### Chapter 7, Survey Sampling

#### Solution 8

#### (a)

Dividing by, , standard error of the sample mean:

Now we can approximate it by standard normal using Central Limit Theorem:

Since we are given that

Thus we get

Now since this is a proportion, we know for dichotmous case, and also ignoring the finite population correction:

which gives us:

Thus the value of delta, which makes the probability of the difference between sample mean and the population mean more than apart equals to , is .

#### (b)

The 95% confidence interval is .

Ignoring the finite population correction, , where is the population variance, which in this case(dichotomous) is . Thus .

Thus the 95% confidence interval becomes , Or

Clearly the interval contains

Note: In both parts, is used instead of because we can compute it using the population variance . Thus we need not to use as an approxiamtion for . In other words, use only when we can not compute .

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