# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### (a)

Let $\, t \,$ denote the probability of answerting yes if Question 2 was asked. Similar to the problem-28, we have:

Thus we can estimate r, by $\, R = Qp + t(1-p) \,$(Note that $\, t \,$ and $\, p \,$ and constant for an experiment. Thus $\, Q = \frac {R-t(1-p)} p \,$.

#### (b)

Thus $\, \Exp(Q) = q \,$. It follows that $\, Q \,$ is unbiased in estimating $\, q \,$.

#### (c)

$$\tag*{\blacksquare}$$