What is the difference between a t-distribution and a Z-distribution?

What is the difference between a t-distribution and a Z-distribution?

• A. The t-distribution has heavier tails and depends on degrees of freedom; Z is standard normal with known variance.
• B. The Z-distribution has heavier tails than the t-distribution for large samples.
• C. They are identical for all sample sizes.
• D. The Z-distribution changes with sample size.

Answer: (A)

The key idea is how variance is treated and how sample size affects the distribution. If the population variance is known, standardizing a statistic leads to a standard normal distribution. When the population variance is unknown and you estimate it from the sample, the statistic follows a t distribution with degrees of freedom typically equal to n−1. The t distribution has heavier tails than the standard normal, meaning more probability in the extremes to reflect the extra uncertainty from estimating the variance. As the sample size grows, that uncertainty diminishes and the t distribution converges to the standard normal. The Z distribution, on the other hand, is fixed: a standard normal with mean 0 and variance 1, not changing with sample size. So the description in terms of heavier tails and dependence on degrees of freedom for the t, versus a standard normal with known variance for Z, best captures the difference.

The key idea is how variance is treated and how sample size affects the distribution. If the population variance is known, standardizing a statistic leads to a standard normal distribution. When the population variance is unknown and you estimate it from the sample, the statistic follows a t distribution with degrees of freedom typically equal to n−1. The t distribution has heavier tails than the standard normal, meaning more probability in the extremes to reflect the extra uncertainty from estimating the variance. As the sample size grows, that uncertainty diminishes and the t distribution converges to the standard normal. The Z distribution, on the other hand, is fixed: a standard normal with mean 0 and variance 1, not changing with sample size. So the description in terms of heavier tails and dependence on degrees of freedom for the t, versus a standard normal with known variance for Z, best captures the difference.