4.9 Justifying a Claim Based on a Confidence Interval for a Difference of Two Means

Confidence interval for a difference of means

To compare two population means, construct a two-sample t-interval for μ₁ − μ₂: (x̄₁ − x̄₂) ± t* · SE, where SE = √(s₁²/n₁ + s₂²/n₂). Use df from technology (or the conservative smaller of n₁−1, n₂−1). Conditions: random samples/assignment, independence (within and between groups, 10% condition if sampling without replacement), and normality (large samples, or nearly normal populations/no strong skew/outliers for small samples).

Interpreting the interval

State: 'We are C% confident the interval (a, b) captures the true difference in means, μ₁ − μ₂.' If 0 is not in the interval, this is evidence of a significant difference between the two means — and the sign of the endpoints tells you which group has the larger mean. If 0 is in the interval, there isn't convincing evidence of a difference.

Using the interval to justify a claim

A confidence interval can be used like a two-sided test: if a claimed difference (often 0) falls outside the interval, reject that claim at the corresponding significance level; if it falls inside, the data are consistent with that claim. Always tie the conclusion back to the context of the problem and note that conclusions apply only to the difference μ₁ − μ�2, not individual values.

Key terms

Two-sample t-interval formula

(x̄₁ − x̄₂) ± t* · √(s₁²/n₁ + s₂²/n₂), estimating the true difference μ₁ − μ₂.

Conditions for inference on μ₁ − μ₂

Random samples/assignment, independence (within and between groups), and approximately normal populations or large sample sizes.

Interpreting a confidence interval

'We are C% confident that the interval (a, b) captures the true difference in population means, μ₁ − μ₂.'

Using CI to evaluate a claimed difference

If the claimed value (often 0) lies outside the interval, reject that claim; if inside, the claim is plausible.

0 not in the interval for μ₁ − μ₂

Indicates convincing evidence of a real difference between the two population means, with the sign showing which mean is larger.

Try an AP-style question

A 95% CI for μ₁ − μ₂ is (1.2, 4.8). What does this suggest?

1. No evidence of any difference
2. Convincing evidence that μ₁ is greater than μ₂
3. Convincing evidence that μ₂ is greater than μ₁
4. μ₁ − μ₂ is exactly 3