Circle $F$ is given by the equation $x^{2}+(y-2)^{2}=4$. Which point does not lie on the circle?

Step 1 ：The equation of the circle is given by \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius. In this case, the center of the circle is at \((0, 2)\) and the radius is \(2\).

Step 2 ：To check if a point lies on the circle, we can substitute the coordinates of the point into the equation of the circle. If the equation holds true, then the point lies on the circle. If not, the point does not lie on the circle.

Step 3 ：The point that does not lie on the circle can be determined by substituting the coordinates of the point into the equation of the circle. For example, the point \((0, 0)\) does not lie on the circle because substituting \((0, 0)\) into the equation of the circle does not satisfy the equation.

Step 4 ：\(\boxed{\text{The point (0, 0) does not lie on the circle.}}\)