What is the main difference between a z-test and a t-test?

• A. Z-test uses known population variance and large samples.
• B. Z-test uses sample variance and small samples.
• C. Z-test uses known population variance and large samples; t-test uses sample variance and is appropriate for small samples.
• D. Z-test and t-test are identical.

Answer: (C)

The key idea is how much we know about the population variance and how that uncertainty is handled in the test. A z-test relies on knowing the true population variance, sigma^2, so you standardize the difference between the sample mean and the hypothesized mean with sigma/sqrt(n) and compare to the standard normal distribution. This approach presumes sigma^2 is known and works best when the sample is large enough that the sampling distribution of the mean is well approximated by normality. The t-test does not assume knowledge of the population variance. Instead, you estimate the variance from the sample (using s^2) and use the t distribution with n-1 degrees of freedom to account for the extra uncertainty in that estimate. Because the t distribution has heavier tails, it gives more conservative results for smaller samples. As sample size grows, the t distribution approaches the normal distribution, so the two tests yield similar conclusions. So the main difference is known population variance with large samples for the z-test versus estimating variance from the sample and using the t distribution, especially for small samples.