When should you use a z-test versus a t-test for a mean comparison?

When should you use a z-test versus a t-test for a mean comparison?

• A. Use a t-test when σ is known and n is large; use a z-test when σ is unknown and estimated from the sample.
• B. Use a z-test when σ is known and n is large; use a t-test when σ is unknown and estimated from the sample.
• C. Use a z-test only for proportions, and a t-test only for means.
• D. Use a t-test when n > 100 regardless of σ.

Answer: (B)

The key idea is whether you know the population standard deviation. If sigma is known, the distribution of the sample mean is normal with standard error sigma divided by the square root of n, so you can use a z-test to assess a mean. If sigma is unknown, you estimate it from the data with s. Replacing sigma with s adds extra uncertainty, and the appropriate reference distribution becomes the t distribution with df = n − 1, which accounts for that extra variability. So, the recommended approach is to use a z-test when sigma is known and you have a large enough sample for the normal approximation to hold, and to use a t-test when sigma is unknown and is estimated from the sample. The other statements aren’t correct: z-test is not limited to proportions and can be used for means when sigma is known; the idea that you should always use a t-test merely because n is large, or that n > 100 dictates t-test regardless of sigma, isn’t accurate.

The key idea is whether you know the population standard deviation. If sigma is known, the distribution of the sample mean is normal with standard error sigma divided by the square root of n, so you can use a z-test to assess a mean. If sigma is unknown, you estimate it from the data with s. Replacing sigma with s adds extra uncertainty, and the appropriate reference distribution becomes the t distribution with df = n − 1, which accounts for that extra variability.

So, the recommended approach is to use a z-test when sigma is known and you have a large enough sample for the normal approximation to hold, and to use a t-test when sigma is unknown and is estimated from the sample. The other statements aren’t correct: z-test is not limited to proportions and can be used for means when sigma is known; the idea that you should always use a t-test merely because n is large, or that n > 100 dictates t-test regardless of sigma, isn’t accurate.