Confidence intervals: study guide

Key ideas

Inference draws conclusions about a population from a sample. A confidence interval gives a range of plausible values, and a test asks whether a sample result is far enough from a claimed value to call it unlikely.

• A confidence interval for a mean is x̄ ± z × σ ÷ √n, with z = 1.96 for 95 % confidence and 2.58 for 99 %. The half-width is the margin of error.
• To reach a margin of error E, the sample size must be n = (z × σ ÷ E)², rounded up.
• The test statistic for a mean is z = (x̄ − μ₀) ÷ (σ ÷ √n). A value beyond ±1.96 is significant at the 5 % level.
• For a proportion, the margin is z × √(p̂(1 − p̂) ÷ n). Larger samples give narrower intervals.

Key equations

• CI = x̄ ± z × σ ÷ √n
• n = (z × σ ÷ E)²
• z = (x̄ − μ₀) ÷ (σ ÷ √n)
• margin for a proportion = z × √(p̂(1 − p̂) ÷ n)

Worked example

A sample of 100 has a mean of 52 and σ = 10, so SE = 1 and the 95 % interval is 52 ± 1.96, or 50.04 to 53.96. To cut the margin to 2 with σ = 10, n = (1.96 × 10 ÷ 2)² = 96.04, so 97 are needed. Testing against μ₀ = 50 gives z = (52 − 50) ÷ 1 = 2.0, which is beyond 1.96, so the result is significant. For p̂ = 0.4 with n = 400 the margin is 1.96 × √(0.24 ÷ 400) = 0.048.

Common mistakes

• Interpreting a 95 % interval as meaning that 95 % of the data lie inside it. It is about the plausible values of the mean.
• Rounding the sample size down. It must be rounded up to the next whole number.
• Using the standard deviation in place of the standard error, σ ÷ √n, for the interval.

Practice

Test yourself with practice questions on this topic: Confidence intervals