What is the difference between a z-interval and a t-interval?

What is the difference between a z-interval and a t-interval?

• A. Z uses known σ; T uses sample s with df = n−1.
• B. Z uses sample s; T uses known σ.
• C. Z and T are identical in all cases.
• D. T uses population standard deviation; Z uses sample standard deviation.

Answer: (A)

The main idea is that the two intervals are built from different knowledge about variability. If the population standard deviation is known (σ), you use the z-interval, which centers on the sample mean and uses σ/√n with a z critical value. If σ is unknown, you estimate it with the sample standard deviation (s) and switch to the t-distribution with n−1 degrees of freedom, giving the margin of error as t_{α/2, n−1} * (s/√n). The t-interval accounts for extra uncertainty from estimating σ with s, so it’s typically wider, especially for small samples; as n grows, the t distribution approaches the normal, and the two intervals become very similar.

The main idea is that the two intervals are built from different knowledge about variability. If the population standard deviation is known (σ), you use the z-interval, which centers on the sample mean and uses σ/√n with a z critical value. If σ is unknown, you estimate it with the sample standard deviation (s) and switch to the t-distribution with n−1 degrees of freedom, giving the margin of error as t_{α/2, n−1} * (s/√n). The t-interval accounts for extra uncertainty from estimating σ with s, so it’s typically wider, especially for small samples; as n grows, the t distribution approaches the normal, and the two intervals become very similar.