4.8 Confidence Intervals for the Difference Between Two Means

Setting up the two-sample t-interval

To estimate the true difference between two population means, use a two-sample t-interval: (x̄₁ - x̄₂) ± t* · SE, where SE = √(s₁²/n₁ + s₂²/n₂). Check conditions first: random samples/assignment, independent samples (and independence within each), and Normal/Large Counts (each population approximately Normal or n₁, n₂ ≥ 30, or graphs show no strong skew/outliers).

Degrees of freedom and t*

Calculators use a complex Welch-Satterthwaite formula for degrees of freedom; by hand, use the smaller of n₁-1 and n₂-1 as a conservative estimate. Look up t* from the t-distribution using the confidence level and df, then multiply by SE to get the margin of error.

Interpreting the interval

Interpret as: 'We are C% confident that the true difference in means (μ₁ - μ₂) is between [lower, upper].' If the interval contains 0, there's no convincing evidence of a difference between the two population means. If it doesn't contain 0, the sign of the interval tells you which group has the larger mean.

Key terms

Two-sample t-interval formula

(x̄₁ - x̄₂) ± t* · √(s₁²/n₁ + s₂²/n₂)

Conditions for two-sample t-interval

Random, Independent (samples and within samples), and Normal/Large Counts (Normal populations or n ≥ 30, or no strong skew/outliers).

Conservative df estimate

Use the smaller of (n₁ - 1) and (n₂ - 1) when calculating by hand.

Interpreting an interval containing 0

No convincing evidence of a difference between the two population means.

Interpretation template

'We are C% confident that the true difference in means (μ₁ - μ₂) is between [lower] and [upper].'

Try an AP-style question

s₁ = 6 (n₁ = 36) and s₂ = 8 (n₂ = 64). What is the standard error of x̄₁ − x̄₂?

1. 10
2. About 0.25
3. 2
4. About 1.41