Exercise 8. Plotting the standard error

The standard error estimates obtained from the Monte Carlo simulation, the theoretical prediction, and the estimate of the theoretical prediction are all very close, which tells us that the theory is working. This gives us a practical approach to knowing the typical error we will make if we predict \(p\) with \(\hat{X}\). The theoretical result gives us an idea of how large a sample size is required to obtain the precision we need. Earlier we learned that the largest standard errors occur for \(p=0.5\).

Create a plot of the largest standard error for \(N\) ranging from 100 to 5,000. Based on this plot, how large does the sample size have to be to have a standard error of about 1%?

N <- seq(100, 5000, len = 100)
p <- 0.5
se <- sqrt(p*(1-p)/N)

Possible Answers

100
500
2,500
4,000