Building a One-Sample t-Interval

A confidence interval for a mean follows the same logic as one for a proportion: take your best estimate, then add and subtract a margin of error that accounts for sampling variability. The estimate is x-bar, the sample mean. The margin of error is a critical value times the standard error of that estimate. Nothing about that structure changes when you move from proportions to means.

What changes is the standard error. For a proportion you know the population spread is tied to p itself, so you can compute it directly. For a mean, the population standard deviation sigma is almost never known, so you estimate it with the sample standard deviation s. The standard error becomes s divided by the square root of n. Because you swapped in an estimate for sigma, the sampling distribution of x-bar is no longer exactly normal. It follows a t-distribution, which has slightly heavier tails to reflect that extra uncertainty. The t-distribution depends on degrees of freedom, df = n minus 1, and as df grows large, t gets very close to the normal curve.

Building the interval has three ingredients: x-bar, the standard error s over √n, and a critical value t·pulled from a t-table using your confidence level and df. Multiply t·by the standard error to get the margin of error, then write x-bar plus or minus that number. This sets up everything that follows in the unit: checking conditions like randomness and either normality or a large enough sample size, and later running the matching hypothesis test with the same t machinery.