Formal confidence intervals

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What this lesson covers

  • For a mean: x̄ ± 1.96 × s ÷ √n.
  • Bootstrap intervals use the middle 95%.
  • Larger samples give narrower intervals.
Worked example

A mean

  • Mean 170 cm, s = 10, n = 100.
  1. 1.96 × 10 ÷ √100 = 1.96
  2. 170 − 1.96 = 168.04
  3. 170 + 1.96 = 171.96
  4. 168.04 to 171.96 cm
Key idea

How to say it

  • We are fairly sure the population mean is between 168.04 and 171.96 cm.
  • The interval is about the mean, not individual people.

Common mistakes

  • Saying 95% of people lie in the interval.
    It estimates the population mean.
  • Forgetting the square root.
    Divide s by √n.

Your turn

  • 1. Mean 52, s = 12, n = 36. Find the 95% interval.
Show answer

1. 1.96 × 2 = 3.92, so 48.08 to 55.92

Summary

What to take away

  • The interval estimates a population value.
  • Quadruple n to halve the width.

More in AS 91582

Sampling variability and bootstrappingNotesFormal confidence intervalsNotesComparing populationsNotesJustifying an inferenceNotes

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