4.3 Additional consideration

Traditionally, decision-makers would choose between z-tests and t-tests based on two main factors: a) Whether the population standard deviation was known (z-test) or unknown (t-test), and b) The size of the sample (when n < 30, they would choose to use t-test). This approach was rooted in the different properties of the normal distribution and t-distribution, particularly for smaller sample sizes.

However, modern statistical practice often favors using t-tests regardless of sample size when the population standard deviation is unknown. This shift occurred because as sample sizes increase, the t-distribution closely resembles the normal distribution (z-distribution), as shown in Figure 2. For small samples, the t-test accounts for the extra uncertainty from estimating the population standard deviation. For large samples, it gives nearly identical results to a z-test. This consistent use of t-tests simplifies statistical analysis, eliminating the need to decide when to switch from a t-test to a z-test based on sample size. It also provides a slightly more conservative estimate for smaller samples, reducing the risk of false positive results. With today’s statistical software easily handling t-tests for any sample size, this approach combines mathematical accuracy with practical simplicity across various research scenarios, moving beyond the traditional "sample size of 30" rule.

Figure 6 Z-distribution vs T-distribution with different sample size