Question

Which of the following confidence intervals for ˆp, taken from the same population, will be the smallest?

A. 90% confidence, n = 200

B. 90% confidence, n = 50

C. 99% confidence, n = 200

D. 99% confidence, n = 50

Answer 1

Here we are considering confidence interval for population proportion. So the formula is ,

- E < P < + E

Where,

Where,

So margin of error = E = Zc * SE

For 90% confidence interval:

c = 0.90,

Zc = -------- (using excel formula " =norm.s.inv( 0.95)" )

For 99% confidence interval:

c = 0.99,

Zc = ------------( using excel formula " =norm.s.inv(0.995)" )

So here we can see that critical value for 99% confidence interval is greater than for 90% .

Hence value of margin of error will be high for 99% confidence interval than the 90% confidence interval keeping standard error constant.

Now ,

If sample size is increases, value of standard error will decreases and vice versa.

So keeping critical value constant, value of margin of error will be high for smaller sample size.

e.g. In 90% confidence interval, SE is higher with sample size n =50 than n =200. Hence margin of error will be higher for sample size n =50 than n =200 .

As, margin of error is subtracted from point estimate value of proportion, the length of confidence interval is dependent on value of margin of error. Higher value of margin of error will increase length of confidence interval and vice versa.
So to find smallest confidence interval, we must find smallest value of margin of error.

And smallest value of margin of error will get with higher sample size and smaller critical value.

Among the four combinations :

A. 90% confidence, n = 200

B. 90% confidence, n = 50

C. 99% confidence, n = 200

D. 99% confidence, n = 50

90% confidence interval with sample size n =200 , will have smallest value of margin of error. Hence this will also have the smallest confidence interval.

So the correct option is A.