used

• Statistical Inference: We made logical deductions from averages without doing unnecessary calculations
• Pattern Recognition: We quickly identified the key values needed rather than analyzing every data point
• Mathematical Necessity: We recognized when averages mathematically require certain outcomes

Strategic Insights

1. The Average Principle: Averages only tell us about central tendency, not extremes (minimum/maximum values)
2. The Comparison Principle: When comparing two items, we only need to know if they're different, not by how much
3. The Mathematical Boundary Principle: When an average exceeds a whole number over many instances, at least some instances must exceed that whole number

Efficiency Gains

• We eliminated all 40-year calculations
• We avoided checking every station and month unnecessarily
• We focused only on the critical values that determined each statement's truth

Common Mistakes We Avoided

• Assuming an average guarantees a minimum value (Statement 2)
• Over-calculating when simple comparisons would suffice (Statement 1)
• Missing mathematical necessities by not thinking about what an average implies (Statement 3)

This problem demonstrates how understanding statistical principles can dramatically simplify data analysis questions, allowing us to focus on logical reasoning rather than extensive calculations.