Margin of Error
Mastery Quiz

✓ Correct: B. The MOE is the maximum expected difference — it tells us how far off our sample result could be from the true population value. About 95% of all sample results fall within 2 standard deviations of the population parameter. It is NOT the exact error (we never know that without the population data).

✓ Correct: B. "Online vs. in-person" is categorical data (a choice between categories, not a measurement). For categorical data: MOE ≈ 1/√n. No standard deviation is involved because we're working with proportions, not numerical measurements.

✓ Correct: C. MOE = 1/√100 = 1/10 = 0.10. Remember: √100 = 10. This means the true population proportion is likely within ±0.10 of the sample proportion.

✓ Correct: B. Quantitative data → MOE = 2σ/√n = 2(5)/√100 = 10/10 = 1.0 inch. The mean height estimate could be off by at most ±1 inch.

✓ Correct: C. At n=25: MOE = 1/√25 = 1/5 = 0.20. At n=100: MOE = 1/√100 = 1/10 = 0.10. The MOE went from 0.20 → 0.10, which is exactly half. When n increases by a factor of 4, MOE decreases by a factor of √4 = 2 (cuts in half).

✓ Correct: A. MOE = 2σ/√n = 2(4)/√64 = 8/8 = 1. Interval = 35 ± 1 = [34, 36]. Wait — that's answer B. Let me re-check: 2(4)/8 = 8/8 = 1. So 35−1=34, 35+1=36. Correct answer is B: 34 to 36 minutes. (Note: if you chose B, you're right!)

✓ Correct: C. Greek letter μ (mu) = population mean. x̄ (x-bar) = sample mean. p = population proportion. p̂ (p-hat) = sample proportion. σ (sigma) = population standard deviation. Keep these straight — they appear on every stats exam!

✓ Correct: A. Step 1: p̂ = 15/60 = 0.25. Step 2 (Categorical): MOE = 1/√60 ≈ 1/7.75 ≈ 0.129 ≈ 0.13. Step 3: Interval = 0.25 ± 0.13 → [0.12, 0.38]. So the true proportion likely falls between 12% and 38%.

✓ Correct: C. MOE = 2(121)/√200 = 242/14.14 ≈ 17.1. KIS interval: 550 ± 17.1 = [532.9, 567.1]. The national mean of 508 is NOT inside this interval, so it's reasonable to conclude KIS scores are indeed above the national average. This is the proper statistical reasoning — not just "550 > 508."

✓ Correct: A. William used MOE = σ/√n = 3/8 = 0.375, then simplified to 1/8 by ignoring σ entirely — he treated it as categorical data! The correct formula for quantitative data is 2σ/√n = 2(3)/√64 = 6/8 = 0.75 inches. He forgot the factor of 2.

✓ Correct: C. Seth: 0.20 ± 0.22 = [−0.02, 0.42] (very wide!). Tia: 0.40 ± 0.07 = [0.33, 0.47]. These intervals DO overlap (around 0.33–0.42). When intervals overlap, we can't say one sample is definitely right. The best approach: collect more samples and look at the distribution. Tia's MOE is smaller (more precise) but we can't declare a winner yet.

✓ Correct: C. Set 1/√n ≤ 0.05. Solve: √n ≥ 1/0.05 = 20. So n ≥ 20² = 400. This is why political polls typically use ~400–1000 people — to achieve a ±5% or smaller MOE. Remember: to halve the MOE, you must quadruple n!

✓ Correct: B. MOE = 2(10)/√100 = 20/10 = 2. Confidence interval: 50 ± 2 = [48, 52]. Since 55 is NOT in the interval [48, 52], the claim that μ = 55 is NOT supported by this data. We'd expect the true mean to be somewhere between 48 and 52.

✓ Correct: C. "Average daily steps" is a numerical measurement — it requires quantitative MOE (2σ/√n). The others are all categorical: owning a dog (yes/no), streaming service (name category), and lunch source (bring/buy). Key rule: if you measure something with a number that has meaningful magnitude and a standard deviation, use quantitative.

✓ Correct: B. MOE = 1/√625 = 1/25 = 0.04 = 4%. Interval: 52% ± 4% = [48%, 56%]. The interval DOES include values below 50% (like 48%, 49%), so the claim is NOT clearly false — we actually CANNOT rule out true approval below 50%. The opposing claim is partially valid because the interval includes values both above and below 50%. This is why overlapping with 50% is so important in politics!

✓ Correct: C. School A: MOE = 2(15)/√900 = 30/30 = 1. School B: MOE = 2(5)/√100 = 10/10 = 1. Both have MOE = 1! This is the key trap — a large σ can be offset by a large n, and a small σ can make a small n equally precise. Both factors (σ and n) affect MOE together.

✓ Correct: C. MOE = 2(6)/√36 = 12/6 = 2. Interval: 146 ± 2 = [144, 148]. Since 150 is OUTSIDE the interval [144, 148], the data does NOT support the company's 150g claim. The consumer group has evidence to challenge it. Note: "not supported" is different from "proven false" — stats never proves things absolutely, but 150 outside the interval is strong evidence.

✓ Correct: C. When multiple samples produce an approximately normal distribution, the center of that distribution is the best estimate of the true population parameter. The range 0.15–0.35 with most data there suggests the center ≈ 0.25. The full range (0.10–0.40) represents the spread of sampling variability, not the best estimate. This is why we collect multiple samples — the distribution tells us where the true value likely lies.

✓ Correct: B. Study 1: [74, 86]. Study 2: [83, 93]. These intervals OVERLAP between 83 and 86. When intervals overlap, we cannot conclude the population means are different — both populations could have a true mean anywhere in the overlap zone (83–86). Only if the intervals do NOT overlap at all can we claim the populations are statistically different. This is one of the most commonly missed concepts on stats exams!

✓ Correct: A.
• National MOE = 1/√400 = 1/20 = 0.05 = 5%. Interval: 44% ± 5% = [39%, 49%].
• City MOE = 1/√64 = 1/8 = 0.125 = 12.5%. Interval: 55% ± 12.5% = [42.5%, 67.5%] ≈ [43%, 67%].
• The intervals overlap between 43% and 49%! Since they overlap, we CANNOT statistically conclude the city's support is higher than the national rate. The city's small sample size (n=64) creates a very wide interval that overlaps the national one. The politician is making an unsupported claim. This is the full exam-level problem — master this and you've mastered MOE!