A political poll estimates the proportion of voters favoring a measure using a random sample of $100$ respondents. The polling firm suggests increasing the sample size to $400$ respondents. Will this change improve the precision of the estimated proportion? Choose the best answer.

Yes, the standard error will become $\frac{1}{2}$ times the original, so the estimate is more precise

Yes, the standard error will become $2$ times the original, so the estimate is less precise

No, the standard error will become $\frac{\sqrt{2}}{2}$ times the original, which is essentially unchanged

No, sample size has no effect on the standard error for proportions