Is it true that if you increase the confidence interval then the margin of error of a confidence interval would also increase since larger confidence equals larger interval?
Is it also true that quadrupling the sample size would make the margin of error of a confidence interval smaller?
An increase in the interval means more uncertainty in the estimate as a general feature, so yes larger interval means larger margin of error probabilistically in general.
In a typical application, quadrupling would half the interval if you were going to use something like a t-statistic or a asymptotic Normal statistic.
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