### Chapter 7, Survey Sampling

#### Solution 18

Denoting the random variables from the two surveys as and . Also denoting the the confidence intervals by and respectively. Denoting the probability that the condidence interval contains the respective population mean by and .

Since the surveys are independent, it means that the random variables, say will also be independent. Thus condidence intervals are also independent. It follows that the probability of both of them containing the respective population mean:

.

And the probability that both of them do not contain the respective population mean is .

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