Justifying a Claim About Two Means

A confidence interval for mu1 - mu2 is a range of plausible values for the true difference between two population means. Zero plays a special role: it stands for 'no difference at all.' If zero falls inside the interval, no difference is one of the plausible values, so you cannot claim the two means are different. If the whole interval sits on one side of zero, every plausible value says the means differ, and in the same direction, so you can justify the claim.

The write-up follows a pattern: state what the interval is (a range for mu1 - mu2, built from sample data, at some confidence level), check where zero falls relative to it, then connect that back to the original claim in context. An interval of -1.4 to 6.8 contains zero, so you cannot conclude the means differ. An interval of 2.1 to 8.4 excludes zero and lies entirely above it, so you can conclude mu1 is larger than mu2, with the size of the effect somewhere between 2.1 and 8.4 units.

This sits right after building the interval itself, using a t-distribution because the population standard deviations are unknown. The interval does the estimating; 4.8 is the step where you turn that estimate into a verdict on the claim, the same logic used for proportions but now applied to averages and widened by the t rather than z critical value.