Sample size for large population
What is the minimum sample size to explore the population of a city?
Does Z score then apply since the size might be larger than 50?
Determining the minimum sample size to explore the population of a city depends on various factors, such as the level of confidence you want in your results, the margin of error you're willing to accept, the variability within the population, and the specific parameter you're trying to estimate (mean, proportion, etc.). There's no one-size-fits-all answer, as the required sample size will depend on these considerations.
The "rule of thumb" that you may use the z-distribution if your sample size is larger than 30 (or 50, as different sources might suggest different thresholds) is an oversimplification. This guideline comes from the Central Limit Theorem, which states that the distribution of the sample mean will be approximately normal if the sample size is sufficiently large, regardless of the shape of the population distribution.
However, this doesn't mean that the population variance is known, and the choice between the z-distribution and the t-distribution isn't simply a matter of sample size. Rather, it's about whether you know the population variance (use the z-distribution) or have to estimate it from the sample (use the t-distribution).
In practice, the distinction between the z- and t-distributions becomes less important as the sample size grows, because the t-distribution approaches the standard normal (z) distribution as the degrees of freedom increase (degrees of freedom is related to the sample size).
I hope this clears your doubts
