### Chapter 7, Survey Sampling

#### Solution 18

Denoting the random variables from the two surveys as $\, X \,$ and $\, Y \,$. Also denoting the the confidence intervals by $\, C_{X} \,$ and $\, C_{Y} \,$ respectively. Denoting the probability that the condidence interval contains the respective population mean by $\, P_X \,$ and $\, P_Y \,$.

Since the surveys are independent, it means that the random variables, say $\, X \text{ and } Y \,$ will also be independent. Thus condidence intervals are also independent. It follows that the probability of both of them containing the respective population mean:

$P_X \times P_Y = \, 0.9 \times 0.9 = 0.81 \,$.

And the probability that both of them do not contain the respective population mean is $\, (1-0.9) \times (1-0.9) = 0.01 \,$.

$$\tag*{$\blacksquare$} $$