Confidence Intervals for a Mean

The population standard deviation is almost never known. Estimating it from the sample adds uncertainty, and the t distribution accounts for it.

The t distribution

It is bell-shaped like the normal but with heavier tails, giving wider intervals. As the sample grows it approaches the normal.

Degrees of freedom

For one sample, df = n − 1. Smaller df means heavier tails and a wider interval.

Conditions

Random sampling, independence with the 10% condition, and either a large sample or a roughly normal population.

The interval

The sample mean plus or minus t times s ÷ √n. The form is estimate plus or minus margin of error, as always.

Interpreting it

The confidence level describes the method: about 95% of intervals built this way capture the true mean.

Why t rather than z

When the population standard deviation is unknown and estimated from the sample, the extra uncertainty makes the distribution heavier-tailed. The t distribution accounts for it, and it is used essentially always in practice.

Degrees of freedom

For a one-sample t interval, df = n − 1. One degree is lost because the sample mean was estimated from the data. As n grows, t approaches z, which is why the distinction fades for large samples.

The normality condition

Either the population is roughly normal, or the sample is large enough for the central limit theorem. For small samples the data should be plotted and checked for strong skew or outliers.

Interpret in context

"We are 95% confident that the true mean commute time for all employees is between 22 and 28 minutes." Naming the parameter, the population and the units is what the exam is grading.