confidence interval interpretation for new parameters In the book "all of statistics" the author brings the following interpretation of confidence intervals with the following exmaple:
Interpretation:
"On day 1, you collect data and construct a 95 percent confidence interval for a parameter $\theta_1$. On day 2, you collect new data and construct a 95 percent confidence interval for an unrelated parameter $\theta_2$ . On day 3, you collect new data and construct a 95 percent confidence interval for an unrelated parameter $\theta_3$ . You continue this way constructing confidence intervals for a sequence of unrelated parameters $\theta_1$ ,$\theta_2$ , . . . Then 95 percent of your intervals will trap the true parameter value. There is no need to introduce the idea of repeating the same experiment over and over."
Example:
"Every day, newspapers report opinion polls. For example, they might say that "83 percent of the population favour arming pilots with guns." Usually, you will see a statement like "this poll is accurate to within 4 points.95 percent of the time." They are saying that 83 ± 4 is a 95 percent confidence interval for the true but unknown proportion p of people who favour arming pilots with guns. If you form a confidence interval this way every day for the rest of your life, 95 percent of your intervals will contain the true parameter. This is true even though you are estimating a different quantity (a different poll question) every day."
can someone explain why this is true? Why can we know 95 percent of the intervals contains the true parameter even though we are estimating a different quantity?
 A: Of course we need to assume that the model assumptions always hold.
Now consider different $\beta$-level (say $\beta=0.95$) confidence intervals $C_1,\ldots,C_n$ for parameters $\theta_1,\ldots,\theta_n$. I also assume that the confidence level is precise for all intervals, i.e., $P\{\theta_i\in C_i\}=\beta,\ i=1,\ldots,n$, and that data for all the different intervals are independent.
I can then define binary (0-1) random variables $I_1,\ldots,I_n$ so that $I_i=1$ if $\theta_i\in C_i$, i.e., the confidence interval $i$ catches true parameter $\theta_i$.
This defines an independent sequence of Bernoulli-experiments with probability $\beta$ for observing 1. The relative frequency of successes of the confidence interval then behaves as the standard estimator for a probability $p$ (here equal to $\beta$) in such an experiment, which is the relative frequency of observing 1 (equal to the arithmetic mean of the zeroes and ones). The Laws of Large Numbers then state that this will converge (in probabilistic terms) toward $\beta$ (it will fluctuate around $\beta$ with ever smaller variance for growing $n$).
