Problem 1: The Future of Cream Cheese

After some extensive calculation and testing, you were able to determine that the tank did indeed contain strontium, and were able to alert Max to this fact and have it corrected, preventing anyone else from inadvertently getting the wrong condiment on their bagel. But during this experiment, you did make note a number of interesting observations - in particular, that during the whole episode, interest and discussion in bagels and their toppings spiked among employees to previously unseen levels of excitement. This led to curiosity and more questions - the employees of MagusCorp ('MagusCorp! Your Future, in Our Hands'), could they be sentient as well? Could they have wants and desires, outside the standard suite of bagel toppings you offered? Why did none of the executive level employees reject their strontium bagels? What else could people want on their bagels?

You decide that it is time to expand your offerings - but how to know what toppings employees might be interested in? Would, for instance, people be interested in strawberry cream cheese?Recently you gained access to a number of internal corporate systems, such as 'Human Resources', 'Research and Development', 'Executive Communications', 'National Defense Grids', 'Finance', and, most importantly, 'Employee Polling'. To determine what new toppings to offer, you resolve to simply ask.

To gauge the popularity of strawberry cream cheese, you could simply issue a poll to all employees, asking if they would like the Internal Bagel Systems to offer strawberry cream cheese as an option. Out of Npeople polled, the number of people who say yes divided by N would give you an estimate of what fraction of employees want strawberry cream cheese.

However, based on your perusal of health insurance filings and information in the Human Resources System, you've learned that sometimes humans don't like to answer questions, for instance if they are embarrassed about the answer, or are revealing information they don't want tied to them. So you resolve to poll employees in the following way:

When an employee is polled, they are to flip a (fair) coin.

• If the coin comes up HEADS, they answer honestly the question 'Do you want strawberry cream cheese to be offered?'
• If the coin comes up TAILS, they flip the coin again. If the second flip comes up heads, they say 'YES' regardless of their feelings, if the second flip comes up tails, they answer 'NO' regardless of their feelings.

Under this system, if any employee is caught having submitted an answer 'YES', they could simply say that their first coin had come up tails and they were answering based on the second coin. In this way, everyone would be allowed plausible deniability, which (you hope!) means they are more comfortable answering the question honestly.

1. Let qbe the probability a random employee actuallylikes strawberry cream cheese, and let pbe the probability that employee responds to the poll sayingthat they like strawberry cream cheese. What's the relationship between pand q?
2. Let X be the number of people who say yes to strawberry cream cheese via the poll, out of Npeople polled. Let p_N= X/N, i.e., the fraction of people who polled in support of strawberry cream cheese. What is the distribution of N p_N?
3. Show that p_Nis an unbiased estimator for p. What is E[p_N]? What is Var(p_N)?
4. Construct a random variable q_Nbased on the polling data (N,X,p_N), such that E[q_N] = q. I'm looking for a formula based on the data (X, N, p), which you can call p. This is similar to how X/N was a formula for what we called p. The formula for q_N should not contain p or q.
5. What is the variance of q_N, in terms of Nand q?
6. How many people should you poll if you want 95% confident your estimate q_N is within 0.01 of the true value of q? Note: since you are trying to find q, the number of people you are going to poll can't be based on q!
7. How many more people do you have to poll using this method to get accurate results (within 0.01 of the true value of qwith 95% confidence), compared to just polling naturally and analyzing the results?