NCEA Level 3Confidence intervalsAS 91584

Confidence intervals, claims, and why overlap fails

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

From one sample to the population

  • The true population value is unseen.
  • An interval gives believable values.
  • Form it, interpret it, then claim.
Worked example

Forming the interval

  • An invented council survey.
  • Estimate ± margin of error.
  1. n = 625, 56% support
  2. MOE ≈ 1/√n = 1/√625
  3. = 1/25 = 0.04 = 4%exact
  4. 56 − 4 = 52
  5. 56 + 4 = 60
  6. 52% to 60%
Key idea

Three things to name

  • The population: everyone, not the sample.
  • The variable: what was measured.
  • The parameter: the proportion.
PartHere
Populationresidents of the region
Variablesupports the bus route
Parameterthe proportion
Worked example

The interpretation

  • I am fairly sure that the proportion...
  • ...of all residents of the region...
  • ...who support the route is 52% to 60%.
Worked example

Then a separate claim

  • The whole interval is above 50%.
  • Claim: a majority support the route.
  • Interpretation and claim, kept apart.
Says
Interpretationwhere the proportion is
Claima majority support it
Becauseall above 50%
Key idea

What "fairly sure" means

  • Roughly a 95% confidence interval.
  • About 95% of such intervals catch the truth.
  • Sampling variation only, not bias.
Worked example

Comparing two towns

  • Two separate samples.
  • Each has its own 1/√n.
  1. A: n = 400, 54% support
  2. B: n = 625, 46% support
  3. A: MOE ≈ 1/√400 = 1/20 = 5%
  4. B: MOE ≈ 1/√625 = 1/25 = 4%
  5. A: 54 − 5 = 49, 54 + 5 = 59
  6. B: 46 − 4 = 42, 46 + 4 = 50
Worked example

The intervals overlap. So what?

  • Town A: 49% to 59%.
  • Town B: 42% to 50%.
  • Overlap reasoning: not accepted.
The intervals overlap. So what?
Worked example

One interval for the difference

  • Average MOE, times 1.5.
  • Then difference ± margin.
  1. A: MOE 5%, B: MOE 4%
  2. average = (5 + 4)/2 = 4.5%
  3. MOE ≈ 1.5 × 4.5 = 6.75%two samples
  4. A − B = 54 − 46 = 8 points
  5. 8 − 6.75 = 1.25
  6. 8 + 6.75 = 14.75
  7. 1.25 to 14.75 points
Worked example

Interpret the difference, then claim

  • Name both populations and the variable.
  • Town A higher by 1.25 to 14.75 points.
  • All above zero: support is higher in A.
Worked example

Overlap versus the difference interval

  • Overlap: two intervals touch.
  • Difference: one gap, clear of zero.
  • Build the difference interval.
OverlapDifference CI
Looks attwo intervalsone difference
Herethey overlap1.25 to 14.75
Saysno difference?Town A is higher
Acceptednoyes
Key idea

Where is zero?

  • All above or all below: claim a difference.
  • Zero inside: no claim either way.
  • That does not prove they are equal.
IntervalClaim
All above zerofirst group higher
All below zerosecond group higher
Zero insideno claim either way
Key idea

Where the marks are

  • Achieved: form the interval.
  • Merit: an interpretation or a claim.
  • Excellence: difference, interpret, claim.
Summary

What to take away

  • Form it, interpret it in context, then claim separately.
Intervalestimate ± MOE
Interpretpopulation, variable
Parameterthe proportion
Thena separate claim
Comparedifference interval
Neveroverlap reasoning
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Pick the rule, then write the interval.
  • Interpret, then claim.
GivenFind
1n = 2500, 62% say yesinterpret
2same poll: 48% v 40%A − B
3625: 52%, 2500: 48%a claim?
4A 46%, B 53%, 625 eachdiffer?
Show answer

1. MOE 1/√2500 = 2%; 60% to 64%. I am fairly sure the proportion of all residents of the region who support the pool is between 60% and 64%. 2. Same sample: 2/√2500 = 4%; difference 8: 4 to 12 points. All above zero, so site A is preferred. 3. Two samples: 1.5 × (4% + 2%)/2 = 4.5%; difference 4: −0.5 to 8.5 points. Zero is inside, so no claim of a difference between teenagers and adults. 4. Yes. The intervals 42% to 50% and 49% to 57% overlap, but the difference MOE is 1.5 × 4% = 6%, so the difference 7 gives 1 to 13 points, all above zero: support is higher in Town B.

Worked example

The answers: questions 1 and 2

  1. 1. MOE ≈ 1/√2500 = 1/50 = 2%one proportion
  2. 62 − 2 = 60, 62 + 2 = 64
  3. 60% to 64% of all residents
  4. 2. MOE ≈ 2/√2500 = 2/50 = 4%same sample
  5. 48 − 40 = 8 pointsthe difference
  6. 8 − 4 = 4, 8 + 4 = 12
  7. 4 to 12 points: site Aall above zero
Worked example

The answers: questions 3 and 4

  1. 3. 1/√625 = 4%, 1/√2500 = 2%two samples
  2. MOE ≈ 1.5 × 3 = 4.5%average (4 + 2)/2 = 3
  3. 52 − 48 = 4 pointsthe difference
  4. 4 − 4.5 = −0.5, 4 + 4.5 = 8.5
  5. zero inside: no claim
  6. 4. 1/√625 = 4% for each town
  7. MOE ≈ 1.5 × 4 = 6%two samples
  8. 53 − 46 = 7 pointsB − A
  9. 7 − 6 = 1, 7 + 6 = 13all above 0: B higher

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