A survey was carried out in which those sampled were asked to circle either NO or YES on a paper questionnaire, according to the following instructions: in private, spin a "wheel of fortune" which is equally likely to end up pointing to any integer from 1 to 100. If the result is a 40 or less, respond to Q1, otherwise respond to Q2.

Q1 Flip a fair coin (again in complete privacy) --- do you get HEADS?
Q2 Have you ever shoplifted?

Of the 500 people randomly sampled from the target population, respond YES. Provide answers to the following to two decimal places.

(a) Give an estimate for the proportion of this population that has ever shoplifted.

(b) Give a standard error for your estimate in (a).

Now, you and a colleague plan to replicate your study in a new population. Your colleague proposes replacing the '40' in the instructions with '20.' You see his point in terms of trying to get a more precise estimate. However, you think that '20' is really "pushing it." In particular, you posit that with '40,' everyone in the population would comply perfectly with the instructions. But with '20,' you guess that 30% of shoplifters would lie if asked Q2. (Whereas no matter what, you expect that non-shoplifters would always answer Q2 honestly.)

(c) If your suppositions happen to be correct, how much bias would the estimation procedure have if '20' is used? To make this question more specific, say the proportion you are trying to estimate --- the proportion of the (new) population that have shoplifted --- happens to be . Then at what value would the sampling distribution of the estimator be centered?