Confidence intervals: 7 exam questions solved step by step

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What this lesson covers

  • A confidence interval is interpreted by naming the population, the variable and the parameter.
  • Comparing groups uses an interval for the difference, not whether two intervals overlap.
Key idea

7 questions, one method

  • 1. Form it: result ± margin of error.
  • 2. Interpret: population, variable.
  • 3. Claim: a separate sentence, a reason.
  • 4. Two groups: interval for the difference.
Your turn

Question 1 · Achieved

  • An invented survey of 1600 secondary
  • students in a region: "63% have a
  • part-time job." Form a confidence
  • interval and interpret it.
Worked example

Q1: 63% of 1600

  • Form it first.
  • Then name three things.
  1. MOE ≈ 1/√1600 = 1/40 = 2.5%
  2. 63 − 2.5 = 60.5
  3. 63 + 2.5 = 65.5
  4. 60.5% to 65.5%
  5. all secondary studentspopulation
  6. has a part-time jobvariable
  7. the proportion who doparameter
Your turn

Question 2 · Achieved

  • An invented poll of adults: "38% want a
  • four-day school week (margin of error
  • 3.5%)." Form the interval. Is it fewer
  • than half? About how many were asked?
Worked example

Q2: 38% ± 3.5%

  • The margin is given.
  • Claim from the whole range.
  1. 38 − 3.5 = 34.5
  2. 38 + 3.5 = 41.5
  3. 34.5% to 41.5%
  4. all below 50%the reason
  5. fewer than half want itthe claim
  6. 1/√n = 0.035
  7. n = 1 ÷ 0.035² ≈ 816about 816 adults
Your turn

Question 3 · Merit

  • An invented survey of 900 residents of a
  • beach town: "54% support a dog-free
  • summer rule." Interpret the interval,
  • then make a claim.
Worked example

Q3: 54% of 900

  • Interpretation first.
  • Claim second, with a reason.
  1. MOE ≈ 1/√900 = 1/30one proportion
  2. = 0.0333 ≈ 3.3%4 d.p.
  3. 54 − 3.3 = 50.7
  4. 54 + 3.3 = 57.3
  5. 50.7% to 57.3%all residents
  6. all above 50%the reason
  7. a majority support the rule
Your turn

Question 4 · Merit

  • An invented headline: "Most residents
  • back the new stadium plan." The survey:
  • 52% of 1089 residents.
  • Careful: 52% is more than half.
Worked example

Q4: is it most residents?

  • The sample is not the claim.
  • Where is 50%?
  1. 52% > 50%, so mostthe sample only
  2. MOE ≈ 1/√1089 = 1/33
  3. = 0.0303 ≈ 3.0%4 d.p.
  4. 52 − 3 = 49, 52 + 3 = 55
  5. 49% to 55%
  6. 50% is insideunder or over half
  7. "most" is not justified
Your turn

Question 5 · Merit, in context

  • An invented poll of 1600 town residents
  • on a new skate park: 41% prefer site A,
  • 33% prefer site B. Is site A preferred?
Worked example

Q5: A − B, one poll

  • Same sample: 2 × 1/√n.
  • Then where zero sits.
  1. MOE ≈ 2 × 1/√1600same sample
  2. = 2/40 = 0.05 = 5%
  3. 41 − 33 = 8 pointsthe difference
  4. 8 − 5 = 3, 8 + 5 = 13
  5. 3 to 13 pointssite A ahead by
  6. all above zerothe reason
  7. site A is preferred
Your turn

Question 6 · Excellence

  • An invented report, two random samples.
  • North: 55% of 400 back a new cycleway.
  • South: 48% of 1600 back it.
  • "The intervals overlap: no difference."
  • Evaluate the claim. Pause and try it.
Worked example

Q6: the two intervals

  • They do overlap.
  • Overlap is not the test.
  1. North: 1/√400 = 1/20 = 5%
  2. 55 ± 5: 50% to 60%
  3. South: 1/√1600 = 1/40 = 2.5%
  4. 48 ± 2.5: 45.5% to 50.5%
  5. overlap: 50% to 50.5%
  6. overlap, so no differencenot the test
Worked example

Q6: the difference interval

  • Two samples: 1.5 × average.
  • One interval, clear of zero?
  1. North 5%, South 2.5%
  2. average = (5 + 2.5)/2 = 3.75%
  3. MOE ≈ 1.5 × 3.75 = 5.625%two samples
  4. 55 − 48 = 7 pointsthe difference
  5. 7 − 5.625 = 1.375
  6. 7 + 5.625 = 12.625
  7. 1.375 to 12.625 pointsnorth higher by
  8. all above zero: north higherthe claim
  9. the report is wrong
Your turn

Question 7 · Excellence

  • An invented report on a library levy:
  • 50% of 1600 over-60s support it;
  • 47% of 2500 under-30s support it.
  • "Older people are more supportive."
  • Is the claim justified? Pause and try it.
Worked example

Q7: the margin of error

  • 50 > 47 in the samples only.
  • Two samples: 1.5 × average.
  1. 50% > 47%, so older higherthe samples only
  2. over-60s: 1/√1600 = 2.5%
  3. under-30s: 1/√2500 = 2%
  4. average = (2.5 + 2)/2 = 2.25%
  5. MOE ≈ 1.5 × 2.25 = 3.375%two samples
Worked example

Q7: is zero inside?

  • Interpret the whole range.
  • Then judge the claim.
  1. MOE ≈ 3.375%
  2. 50 − 47 = 3 pointsthe difference
  3. 3 − 3.375 = −0.375
  4. 3 + 3.375 = 6.375
  5. −0.375 to 6.375 points
  6. zero is insidelower, same or higher
  7. claim not justified
  8. not proof they are equal
Summary

The method that solved all seven

  • Form it, interpret it, then claim separately, with a reason.
StepHow
Form itresult ± margin of error
Interpretpopulation, variable
Then claima separate sentence, a reason
Two groupsinterval for the difference
Zero insideno claim either way
Overlapnever the test

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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