In part (a), we know and as well as (known from the structure of the randomized device). Thus we can get .
Let if member of the random sample answers yes. Then . Thus .
Lets propose as an estimate of . Since it is a linear expression in , it follows that which is same as .
Since is a bernoulli’s trial, .
We have . Now, . Thus .
Putting the values in , we get . Approximating , we get: .
TODO: Not sure how if the extra term can be ignored using finite population correction as expected from the answer.