As stated in the online notes holding everything else equal

As stated in the online notes, holding everything else equal, a four fold increase in sample size alone (as when the sample size increases from 100 to 400) will cut the margin of error in half. Remembering that the margin of error = z\'SE (the margin of error is the distance on either side of the sample value that our confidence interval covers). What is the reason why the margin of error is cut in half?

Solution

the reason that the margin of error is cut in half is because when you have a sample size more bigger

you are reducing the error because the sample is increasing

so the sample will be more similar to the population

that is the reason