Question

to increase loudness of the sound by four times, by how much should the amplitude of vibration be changed?

Answer 1

Amplitude is directly proportional to square of loudness,
that is amplitude = loudness$^{2}$
from this we can derive that 
loudness = $\sqrt {amplitude}$
therefore,
let x be the initial loudness of sound
4x = amplitude
amplitude= $\sqrt {4}$
amplitude = 2