This is based on the principles of sampling and the law of large numbers. When you have a larger sample size 1,600 voters it provides a more reliable estimate of the true population proportion of p-value. The law of large numbers of states if the sample size increases, the sample statistics in this case, has the same proportion which is more likely to get closer to the true population parameter. However, if the larger samples reduce the variability and sampling error, the more likely it produces an estimate that is closer to the true value which in summary the sample proportion form the same of 1,600 voters which is more likely to have a better estimate from the true population with the sample of 400 voters
