The figure illustrates two sampling distributions for sample proportions when the population proportion p = 0.50.
a. Find the standard deviation for the sampling distribution of the sample proportion with (i) n = 100 and (ii) n = 1000.
b. Explain why the sample proportion would be very likely (as the figure suggests) to fall (i) between 0.35 and 0.65 when n = 100, and (ii) between 0.45 and 0.55 when n = 1000. (Recall that for an approximately normal distribution, nearly the entire distribution is within 3 standard deviations of the mean.)
c. Explain how the results in part b indicate that the sample proportion tends to more precisely estimate the population proportion when the sample size is larger.