Linear Combination of Normals
Suppose that X and Y are independent normal random variables.

Let X ~ N(m_{1}, s_{1}^{2}) and Y ~ N(m_{2}, s_{2}^{2}), then X + Y is a normal random variable with mean m_{1 } + m_{2 }and variance s_{1}^{2 } + s_{2}^{2}.
We can go a bit further: if a and b are constants then:

aX + bY ~ N(am_{1 } + bm_{2} , a^{2}s_{1}^{2} + b^{2}s_{2}^{2})
The Central Limit Theorem
The following is an important result known as the central limit theorem:
If X_{1}, … X_{n} is are independent random variables random sample from any distribution which has mean m and variance s^{2}, then the distribution of X_{1}+X_{2}+…+X_{n} is approximately normal with mean nm and variance ns^{2}.
In particular, the distribution of the sample mean, which is (X_{1} + X_{2} +…+ X_{n})/n, is approximately normal with mean m and variance s^{2}/n (since we have multiplied X_{1}+X_{2}+…+X_{n} by (1/n) and multiplying by a constant multiplies the mean by that constant and the variance by the constant squared). This important result will be used in constructing confidence intervals.