Module 06 · Confidence intervals
Confidence intervals with t.
When the population standard deviation is unknown, estimate it with the sample standard deviation s. The t distribution accounts for this extra uncertainty.
See the interval take shape
Estimated spread · sInterval-construction illustration: a t reference curve centered at x̄, with scale set by s / √n. This is not a probability distribution for the unknown population mean μ.
The 95 percent t confidence interval is 45.87 to 54.13, with standard error 2, critical value 2.064, and 24 degrees of freedom.
The idea
The t distribution has heavier tails than the standard normal. With n − 1 degrees of freedom, its critical value accounts for estimating the population spread from the sample.
As n grows, t* approaches z*. For the same numerical spread, sample size, and confidence level, the t interval is wider than the z interval.
TRY THIS Start with a small sample
Set n to 5 and look at the critical value. Then increase n toward 200. Notice how the interval narrows and t* approaches the normal critical value of about 1.96 at 95% confidence.
Assumptions, notation & reference
Use an independent random sample. The population standard deviation σ is unknown and is estimated by the sample standard deviation s. The t interval is exact for normal populations. With small samples, strong skewness or outliers can make the method unreliable.
Let C be the confidence level as a proportion and α = 1 − C. The positive critical value is t* = F⁻¹(1 − α/2) for a t distribution with n − 1 degrees of freedom. The estimated standard error is s / √n. Tail labels refer to the entire tails, including areas beyond the displayed range. The plot height is normalized for comparison.
At 95% confidence, the interval procedure contains the fixed population mean in about 95% of repeated samples under its assumptions. This is not a 95% probability statement about μ after this particular interval has been observed.
Reference: NIST / SEMATECH e-Handbook: Confidence limits for the mean ↗