Democracy Data

Case Study 3: Polling Margin of Error (MoE) Pre-Election Polling

A major state Senate pre-election poll lists Candidate Red at 49% and Candidate Blue at 46%. The poll notes a Margin of Error of ±3.5%.

Task: Calculate the actual interval range of support for both candidates. Explain whether Candidate Blue could mathematically be ahead based on these ranges.

Workspace (Calculate both ranges and check for overlap)

Candidate Red Range:

Candidate Blue Range:

Problem 3 Checklist

• Find Candidate Red's lower bound: \(49\% - 3.5\%\).
• Find Candidate Red's upper bound: \(49\% + 3.5\%\).
• Find Candidate Blue's lower bound: \(46\% - 3.5\%\).
• Find Candidate Blue's upper bound: \(46\% + 3.5\%\).
• Compare bounds. Do the ranges overlap? Explain what this overlap means.

Critical Thinking:

"Statistics in public policy aren't static—they represent margins of possibility that guide governing coalitions."

© 2026 iCivics Democracy Data Series Midterm Math Practice Page 2 of 2

1. Fraction representation: \(\frac{122,000,000}{264,000,000} \text{ or } \frac{122}{264}\)
2. Divide: \(122 \div 264 = 0.462121...\)
3. Round to 3 decimal places: \(0.462\)
4. Convert to percentage: \(0.462 \times 100 = 46.2\%\)

Turnout Rate Final Answer: 46.2%

Comparison Step: Students compare 46.2% to 2018's 50.1%. They should note turnout decreased by 3.9 percentage points, but remained extremely high compared to the historical average (usually under 40%).

Case Study 2 Solution: House Margin Shift Problem 2 Answer

Step-by-Step Math Solution:

Democrats (Blue): Start: 222 seats | End: 213 seats
Calculation: \(213 - 222 = -9\)
Net Change: -9 Seats

Republicans (Red): Start: 213 seats | End: 222 seats
Calculation: \(222 - 213 = +9\)
Net Change: +9 Seats

Key Insight: The sum of the changes is zero (\(-9 + 9 = 0\)), demonstrating the law of conservation of seats in a fixed-size legislature (435 seats total). The Republican party took control of the House because they passed the 218-seat threshold.

Case Study 3 Solution: Polling Margin of Error Problem 3 Answer

Step-by-Step Math Solution:

Candidate Red (49% ± 3.5%): Lower Bound: \(49\% - 3.5\% = 45.5\%\)
Upper Bound: \(49\% + 3.5\% = 52.5\%\)
Interval: 45.5% to 52.5%

Candidate Blue (46% ± 3.5%): Lower Bound: \(46\% - 3.5\% = 42.5\%\)
Upper Bound: \(46\% + 3.5\% = 49.5\%\)
Interval: 42.5% to 49.5%

The Overlap Explanation: Since Candidate Red's range drops as low as 45.5% and Candidate Blue's range reaches as high as 49.5%, the intervals overlap significantly (from 45.5% to 49.5%). This means Candidate Blue could mathematically lead.

Classroom Debrief Discussion Starters

• "If turnout is so important to democratic legitimacy, what math-driven systems can states use to encourage it?"
• "Why are narrow margins of control (like a 9-seat shift) challenging for passing laws in Congress?"
• "How should news media report polling so voters understand that percentages are ranges, not solid facts?"

© 2026 iCivics Democracy Data Series Teacher Keys & Guides Page 2 of 2