Confidence intervals and sample size.Are confidence intervals wider if sample size is larger? Similarly are confidence intervals narrower if sample size is smaller?

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elilsonoulp2l · Accepted Answer

Step 1Here is a specific example. Suppose you have n independent observations       X    i   from   N  (  μ  ,      σ    0    2    )  ,, where you want a 95% confidence interval (CI) for unknown   μ. Also suppose       σ    0    2   is known, and denote the mean of the observations as             X      ¯        ..Then the confidence interval is             X      ¯        ±  1.96      σ    0        /        n    . The &#039;margin of error&#039; of this CI is   1.96      σ    0        /        n   and its total width is twice the margin of error.Step 2For simplicity, suppose       σ    0    =  1.. First, let   n  =  4.. Then the margin of error is   1.96      /    2  ≈  1. But if we increase the sample size to   n  =  16, then the margin of error is   1.96      /    4  ≈  1      /    2..For some situations the math is a little less obvious, but this same relationship holds in almost all practical situations. Very roughly, the idea is that more data gives more information and that means we can more precisely estimate   μ  .. The smaller the margin of error the more precise the estimate.

Confidence intervals and sample size. Are confidence intervals wider if sample size is larger? Similarly are confidence intervals narrower if sample size is smaller?

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