According to what I have learned there is no minimum sample size for a t-test. In fact the t-test is suitable for cases where the $n$ sample size is: 3 and more.

<sub>Even $n=2$ would work.</sub>

A paired t-test on observations $\{X_{1i}\}_{i=1}^n$ and $\{X_{2i}\}_{i=1}^n$ **is the same** as a one-sample t test on differences. [*](https://stats.stackexchange.com/a/511216/294316) 

You choose t-test well, since you don't know the $\sigma$ of the population, i.e. z-test would not work for your case unless you somehow interact with the God and get $\sigma$ from it.

In other words, you should do t-test as you do, where t-distribution is a sample distribution behind the sampling.
This distribution further assumes you relay on sample SD $S$ (standard deviation of the sample).

Since $S$ brings uncertainty, unless $n$ is big (where we usually assume $\sigma \approx S$) we decrease the degrees of freedom.

$$
\frac{\bar{x}-\mu}{\sigma / \sqrt{n}} \sim z \longrightarrow \frac{\bar{x}-\mu}{s / \sqrt{n}} \sim t_{\mathrm{n}-1}
$$

Once we calculate the sample mean $\bar X$ we can estimate the confidence interval.

$$
\bar X \pm t \frac{S}{\sqrt{n}}
$$

Where $t$ you can get in R via the 95% confidence interval rule:

```R
t = qt(0.975,df=n-1)
```





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[Ref](https://stats.stackexchange.com/a/511216/294316)