A political consultant wants to sample residents of a county to estimate what proportion of all the county's residents support a certain candidate. They will use the sample data to construct a one-sample
\[z\] interval for the proportion, and they are considering two sample sizes: a large sample of
\[n=900\] for more accurate results, or a small sample of
\[n=100\] to save time and money.
Assuming the sample proportion is the same in each sample, what is true about the margins of error from these two samples?