When should you use the t-distribution instead of the z-distribution?

When should you use the t-distribution instead of the z-distribution?

• A. When the population standard deviation is unknown and the sample size is small
• B. When the population standard deviation is known
• C. Only for population means exactly known
• D. When data are non-normal and the sample size is large

Answer: (B)

Use the t-distribution whenever you don’t know the population standard deviation and you have to estimate it from your sample, especially when the sample size is small. The t distribution accounts for the extra uncertainty that comes from using the sample standard deviation in place of the true sigma. It’s centered like the normal distribution but has heavier tails, reflecting that added variability. The exact shape depends on degrees of freedom (n minus 1); as the sample size grows, the t distribution looks more like the standard normal, which is why large samples often use z even if sigma isn’t known. So, when sigma is known, you’d use z. When sigma is unknown and n is small (or moderate), you switch to t. The other scenarios—such as needing z for a known sigma or using t only when means are precisely known or data are non-normal with large samples—don’t align with the standard usage of the t distribution.

Use the t-distribution whenever you don’t know the population standard deviation and you have to estimate it from your sample, especially when the sample size is small. The t distribution accounts for the extra uncertainty that comes from using the sample standard deviation in place of the true sigma. It’s centered like the normal distribution but has heavier tails, reflecting that added variability. The exact shape depends on degrees of freedom (n minus 1); as the sample size grows, the t distribution looks more like the standard normal, which is why large samples often use z even if sigma isn’t known.

So, when sigma is known, you’d use z. When sigma is unknown and n is small (or moderate), you switch to t. The other scenarios—such as needing z for a known sigma or using t only when means are precisely known or data are non-normal with large samples—don’t align with the standard usage of the t distribution.