In repeating confidence interval experiments, are we allowed to take samples of different size every time? Because a confidence interval of 95% means that if the sampling process is repeated infinite times, 95% of all the intervals obtained will have the parameter of interest.
So wrt this process of repeating infinite times - do we have to repeat with the same sample size or can we vary it?
As it is always the case, the answer is: it depends. Let me elaborate. Suppose that $X_1,...,X_n$ are iid $N(\mu,\sigma^2)$. Then, one can show that $$ \sqrt{n-1}\dfrac{\overline{X}_n-\mu}{\hat{\sigma}}\sim T_{n-1},\quad\forall n\geq 2. $$ where $\overline{X}_n=\frac{1}{n}\sum_{i=1}^nX_i$, $\hat{\sigma}^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X}_n)$ and $T_{n-1}$ has a Students T distribution with $n-1$ degrees of freedom. This say that, for any sample size, the confidence interval $$ \overline{X}_n\pm t_{(n-1),\alpha/2}\dfrac{\hat{\sigma}}{\sqrt{n-1}}, $$ where $P(T_{n-1}\geq t_{(n-1),\alpha/2})=\alpha/2$, will cover $\mu$ with probability $1-\alpha$ (see "In all likelihood" by Yudi Pawitan). So $n$ is not important because $X_i\sim N(\mu,\sigma^2)$. (Of course it is important for calculating each term, but the main thing is the probability of covering $\mu$).
On the other hand, if you use an asymptotic interval as Walds confidence interval: $$ \hat{\theta}_n\pm z_{\alpha/2}\sqrt{I(\hat{\theta}_n)}, $$ where $P(N(0,1)\geq z_{\alpha/2})=\alpha/2$, $\hat{\theta}$ is the maximum likelihood estimator and $I(\theta)$ is the Fisher information, then $n$ does matter, since the level of approximation to the normal is improved with $n$. (See again the book of Pawitan and the Beery-Essen theorem in "Approximation theorems of mathematical statistics" by Robert Serfling). In this case, if you use different $n$'s you will see better or worse intervals depending on $n$.