TA Hayden and TA Jiayi were deciding who has to mark an assignment first, and chose to flip a coin to decide. They suspect the coin they are using is biased. They decide to test the coin to make inference about the true proportion of heads.

(a) If they want the estimate of the population proportion to be correct within 0.05 with 95% confidence, how many times should they flip the coin, assuming p = 0.5?

(b) Suppose Hayden and Jiayi decide to flip the coin 121 times and observed 87 heads among the 121 results. What is the resulting 95% confidence interval for the true proportion of heads? As always, make sure to state all necessary assumptions and conditions for the confidence interval to get full marks.

(c) After taking the sample and calculating the confidence interval above, Jiayi makes the following claim: "There is a 95% chance that the true proportion is within this confidence interval." Explain what is wrong with this claim and what the 95% confidence interval actually means in this context.

(d) Hayden is not satisfied with a 95% confidence interval, and wants a higher confidence level. What will happen to the confidence interval if they use a higher confidence level than 95%?

(e) Hayden wants to be 100% confident about the true proportion of heads. He proposes making a 100% confidence interval for this proportion. What would a 100% confidence interval for the true proportion heads be? Why is making a 100% confidence interval a bad idea?