[blank frank]

No. Think about the case of 2 standard normals at 0.05. The CI for either is -1.96 to 1.96. Add them together your way and you are adding two intervals at alpha=0.224. That gives you (-0.76,.76)+(-0.76,0.76)=(-1.52,1.52). That indicates a lower variance, when the variance should be higher. In fact, the variance of the sum should be 2 (the sum of the variances), giving a standard deviation of sqrt(2). So the answer should be -/+ 1.96*sqrt(2) = -/+2.77.

The way to get the answer is to square the margins of error, add them together, and take the square root. Sort of a Pythagorean thing. With the sum of two standard normals, that gives us sqrt(1.96^2+1.96^2) = 2.77, the correct answer.