Margin of error: 7 exam questions solved step by step

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What this lesson covers

  • The margin of error measures sampling variation only, not bias or other errors.
  • The 1/√n rules of thumb are not in the booklet, so they must be learned.
Key idea

7 questions, one method

  • 1. One result, same sample, or two samples?
  • 2. The rule: 1/√n, 2 × 1/√n, 1.5 × average.
  • 3. Result ± margin: work both ends.
  • 4. Say it about the population.
Your turn

Question 1 · Achieved

  • An invented poll of 1600 voters in a city:
  • "47% support a new stadium."
  • Find the margin of error, then the
  • confidence interval, and interpret it.
Worked example

Q1: 47% of 1600

  • One result: 1/√n.
  • Interval, then the sentence.
  1. MOE ≈ 1/√1600one proportion
  2. = 1/40 = 0.025 = 2.5%
  3. 47 − 2.5 = 44.5
  4. 47 + 2.5 = 49.5
  5. 44.5% to 49.5%
  6. of all voters in the citythe population
Your turn

Question 2 · Achieved

  • An invented survey of 850 adults:
  • "58% would use a new ferry service."
  • Form the confidence interval.
  • Careful how you add the margin.
Worked example

Q2: 58% of 850

  • Not a square: 4 d.p.
  • Points, not % of 58.
  1. MOE ≈ 1/√850one proportion
  2. = 0.0343 ≈ 3.4%4 d.p.
  3. 3.4% of 58 = 1.97not a % of the result
  4. 58 − 3.4 = 54.6
  5. 58 + 3.4 = 61.4
  6. 54.6% to 61.4%
Your turn

Question 3 · Merit

  • An invented poll of 1225 voters:
  • "Option A 46%, Option B 35%."
  • Find an interval for the difference.
  • Is Option A ahead?
Worked example

Q3: A − B, one poll

  • Same sample: double it.
  • Is zero inside?
  1. A: 46%, B: 35%same 1225 voters
  2. MOE ≈ 2 × 1/√1225same sample
  3. = 2/35 = 0.0571 ≈ 5.7%
  4. 46 − 35 = 11 pointsthe difference
  5. 11 − 5.7 = 5.3
  6. 11 + 5.7 = 16.7
  7. 5.3 to 16.7 points
  8. all above zero: A is aheadthe claim
Your turn

Question 4 · Merit

  • An invented survey, two random samples:
  • 400 students: 62% play sport weekly.
  • 2500 adults: 45% play sport weekly.
  • Find an interval for the difference.
Worked example

Q4: students − adults

  • Two samples: 1.5 × average.
  • Each n has its own 1/√n.
  1. students: 1/√400 = 1/20 = 5%
  2. adults: 1/√2500 = 1/50 = 2%
  3. average = (5 + 2)/2 = 3.5%
  4. MOE ≈ 1.5 × 3.5 = 5.25%two samples
  5. 62 − 45 = 17 pointsthe difference
  6. 17 − 5.25 = 11.75
  7. 17 + 5.25 = 22.25
  8. 11.75 to 22.25 points
  9. students higher than adults
Your turn

Question 5 · Merit, in context

  • A council plans an invented survey.
  • It wants a margin of error of 2% at most.
  • How many residents must it ask?
  • And for a margin of 1%?
Worked example

Q5: how big a sample?

  • Set 1/√n = the target.
  • Then square.
  1. 1/√n = 0.02MOE = 2%
  2. √n = 1 ÷ 0.02 = 50
  3. n = 50² = 2500
  4. at least 2500 residents
  5. 1/√n = 0.01, √n = 100
  6. n = 100² = 10000
  7. 10000 residents for 1%
  8. 10000 ÷ 2500 = 4half the MOE: 4 × n
Your turn

Question 6 · Excellence

  • An invented news site's online poll:
  • "1000 readers answered; 8% ride an
  • e-scooter weekly (margin of error 3.2%)."
  • Evaluate the margin of error.
Worked example

Q6: does 3.2% apply?

  • Sum first, then the rule.
  • Then how it was sampled.
  1. 1/√1000 = 0.0316 ≈ 3.2%arithmetic right
  2. 8 ± 3.2: 4.8% to 11.2%
  3. 8% is outside 30% to 70%
  4. rule overstates the MOEinterval too wide
  5. readers chose to answerself-selected
  6. not random: MOE does not apply
  7. one site, not all adultsthe frame
  8. MOE: sampling error only
  9. 3.2% does not describe it
Your turn

Question 7 · Excellence

  • An invented report: support for a bypass
  • was 44% in March (n = 625) and 48% in
  • September (n = 1600). "The 4-point rise
  • beats our 2.5% margin of error."
  • Is the rise real? Pause and try it.
Worked example

Q7: pick the rule

  • Two polls: two samples.
  • 2.5% is one poll only.
  1. compare 4 with 2.5%one poll's margin
  2. March: 1/√625 = 1/25 = 4%
  3. Sept: 1/√1600 = 1/40 = 2.5%
  4. average = (4 + 2.5)/2 = 3.25%
  5. MOE ≈ 1.5 × 3.25 = 4.875%two samples
Worked example

Q7: is zero inside?

  • Rise ± the right margin.
  • Then judge the claim.
  1. MOE ≈ 4.875%
  2. 48 − 44 = 4 pointsthe rise
  3. 4 − 4.875 = −0.875
  4. 4 + 4.875 = 8.875
  5. −0.875 to 8.875 points
  6. zero is insidefall, none, or rise
  7. claim not justified
  8. not proof support is unchanged
Summary

The method that solved all seven

  • Pick the rule, work both ends, say it about the population.
StepHow
One result1/√n
Same sample2 × 1/√n
Two samples1.5 × average MOE
The intervalresult ± MOE, both ends
Zero insideno claim either way
Conditionsrandom, 30% to 70%

More in AS 91584

Margin of error: the three rules of thumb6 minConfidence intervals, claims, and why overlap fails7 minExperiments vs observational studies: what you can conclude7 minSurveys and bias: sampling frames, non-response and wording6 minEvaluating a statistical report: the full answer7 minMargin of error: 7 exam questions solved step by step10 minConfidence intervals: 7 exam questions solved step by step10 minStudy design: 7 exam questions solved step by step10 minSurveys and bias: 7 exam questions solved step by step10 minEvaluating reports: 7 exam questions solved step by step10 min

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