Sample size in Confidence Intervals. 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?

obojeneqk 2022-09-11 Answered
Sample size in Confidence Intervals
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?
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Answers (1)

Bordenauaa
Answered 2022-09-12 Author has 18 answers
Step 1
As it is always the case, the answer is: it depends. Let me elaborate. Suppose that X 1 , . . . , X n are iid N ( μ , σ 2 ). Then, one can show that
n 1 X ¯ n μ σ ^ T n 1 , n 2.
where X ¯ n = 1 n i = 1 n X i , σ ^ 2 = 1 n 1 i = 1 n ( X i 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
X ¯ n ± t ( n 1 ) , α / 2 σ ^ n 1 ,
where P ( T n 1 t ( n 1 ) , α / 2 ) = α / 2, will cover μ with probability 1 α (see "In all likelihood" by Yudi Pawitan). So n is not important because X i N ( μ , σ 2 ). (Of course it is important for calculating each term, but the main thing is the probability of covering μ).
Step 2
On the other hand, if you use an asymptotic interval as Walds confidence interval:
θ ^ n ± z α / 2 I ( θ ^ n ) ,
where P ( N ( 0 , 1 ) z α / 2 ) = α / 2, θ ^ is the maximum likelihood estimator and I ( θ ) 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.
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