Why would an airplane's nosewheel bounce during landing?
Sounds most like the pilot released back pressure  on the control stick abruptly once the main wheels touchdown. The idea here is that you gently lower the nose wheel to the pavement once the mains are on the ground in order to prevent that kind of a bounce.   As I recall the Cessna 162 has a tendency to bounce on the nose wheel as well as to porpoise if the landing attitude is not done correctly it's just a quirk of that airplane.