# 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:

\, \begin{align*} r = P(yes) \\ &= P(yes \,\vert\, \text{Q1 was asked})P(\text{Q1 was asked}) + P(yes \,\vert\, \text{Q2 was asked})P(\text{Q2 was asked}) \\ &= qp + t(1-p) \\ \end{align*} \,

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)

\, \begin{align*} \Exp Q \\ &= \Exp\Prn{\frac {R-t(1-p)} p} \\ &= \frac 1 {p^2} \Exp(R-t(1-p)) \\ &= \frac 1 {p^2} (\Exp(R)-\Exp(t(1-p))) \\ &= \frac 1 {p^2} (r-t(1-p)) \\ &= q \end{align*} \,

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

#### (c)

\, \begin{align*} \Var(Q) \\ &= \Var(\frac {R-t(1-p)} p) \\ &= \frac 1 {p^2} \Var(R-t(1-p)) \\ &= \frac 1 {p^2} \Var(R) && \text{t and p are constants} \\ &= \frac {r(1-r)} {np^2} && \text{Using Problem 28, part-d} \end{align*} \,
$$\tag*{\blacksquare}$$