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Standard error of the difference between two proportions

I need to estimate

SE[P_0 - 2P_1]

, where

P_0 = 0.2048

and

P_1 = 0.12

.

Here's the question for reference.

I calculated the proportion,

P_0

, from the first sample, which had [; 256 ;] of the variable I'm looking at, in a sample of

1250

P_1

came from the fact that there are

150

out of

1250

in the second sample.

The problem I'm facing is that if I do the transformation first (

P_0 - 2P_1

), I get

-0.0352

, which obviously won't work with the SQRT in the SE formula.

We are also told that

\pi _0 = 0.2

(this is the population proportion).

Where am I going wrong? I can provide more info if needed. Also, apologies if some of my terminology is a bit off.

Thanks!

To clarify what I have already done:

An earlier question asked me to estimate [; SE[P_0] ;], which I did by first calculating the proportion, [;], then putting this value into the SE formula as follows:

SE[P_0] = SE[0.2048] = \sqrt\frac{0.2048(1-0.2048)}{1250} = 0.01141427037

I used this formula as it was how I was taught to find the SE of a sample proportion.

I then applied similar intuition for the next question, by first calculating [; P_1 ;], then the difference between the proportions:

P_1 = \frac{150}{1250} = 0.12

P_0 - 2P_1 = 0.2048 - 2(0.12) = -0.0352

The problem I then face is that if I try to use the same SE formula from above, I get tripped up by the negative:

[; SE[P_0 - 2P_1] = SE[-0.0352] = \sqrt\frac{-0.0352(1--0.0352)}{1250} = \sqrt{-0.00002915...};]