A Gallup poll from January 2018 found that only 39% of adult Americans planned on watching a great deal/fair amount of the Winter Olympic Games. This information can be used to estimate the population proportion of adult Americans who plan to watch the Winter Olympics. 

a. What additional information do you need in order to determine a 95% confidence interval for the probability that a randomly selected American adult plans to watch a great deal of the Winter Olympics? 

b. Consider three possible sample sizes for this study: 250, 1,000, and 4,000. Without performing any calculations, how do you expect the three confidence intervals based on these sample sizes to compare to each other? Explain your reasoning. 

c. Carry out the simulation at each sample size to get the standard deviation of the distribution of sample proportions and use the 2SD method to approximate the margin of error with each of these three sample sizes. Then produce the 95% confidence interval for the each of these three sample sizes. 

d. How do the midpoints of these intervals compare? 

e. How do the widths of these intervals compare? 

f. By how many times must the sample size increase in order to cut the margin of error in half? 

g. The sample size was really 2,228. Calculate the 95% confidence interval for the population proportion of adult Americans who plan to watch a great deal of the Winter Olympics. Does this interval suggest that fewer than half of adult Americans will watch a great deal of the Winter Olympics?