Java 7, 130 bytes
Golfed:
double[]w(int s,int f){double n=s+f,z=1.96,x=z*z,p=s+x/2,d=z*Math.sqrt(s*f/n+x/4),m=1/(n+x);return new double[]{m*(p-d),m*(p+d)};} Ungolfed:
double[] w(int s, int f) { double n = s + f, z = 1.96, x = z * z, p = s + x / 2, d = z * Math.sqrt(s * f / n + x / 4), m = 1 / (n + x); return new double[] { m * (p - d), m * (p + d) }; } [Try it online][1]Try it online
Returns an array of type double of length 2, can probably be golfed more. [1]: https://ideone.com/CHeljM