myLearning

Step 17 · Statistics and probability topics

Estimate a population mean or proportion with a confidence interval, see what controls its width, and say correctly what the confidence level means.

Building an interval

A confidence interval is an estimate plus or minus a margin of error. The margin is a critical value times the standard error. For 95% confidence the critical z value is 1.96.

xˉ±z∗σnp^±z∗p^(1−p^)n\bar{x} \pm z^{*}\frac{\sigma}{\sqrt{n}} \qquad \hat{p} \pm z^{*}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}

Example: A mean

A sample of 64 bolts has mean length 50.2 mm, and σ = 0.8 mm. Find a 95% confidence interval for the mean length.

  1. The standard error is 0.8 ÷ 8 = 0.1.
  2. The margin is 1.96 × 0.1 = 0.196.
  3. Take 50.2 plus and minus 0.196.

AnswerAbout 50.00 to 50.40 mm

Example: A proportion

In a sample of 400 adults, 240 say yes. Find a 95% confidence interval for the share who would say yes.

  1. p̂ = 240 ÷ 400 = 0.6.
  2. SE = √(0.6 × 0.4 ÷ 400), about 0.0245.
  3. The margin is 1.96 times that, about 0.048.

AnswerAbout 0.552 to 0.648

Confidence90%95%99%
z*1.6451.962.576

Margin of error

The margin of error shrinks as the sample grows, but slowly, because it depends on √n. It grows with the confidence level: a surer net must be wider.

Example: Choosing n

How many people must you survey for a 95% margin of error of 3 percentage points, whatever the true share?

  1. The margin is largest when p̂ = 0.5.
  2. Solve 1.96 × √(0.25 ÷ n) = 0.03: √n is about 32.7.
  3. So n is about 1,067.1.

AnswerAbout 1,068 people: round up

Example: Four times the sample

A margin of error is 4 points with n = 500. What is it with n = 2,000?

  1. n is 4 times as big, so √n is 2 times as big.
  2. The margin halves.

Answer2 points

Interpreting confidence

95% confidence describes the method, not one interval. If you took many samples and built an interval from each, about 95% of those intervals would contain the true value. Any one interval either contains it or does not.

Example: Saying it right

A 95% interval for mean nightly sleep is 6.6 to 7.2 hours. What does it mean?

  1. Wrong: a 95% chance that the true mean is in this interval. The true mean is fixed: it is either inside or not.
  2. Wrong: 95% of people sleep between 6.6 and 7.2 hours. The interval is about the mean, not about individuals.
  3. Right: the interval came from a method that captures the true mean 95% of the time.

AnswerWe are 95% confident the true mean sleep is between 6.6 and 7.2 hours

Try it yourself

Work each one on paper first, then open the answer to check.

  1. x̄ = 20, σ = 5 and n = 25. Find the 95% confidence interval for μ.

    Show answer

    18.04 to 21.96

  2. p̂ = 0.5 and n = 100. What is the 95% margin of error?

    Show answer

    About 0.098

  3. Which is wider, a 90% or a 99% interval from the same data?

    Show answer

    The 99% interval

  4. A margin of error is 6 points with n = 200. About what is it with n = 800?

    Show answer

    3 points

  5. A 95% interval for a proportion is 0.42 to 0.50. What was p̂?

    Show answer

    0.46

  6. What is the margin of error of that interval?

    Show answer

    0.04

What next?