# Difficult Survey Sampling question

Question:

A Secretary of State wants to survey the primary owners of motorcycles registered in the state to estimate the proportion who want the license plates redesigned. (Primary owner means that if more than one owner is listed, the person listed first is considered the primary owner.) There are 6,000,000 motorcycles registered. He asks an employee to pick the sample, who then picks a SRS with replacement of 1000 license plate numbers corresponding to the motorcycles. All 1000 license plate numbers are distinct – no duplicates. The employee sends a questionnaire to all of the primary owners for those motorcycles. The response rate is 100%, and 400 of the 1000 respondents report that they favor redesigning the license plates, and the remaining 600 report that they do not favor redesigning the plates. After receiving this data, the employee realizes that some people own multiple motorcycles and could have been included multiple times in the survey. However in fact no owner was selected more than one time.

(a) Is the sample proportion (40%, or 400/1000) an unbiased estimator of the proportion of primary motorcycle owners who want the license plates redesigned? Why or why not?

(b) The employee finds out how many motorcycles each survey respondent has registered in the state, and prepares the following tabulation.

\begin{array}{} No.of motorcycles Owned & No. of Respondents & No.Favoring Redesign& Proportion Favoring \\ \hline 1 & 640 & 240 & 0.375\\ 2 & 200 & 90 & 0.450\\ 3 & 100 & 40 & 0.400\\ 4 & 50 & 25 & 0.500\\ 5 & 10 & 5 & 0.500\\ 6.or.more & 0 \\ Total & 1000 & 400 & 0.400 \end{array}

Given all of this information, estimate the total number of primary motorcycle owners who want the license plates redesigned.

For (b), one might think that the department could determine the total number of motorcycle owners with exactly one registered motorcycle each, and multiply the estimated fraction of those owners who favor redesign by the total number, thereby estimating the total number of one-motorcycle owners who favor redesign; and similarly for two-motorcycle owners, three-motorcycle owners, and so forth. But since that information is not available to us here, we might be able to estimate it from the number of responses from each class of owner--if we can correct for the biases in the number of owners of each class selected. (I'm fairly sure the unbiased estimate of the number of one-motorcycle owners is greater than $64\%$ of the population.)
Above all, however, I'd be alarmed by the extreme unlikelihood of selected $1000$ distinct owners from this pool merely by chance, given the selection method. It seems to me very likely that someone conducting this survey has already made a gross error or has substantially incorrectly described what they were doing. But I suppose that concern is outside the scope of this question.
Presumably, the opinion $R$ of someone doesn't change when their asked multiple times, but it may depend on the number of motorcycles owned $O$. If so, then the expected value of your sample proportion $p$ is $\sum P(O=i)E(X|O=i)$. Therefore, your estimator is biased if there is substantial stratification of opinions based on number of motorcycles owned. In your case, I'd say it looks like you'd have a biased sample, since even if you only have one response from each primary owner, the owners who own more motorcycles are more likely to even be included in the survey. Also, the data indicate that the number of motorcycles owned is a predictor of opinion.