Question

Which is shorter, a 95% z-interval for the mean or a 95% t-interval for the mean? Is one of these always shorter, or does the outcome depend on the sample?

Answer 1

Confidence Intervals for $\mu$

For $n\geq30$

$$\bar{X}\pm z\times\dfrac{s}{\sqrt{n}}$$

Use the Z table for the standard normal distribution.

For $n<30$

$$\bar{X}\pm t\times\dfrac{s}{\sqrt{n}}$$

Use the t table with $df=n-1$

$\alpha=0.05$

$z=1.96$

$df=1, t=12.706$

$df=10, t=2.228$

$df=20, t=2.086$

$df=30, t=2.042$

t-value is greater than z-value.

A 95% z-interval for the mean is always shorter than  a 95% t-interval for the mean.